Physics Book - User contributions [en]

Bohr Model

2017-03-28T18:39:37Z

Sguo70: /* History */

Sherry Guo (Spring 2017)

This page gives basic information about the Bohr model and quantization. It also includes examples using the Bohr model.
==Main Idea==

[[File: Bohrmodel2.png|frame|right|A visualization of the Bohr model and the hydrogen spectrum]] The Bohr model of the atom was proposed by Niels Bohr in 1913, and it was radically different from any other models that previously existed. His model depicted atoms as having negatively charged electrons which orbited as small, positively charged nucleus. Bohr's incorporation of quantum theory was the most groundbreaking part; in this model, electrons can only exist in discrete energy levels, which are quantized. This model is very simplistic and is useful in introducing students to quantum mechanics. Although this model successfully predicts energy levels of the hydrogen atom, it has its shortcomings.
===Bohr's Assumptions===
* Electrons travel in a circular orbit around the nucleus, similar to how planets orbit around the sun.
* The energy that electrons have is related to their distance from the nucleus, or which energy level they occupy. The further away the electron, the more energy it has.
* Electrons can gain or lose energy by jumping from one orbit to another. The energy is quantized - the orbitals have discreet radii, which Bohr called "stationary orbits."
* When an excited electron returns back to its ground state, then it releases the energy that it absorbed, in the form of light. This quantized energy is equal to the difference between the energies of the orbits.
[[File:Stairsbohr.jpg]]

Stairs are a great way to visualize quantized energy. When you're going up stairs, you can only be standing on the steps, and not anywhere in between the steps. Similarly, energy can only be absorbed or emitted in specific quanta.

==History==
[[Image:200px-Niels_Bohr_Date_Unverified_LOC.jpg |right|frame|Niels Bohr]] Ernest Rutherford proposed that the basic structure of an atom is a cloud of electrons which surround a small nucleus. With his experiments, he developed a model that depicted an atom like a solar system, with electrons orbiting around the nucleus. He proposed that the emission spectrum of hydrogen would look more like a smear rather than being made up of distinct lines. However, this was flawed, as it suggested that atoms can emit energy that isn't quantized. [8]
[[File:RutherfordModel2.png|frame|left|Rutherford's model of the atom, which Bohr expanded on]]
Niels Bohr, a physicist from Denmark, was able to explain the emission spectra of hydrogen by improving on the Rutherford model of the atom. In 1911 Bohr traveled to England in order to study the structure of atoms and molecules. There, he attended lectures on electromagnetism and worked with Rutherford and other scientists such as J. J. Thomson. When he returned to Denmark in 1912, Bohr noticed that in the atomic emissions spectrum of hydrogen, only certain colors could be seen. Thus, he theorized that electrons need to be in energy levels that are quantized. He related the energies of the colors he saw to the differences of hydrogen's energy levels. Although this model is not entirely correct, as it only applies to systems where two charged particles orbit each other, it still has many features that are very applicable to physics today. Later on, scientists such as Werner Heisenberg and Erwin Schrödinger worked to improve upon this model.[9]

== Shortcomings of the Bohr Model ==
While the Bohr model is an important predecessor to the current quantum mechanical models of the atom, it doesn't correctly describe some aspects of electron orbitals. It does not provide any reasoning as to why certain spectral lines are brighter than others. Its main shortcoming is that it violates the uncertainty principle, as it considers electrons to have a definite radius and momentum. This model is very basic, and in order to know more specific details about spectra and charge distribution, more calculations must be done. Many of the failures of the Bohr model can be corrected by the [http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hydsch.html#c2 Schrodinger equation]for the hydrogen atom.

==Application==
===A Mathematical Model===
==== The Angular Momentum Quantum ====
<math>ħ = h/2π =1.05*10^{-34} J*s </math>

h is Planck's constant, which is a physical constant that is essential in quantum mechanics.

==== Angular Momentum Is Quantized ====
<math> |\vec L_{trans,C}| = rp = Nħ</math>

p is the electron's momentum
r is the radius of circular orbit
N is an integer (1,2,3, ...)

We can derive the equation for r, the allowed Bohr radii for electron orbits.

1) Electric force the proton exerts on the electron:
<math> F_{el} = {\frac{e^2}{4π ε0 r^2}} </math>

2) Using the momentum principle and curving motion:
<math>|F_{perpendicular}| = {\frac{|p| |v|}{r}} = {\frac{e^2}{4π ε0 r^2}} </math>

3)Substituting in Bohr's conditions:
<math> {\frac{N^2 h*^2}{mr^3}} = {\frac{e^2}{4π ε0 r^2}} </math>

4)Solving for the allowed radii
<math> r = {\frac{4π ε0 h*^2}{me^2}} *N^2</math>
where N = 1,2,3,...

Additionally, the formula for energy of hydrogen atom of different levels is also derived from this model.

E = K + Uelectric

1)
<math> E = {\frac{mv^2}{2}} - {\frac{{\frac{1}{2}}*{\frac{1}{4π ε0}}*{\frac{me^2}{h*}}}{N^2}}</math>

2)
<math> E = {\frac{13.6 eV}{N^2}}</math> where N = 1,2,3
====Wavelengths====
The Debroglie relationship can tell us about the wavelength associated with the electron:
<math>λ = \frac{h}{mv}</math>

This equation is used to help derive the equation for the angular momentum of an electron in orbit
<math> L = \frac{nh}{2π} </math>

===A Computational Model===

[[File:bohr2.gif|upright]]
[[File:bohr3.gif]]

To interact with this glowscript visualization of the Bohr Model, click [http://www.glowscript.org/#/user/matterandinteractions/folder/matterandinteractions/program/08-Bohr-levels here]. You will also find a graph of Energy (eV), Kinetic Energy, and Potential Energy. This visualization shows how the electrons jump from level to level according to the Bohr Model. There is also an energy vs distance graph shown which varies according to these levels. Notice that the further away the electron is, the more energy it has.

==Examples==

===Simple===
Find the magnitude of the translational angular momentum of an electron when a hydrogen atom is in its 2nd excited state above the ground state.

--

We know that the only possible states of the hydrogen atom are those when the electron's translational angular momentum is an integer multiple of ħ.
<math>|\vec L_{trans,c}| = Nħ </math>
For the 2nd excited state, N = 3
Now just plug the numbers in
<math>|\vec L_{trans,c}| = (3)(1.05*10^{-34} J*s) = 3.15*10^{-34} J*s</math>

===Middling===
[6]A hydrogen atom is in state N = 3, where N = 1 is the lowest energy state. What is K+U in electron volts for this atomic hydrogen energy state?

1)
<math>E(3) = {\frac{-13.6 eV}{3^2}}</math> = -1.51 Joules

2)
<math>E(1) = {\frac{-13.6 eV}{1^2}}</math> = -13.6 Joules

3)
K+U = energy of photon = <math>E(1) - E(3) = {\frac{-13.6 eV}{3^2}} - {\frac{-13.6 eV}{1^2}}</math> = 12.09 Joules

Below is the graph of E as the hydrogen atom goes from N = 3 to N = 1.

[[Image:Graphproblem.png]]

===Difficult===
Hydrogen has been detected transitioning from the 101st to the 100th energy levels. What is the wavelength of the radiation? Where in the electromagnetic spectrum is this emission?

To solve this problem, we first need to use formulas derived from Bohr Model of hydrogen atom. It is <math>E = {\frac{-13.6 eV}{N^2}}</math>

Then solve for the wavelength using formula from Electromagnetic Wave Theory.

[[Image:Screen Shot 2015-12-01 at 8.21.56 PM.png|200x250px|]]

This wavelength falls in the microwave portion of the electromagnetic spectrum.

==Connectedness==
I chose this topic because in our chemistry class, we learned a brief introduction to quantum mechanics, and this was one of the things that really interested me. I also really enjoyed the lab report we did involving this topic, and we got to use spectrometers to observe the emission spectrum of hydrogen and other elements. Bohr's discovery of this model was extremely important in the physics and chemistry world, because it laid down the framework for other theories in quantum mechanics. This model has applications to almost everything in our everyday lives - Bohr's notion that atoms emit light with very specific and quantized energies can explain how lasers work, and aid in figuring out what elements galaxies are made of, since every element has its own emission spectrum. Bohr's model has numerous amounts of applications to chemical engineering especially, because it explains why atoms have certain properties, such as conductivity. Every chemical process involves bonds between atoms being broken or formed.

== See also ==

===Further reading===

Matter and Interactions I Modern Mechanics 4th Edition Chapter 11.10

===External links===

https://en.wikipedia.org/wiki/Bohr_model

https://en.wikipedia.org/wiki/Quantization_(physics)

Videos:

https://www.khanacademy.org/science/chemistry/electronic-structure-of-atoms/bohr-model-hydrogen/v/bohr-model-energy-levels

https://www.youtube.com/watch?v=nVW1zDPPZGM

Simulation of Bohr Model:

https://phet.colorado.edu/en/simulation/legacy/hydrogen-atom

Why Bohr's model explains everything around us:

http://scitech.au.dk/en/roemer/apr13/bohrs-model-of-the-atom-explains-science-in-everyday-life/

==References==

This section contains the the references you used while writing this page

[http://pachamamatrust.org/f2/1_K/SP_physics/H4_bohr_model_KSP.htm]
[http://chemistry.about.com/od/atomicstructure/a/bohr-model.htm]
[http://scitech.au.dk/en/roemer/apr13/bohrs-model-of-the-atom-explains-science-in-everyday-life/]
[http://www.encyclopedia.com/topic/Bohr_model.aspx]
[http://ib-physics-ii-6b-e.aspen.high.schoolfusion.us/modules/locker/files/get_group_file.phtml?gid=4397333&fid=18592200]
[https://files.t-square.gatech.edu/access/content/group/gtc-80c7-d8ef-5c05-873d-52adf5b9ce5c/Lecture%20Notes/Fenton/Phys2211_C_2015_fall_week_16_xMonday.pdf]
[http://hyperphysics.phy-astr.gsu.edu/hbase/bohr.html]
[http://web.ihep.su/dbserv/compas/src/bohr13/eng.pdf]
[http://chemed.chem.purdue.edu/genchem/history/bohr.html]
Note: All images on this page are either free for commercial use (with no attribution required) or made by myself.
[[Category:Energy]]

Bohr Model

2017-03-28T18:38:18Z

Sguo70: /* Bohr's Assumptions */

Sherry Guo (Spring 2017)

This page gives basic information about the Bohr model and quantization. It also includes examples using the Bohr model.
==Main Idea==

[[File: Bohrmodel2.png|frame|right|A visualization of the Bohr model and the hydrogen spectrum]] The Bohr model of the atom was proposed by Niels Bohr in 1913, and it was radically different from any other models that previously existed. His model depicted atoms as having negatively charged electrons which orbited as small, positively charged nucleus. Bohr's incorporation of quantum theory was the most groundbreaking part; in this model, electrons can only exist in discrete energy levels, which are quantized. This model is very simplistic and is useful in introducing students to quantum mechanics. Although this model successfully predicts energy levels of the hydrogen atom, it has its shortcomings.
===Bohr's Assumptions===
* Electrons travel in a circular orbit around the nucleus, similar to how planets orbit around the sun.
* The energy that electrons have is related to their distance from the nucleus, or which energy level they occupy. The further away the electron, the more energy it has.
* Electrons can gain or lose energy by jumping from one orbit to another. The energy is quantized - the orbitals have discreet radii, which Bohr called "stationary orbits."
* When an excited electron returns back to its ground state, then it releases the energy that it absorbed, in the form of light. This quantized energy is equal to the difference between the energies of the orbits.
[[File:Stairsbohr.jpg]]

Stairs are a great way to visualize quantized energy. When you're going up stairs, you can only be standing on the steps, and not anywhere in between the steps. Similarly, energy can only be absorbed or emitted in specific quanta.

==History==
[[Image:200px-Niels_Bohr_Date_Unverified_LOC.jpg |right|frame|Niels Bohr]] Ernest Rutherford proposed that the basic structure of an atom is a cloud of electrons which surround a small nucleus. With his experiments, he developed a model that depicted an atom like a solar system, with electrons orbiting around the nucleus. He proposed that the emission spectrum of hydrogen would look more like a smear rather than being made up of distinct lines. However, this was flawed, as it suggested that atoms can emit energy that isn't quantized. [8]
[[File:RutherfordModel2.png|frame|left|Rutherford's model of the atom, which Bohr expanded on]]
Niels Bohr, a physicist from Denmark, was able to explain the emission spectra of hydrogen by improving on the Rutherford model of the atom. In 1911 Bohr traveled to England in order to study the structure of atoms and molecules. There, he attended lectures on electromagnetism and worked with Rutherford and other scientists such as J. J. Thomson. When he returned to Denmark in 1912, Bohr noticed that in the atomic emissions spectrum of hydrogen, only certain colors could be seen. Thus, he theorized that electrons need to be in energy levels that are quantized. He related the energies of the colors he saw to the differences of hydrogen's energy levels. Although this model is not entirely correct, as it only applies to systems where two charged particles orbit each other, it still has many features that are very applicable to physics today. [9]

== Shortcomings of the Bohr Model ==
While the Bohr model is an important predecessor to the current quantum mechanical models of the atom, it doesn't correctly describe some aspects of electron orbitals. It does not provide any reasoning as to why certain spectral lines are brighter than others. Its main shortcoming is that it violates the uncertainty principle, as it considers electrons to have a definite radius and momentum. This model is very basic, and in order to know more specific details about spectra and charge distribution, more calculations must be done. Many of the failures of the Bohr model can be corrected by the [http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hydsch.html#c2 Schrodinger equation]for the hydrogen atom.

==Application==
===A Mathematical Model===
==== The Angular Momentum Quantum ====
<math>ħ = h/2π =1.05*10^{-34} J*s </math>

h is Planck's constant, which is a physical constant that is essential in quantum mechanics.

==== Angular Momentum Is Quantized ====
<math> |\vec L_{trans,C}| = rp = Nħ</math>

p is the electron's momentum
r is the radius of circular orbit
N is an integer (1,2,3, ...)

We can derive the equation for r, the allowed Bohr radii for electron orbits.

1) Electric force the proton exerts on the electron:
<math> F_{el} = {\frac{e^2}{4π ε0 r^2}} </math>

2) Using the momentum principle and curving motion:
<math>|F_{perpendicular}| = {\frac{|p| |v|}{r}} = {\frac{e^2}{4π ε0 r^2}} </math>

3)Substituting in Bohr's conditions:
<math> {\frac{N^2 h*^2}{mr^3}} = {\frac{e^2}{4π ε0 r^2}} </math>

4)Solving for the allowed radii
<math> r = {\frac{4π ε0 h*^2}{me^2}} *N^2</math>
where N = 1,2,3,...

Additionally, the formula for energy of hydrogen atom of different levels is also derived from this model.

E = K + Uelectric

1)
<math> E = {\frac{mv^2}{2}} - {\frac{{\frac{1}{2}}*{\frac{1}{4π ε0}}*{\frac{me^2}{h*}}}{N^2}}</math>

2)
<math> E = {\frac{13.6 eV}{N^2}}</math> where N = 1,2,3
====Wavelengths====
The Debroglie relationship can tell us about the wavelength associated with the electron:
<math>λ = \frac{h}{mv}</math>

This equation is used to help derive the equation for the angular momentum of an electron in orbit
<math> L = \frac{nh}{2π} </math>

===A Computational Model===

[[File:bohr2.gif|upright]]
[[File:bohr3.gif]]

To interact with this glowscript visualization of the Bohr Model, click [http://www.glowscript.org/#/user/matterandinteractions/folder/matterandinteractions/program/08-Bohr-levels here]. You will also find a graph of Energy (eV), Kinetic Energy, and Potential Energy. This visualization shows how the electrons jump from level to level according to the Bohr Model. There is also an energy vs distance graph shown which varies according to these levels. Notice that the further away the electron is, the more energy it has.

==Examples==

===Simple===
Find the magnitude of the translational angular momentum of an electron when a hydrogen atom is in its 2nd excited state above the ground state.

--

We know that the only possible states of the hydrogen atom are those when the electron's translational angular momentum is an integer multiple of ħ.
<math>|\vec L_{trans,c}| = Nħ </math>
For the 2nd excited state, N = 3
Now just plug the numbers in
<math>|\vec L_{trans,c}| = (3)(1.05*10^{-34} J*s) = 3.15*10^{-34} J*s</math>

===Middling===
[6]A hydrogen atom is in state N = 3, where N = 1 is the lowest energy state. What is K+U in electron volts for this atomic hydrogen energy state?

1)
<math>E(3) = {\frac{-13.6 eV}{3^2}}</math> = -1.51 Joules

2)
<math>E(1) = {\frac{-13.6 eV}{1^2}}</math> = -13.6 Joules

3)
K+U = energy of photon = <math>E(1) - E(3) = {\frac{-13.6 eV}{3^2}} - {\frac{-13.6 eV}{1^2}}</math> = 12.09 Joules

Below is the graph of E as the hydrogen atom goes from N = 3 to N = 1.

[[Image:Graphproblem.png]]

===Difficult===
Hydrogen has been detected transitioning from the 101st to the 100th energy levels. What is the wavelength of the radiation? Where in the electromagnetic spectrum is this emission?

To solve this problem, we first need to use formulas derived from Bohr Model of hydrogen atom. It is <math>E = {\frac{-13.6 eV}{N^2}}</math>

Then solve for the wavelength using formula from Electromagnetic Wave Theory.

[[Image:Screen Shot 2015-12-01 at 8.21.56 PM.png|200x250px|]]

This wavelength falls in the microwave portion of the electromagnetic spectrum.

==Connectedness==
I chose this topic because in our chemistry class, we learned a brief introduction to quantum mechanics, and this was one of the things that really interested me. I also really enjoyed the lab report we did involving this topic, and we got to use spectrometers to observe the emission spectrum of hydrogen and other elements. Bohr's discovery of this model was extremely important in the physics and chemistry world, because it laid down the framework for other theories in quantum mechanics. This model has applications to almost everything in our everyday lives - Bohr's notion that atoms emit light with very specific and quantized energies can explain how lasers work, and aid in figuring out what elements galaxies are made of, since every element has its own emission spectrum. Bohr's model has numerous amounts of applications to chemical engineering especially, because it explains why atoms have certain properties, such as conductivity. Every chemical process involves bonds between atoms being broken or formed.

== See also ==

===Further reading===

Matter and Interactions I Modern Mechanics 4th Edition Chapter 11.10

===External links===

https://en.wikipedia.org/wiki/Bohr_model

https://en.wikipedia.org/wiki/Quantization_(physics)

Videos:

https://www.khanacademy.org/science/chemistry/electronic-structure-of-atoms/bohr-model-hydrogen/v/bohr-model-energy-levels

https://www.youtube.com/watch?v=nVW1zDPPZGM

Simulation of Bohr Model:

https://phet.colorado.edu/en/simulation/legacy/hydrogen-atom

Why Bohr's model explains everything around us:

http://scitech.au.dk/en/roemer/apr13/bohrs-model-of-the-atom-explains-science-in-everyday-life/

==References==

This section contains the the references you used while writing this page

[http://pachamamatrust.org/f2/1_K/SP_physics/H4_bohr_model_KSP.htm]
[http://chemistry.about.com/od/atomicstructure/a/bohr-model.htm]
[http://scitech.au.dk/en/roemer/apr13/bohrs-model-of-the-atom-explains-science-in-everyday-life/]
[http://www.encyclopedia.com/topic/Bohr_model.aspx]
[http://ib-physics-ii-6b-e.aspen.high.schoolfusion.us/modules/locker/files/get_group_file.phtml?gid=4397333&fid=18592200]
[https://files.t-square.gatech.edu/access/content/group/gtc-80c7-d8ef-5c05-873d-52adf5b9ce5c/Lecture%20Notes/Fenton/Phys2211_C_2015_fall_week_16_xMonday.pdf]
[http://hyperphysics.phy-astr.gsu.edu/hbase/bohr.html]
[http://web.ihep.su/dbserv/compas/src/bohr13/eng.pdf]
[http://chemed.chem.purdue.edu/genchem/history/bohr.html]
Note: All images on this page are either free for commercial use (with no attribution required) or made by myself.
[[Category:Energy]]

Bohr Model

2017-03-28T18:36:11Z

Sguo70: /* The Angular Momentum Quantum */

Sherry Guo (Spring 2017)

This page gives basic information about the Bohr model and quantization. It also includes examples using the Bohr model.
==Main Idea==

[[File: Bohrmodel2.png|frame|right|A visualization of the Bohr model and the hydrogen spectrum]] The Bohr model of the atom was proposed by Niels Bohr in 1913, and it was radically different from any other models that previously existed. His model depicted atoms as having negatively charged electrons which orbited as small, positively charged nucleus. Bohr's incorporation of quantum theory was the most groundbreaking part; in this model, electrons can only exist in discrete energy levels, which are quantized. This model is very simplistic and is useful in introducing students to quantum mechanics. Although this model successfully predicts energy levels of the hydrogen atom, it has its shortcomings.
===Bohr's Assumptions===
* Electrons travel in a circular orbit around the nucleus, similar to how planets orbit around the sun.
* The energy that electrons have is related to their distance from the nucleus, or which energy level they occupy. The further away the electron, the more energy it has.
* Electrons can gain or lose energy by jumping from one orbit to another. The energy is quantized - the orbitals have discreet radii.
* When an excited electron returns back to its ground state, then it releases the energy that it absorbed, in the form of light. This quantized energy is equal to the difference between the energies of the orbits.
[[File:Stairsbohr.jpg]]

Stairs are a great way to visualize quantized energy. When you're going up stairs, you can only be standing on the steps, and not anywhere in between the steps. Similarly, energy can only be absorbed or emitted in specific quanta.

==History==
[[Image:200px-Niels_Bohr_Date_Unverified_LOC.jpg |right|frame|Niels Bohr]] Ernest Rutherford proposed that the basic structure of an atom is a cloud of electrons which surround a small nucleus. With his experiments, he developed a model that depicted an atom like a solar system, with electrons orbiting around the nucleus. He proposed that the emission spectrum of hydrogen would look more like a smear rather than being made up of distinct lines. However, this was flawed, as it suggested that atoms can emit energy that isn't quantized. [8]
[[File:RutherfordModel2.png|frame|left|Rutherford's model of the atom, which Bohr expanded on]]
Niels Bohr, a physicist from Denmark, was able to explain the emission spectra of hydrogen by improving on the Rutherford model of the atom. In 1911 Bohr traveled to England in order to study the structure of atoms and molecules. There, he attended lectures on electromagnetism and worked with Rutherford and other scientists such as J. J. Thomson. When he returned to Denmark in 1912, Bohr noticed that in the atomic emissions spectrum of hydrogen, only certain colors could be seen. Thus, he theorized that electrons need to be in energy levels that are quantized. He related the energies of the colors he saw to the differences of hydrogen's energy levels. Although this model is not entirely correct, as it only applies to systems where two charged particles orbit each other, it still has many features that are very applicable to physics today. [9]

== Shortcomings of the Bohr Model ==
While the Bohr model is an important predecessor to the current quantum mechanical models of the atom, it doesn't correctly describe some aspects of electron orbitals. It does not provide any reasoning as to why certain spectral lines are brighter than others. Its main shortcoming is that it violates the uncertainty principle, as it considers electrons to have a definite radius and momentum. This model is very basic, and in order to know more specific details about spectra and charge distribution, more calculations must be done. Many of the failures of the Bohr model can be corrected by the [http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hydsch.html#c2 Schrodinger equation]for the hydrogen atom.

==Application==
===A Mathematical Model===
==== The Angular Momentum Quantum ====
<math>ħ = h/2π =1.05*10^{-34} J*s </math>

h is Planck's constant, which is a physical constant that is essential in quantum mechanics.

==== Angular Momentum Is Quantized ====
<math> |\vec L_{trans,C}| = rp = Nħ</math>

p is the electron's momentum
r is the radius of circular orbit
N is an integer (1,2,3, ...)

We can derive the equation for r, the allowed Bohr radii for electron orbits.

1) Electric force the proton exerts on the electron:
<math> F_{el} = {\frac{e^2}{4π ε0 r^2}} </math>

2) Using the momentum principle and curving motion:
<math>|F_{perpendicular}| = {\frac{|p| |v|}{r}} = {\frac{e^2}{4π ε0 r^2}} </math>

3)Substituting in Bohr's conditions:
<math> {\frac{N^2 h*^2}{mr^3}} = {\frac{e^2}{4π ε0 r^2}} </math>

4)Solving for the allowed radii
<math> r = {\frac{4π ε0 h*^2}{me^2}} *N^2</math>
where N = 1,2,3,...

Additionally, the formula for energy of hydrogen atom of different levels is also derived from this model.

E = K + Uelectric

1)
<math> E = {\frac{mv^2}{2}} - {\frac{{\frac{1}{2}}*{\frac{1}{4π ε0}}*{\frac{me^2}{h*}}}{N^2}}</math>

2)
<math> E = {\frac{13.6 eV}{N^2}}</math> where N = 1,2,3
====Wavelengths====
The Debroglie relationship can tell us about the wavelength associated with the electron:
<math>λ = \frac{h}{mv}</math>

This equation is used to help derive the equation for the angular momentum of an electron in orbit
<math> L = \frac{nh}{2π} </math>

===A Computational Model===

[[File:bohr2.gif|upright]]
[[File:bohr3.gif]]

To interact with this glowscript visualization of the Bohr Model, click [http://www.glowscript.org/#/user/matterandinteractions/folder/matterandinteractions/program/08-Bohr-levels here]. You will also find a graph of Energy (eV), Kinetic Energy, and Potential Energy. This visualization shows how the electrons jump from level to level according to the Bohr Model. There is also an energy vs distance graph shown which varies according to these levels. Notice that the further away the electron is, the more energy it has.

==Examples==

===Simple===
Find the magnitude of the translational angular momentum of an electron when a hydrogen atom is in its 2nd excited state above the ground state.

--

We know that the only possible states of the hydrogen atom are those when the electron's translational angular momentum is an integer multiple of ħ.
<math>|\vec L_{trans,c}| = Nħ </math>
For the 2nd excited state, N = 3
Now just plug the numbers in
<math>|\vec L_{trans,c}| = (3)(1.05*10^{-34} J*s) = 3.15*10^{-34} J*s</math>

===Middling===
[6]A hydrogen atom is in state N = 3, where N = 1 is the lowest energy state. What is K+U in electron volts for this atomic hydrogen energy state?

1)
<math>E(3) = {\frac{-13.6 eV}{3^2}}</math> = -1.51 Joules

2)
<math>E(1) = {\frac{-13.6 eV}{1^2}}</math> = -13.6 Joules

3)
K+U = energy of photon = <math>E(1) - E(3) = {\frac{-13.6 eV}{3^2}} - {\frac{-13.6 eV}{1^2}}</math> = 12.09 Joules

Below is the graph of E as the hydrogen atom goes from N = 3 to N = 1.

[[Image:Graphproblem.png]]

===Difficult===
Hydrogen has been detected transitioning from the 101st to the 100th energy levels. What is the wavelength of the radiation? Where in the electromagnetic spectrum is this emission?

To solve this problem, we first need to use formulas derived from Bohr Model of hydrogen atom. It is <math>E = {\frac{-13.6 eV}{N^2}}</math>

Then solve for the wavelength using formula from Electromagnetic Wave Theory.

[[Image:Screen Shot 2015-12-01 at 8.21.56 PM.png|200x250px|]]

This wavelength falls in the microwave portion of the electromagnetic spectrum.

==Connectedness==
I chose this topic because in our chemistry class, we learned a brief introduction to quantum mechanics, and this was one of the things that really interested me. I also really enjoyed the lab report we did involving this topic, and we got to use spectrometers to observe the emission spectrum of hydrogen and other elements. Bohr's discovery of this model was extremely important in the physics and chemistry world, because it laid down the framework for other theories in quantum mechanics. This model has applications to almost everything in our everyday lives - Bohr's notion that atoms emit light with very specific and quantized energies can explain how lasers work, and aid in figuring out what elements galaxies are made of, since every element has its own emission spectrum. Bohr's model has numerous amounts of applications to chemical engineering especially, because it explains why atoms have certain properties, such as conductivity. Every chemical process involves bonds between atoms being broken or formed.

== See also ==

===Further reading===

Matter and Interactions I Modern Mechanics 4th Edition Chapter 11.10

===External links===

https://en.wikipedia.org/wiki/Bohr_model

https://en.wikipedia.org/wiki/Quantization_(physics)

Videos:

https://www.khanacademy.org/science/chemistry/electronic-structure-of-atoms/bohr-model-hydrogen/v/bohr-model-energy-levels

https://www.youtube.com/watch?v=nVW1zDPPZGM

Simulation of Bohr Model:

https://phet.colorado.edu/en/simulation/legacy/hydrogen-atom

Why Bohr's model explains everything around us:

http://scitech.au.dk/en/roemer/apr13/bohrs-model-of-the-atom-explains-science-in-everyday-life/

==References==

This section contains the the references you used while writing this page

[http://pachamamatrust.org/f2/1_K/SP_physics/H4_bohr_model_KSP.htm]
[http://chemistry.about.com/od/atomicstructure/a/bohr-model.htm]
[http://scitech.au.dk/en/roemer/apr13/bohrs-model-of-the-atom-explains-science-in-everyday-life/]
[http://www.encyclopedia.com/topic/Bohr_model.aspx]
[http://ib-physics-ii-6b-e.aspen.high.schoolfusion.us/modules/locker/files/get_group_file.phtml?gid=4397333&fid=18592200]
[https://files.t-square.gatech.edu/access/content/group/gtc-80c7-d8ef-5c05-873d-52adf5b9ce5c/Lecture%20Notes/Fenton/Phys2211_C_2015_fall_week_16_xMonday.pdf]
[http://hyperphysics.phy-astr.gsu.edu/hbase/bohr.html]
[http://web.ihep.su/dbserv/compas/src/bohr13/eng.pdf]
[http://chemed.chem.purdue.edu/genchem/history/bohr.html]
Note: All images on this page are either free for commercial use (with no attribution required) or made by myself.
[[Category:Energy]]

Bohr Model

2017-03-28T18:35:57Z

Sguo70: /* History */

Sherry Guo (Spring 2017)

This page gives basic information about the Bohr model and quantization. It also includes examples using the Bohr model.
==Main Idea==

[[File: Bohrmodel2.png|frame|right|A visualization of the Bohr model and the hydrogen spectrum]] The Bohr model of the atom was proposed by Niels Bohr in 1913, and it was radically different from any other models that previously existed. His model depicted atoms as having negatively charged electrons which orbited as small, positively charged nucleus. Bohr's incorporation of quantum theory was the most groundbreaking part; in this model, electrons can only exist in discrete energy levels, which are quantized. This model is very simplistic and is useful in introducing students to quantum mechanics. Although this model successfully predicts energy levels of the hydrogen atom, it has its shortcomings.
===Bohr's Assumptions===
* Electrons travel in a circular orbit around the nucleus, similar to how planets orbit around the sun.
* The energy that electrons have is related to their distance from the nucleus, or which energy level they occupy. The further away the electron, the more energy it has.
* Electrons can gain or lose energy by jumping from one orbit to another. The energy is quantized - the orbitals have discreet radii.
* When an excited electron returns back to its ground state, then it releases the energy that it absorbed, in the form of light. This quantized energy is equal to the difference between the energies of the orbits.
[[File:Stairsbohr.jpg]]

Stairs are a great way to visualize quantized energy. When you're going up stairs, you can only be standing on the steps, and not anywhere in between the steps. Similarly, energy can only be absorbed or emitted in specific quanta.

==History==
[[Image:200px-Niels_Bohr_Date_Unverified_LOC.jpg |right|frame|Niels Bohr]] Ernest Rutherford proposed that the basic structure of an atom is a cloud of electrons which surround a small nucleus. With his experiments, he developed a model that depicted an atom like a solar system, with electrons orbiting around the nucleus. He proposed that the emission spectrum of hydrogen would look more like a smear rather than being made up of distinct lines. However, this was flawed, as it suggested that atoms can emit energy that isn't quantized. [8]
[[File:RutherfordModel2.png|frame|left|Rutherford's model of the atom, which Bohr expanded on]]
Niels Bohr, a physicist from Denmark, was able to explain the emission spectra of hydrogen by improving on the Rutherford model of the atom. In 1911 Bohr traveled to England in order to study the structure of atoms and molecules. There, he attended lectures on electromagnetism and worked with Rutherford and other scientists such as J. J. Thomson. When he returned to Denmark in 1912, Bohr noticed that in the atomic emissions spectrum of hydrogen, only certain colors could be seen. Thus, he theorized that electrons need to be in energy levels that are quantized. He related the energies of the colors he saw to the differences of hydrogen's energy levels. Although this model is not entirely correct, as it only applies to systems where two charged particles orbit each other, it still has many features that are very applicable to physics today. [9]

== Shortcomings of the Bohr Model ==
While the Bohr model is an important predecessor to the current quantum mechanical models of the atom, it doesn't correctly describe some aspects of electron orbitals. It does not provide any reasoning as to why certain spectral lines are brighter than others. Its main shortcoming is that it violates the uncertainty principle, as it considers electrons to have a definite radius and momentum. This model is very basic, and in order to know more specific details about spectra and charge distribution, more calculations must be done. Many of the failures of the Bohr model can be corrected by the [http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hydsch.html#c2 Schrodinger equation]for the hydrogen atom.

==Application==
===A Mathematical Model===
==== The Angular Momentum Quantum ====
<math>ħ = h/2π =1.05*10^{-34} J*s </math>
h is Planck's constant, which is a physical constant that is essential in quantum mechanics.

==== Angular Momentum Is Quantized ====
<math> |\vec L_{trans,C}| = rp = Nħ</math>

p is the electron's momentum
r is the radius of circular orbit
N is an integer (1,2,3, ...)

We can derive the equation for r, the allowed Bohr radii for electron orbits.

1) Electric force the proton exerts on the electron:
<math> F_{el} = {\frac{e^2}{4π ε0 r^2}} </math>

2) Using the momentum principle and curving motion:
<math>|F_{perpendicular}| = {\frac{|p| |v|}{r}} = {\frac{e^2}{4π ε0 r^2}} </math>

3)Substituting in Bohr's conditions:
<math> {\frac{N^2 h*^2}{mr^3}} = {\frac{e^2}{4π ε0 r^2}} </math>

4)Solving for the allowed radii
<math> r = {\frac{4π ε0 h*^2}{me^2}} *N^2</math>
where N = 1,2,3,...

Additionally, the formula for energy of hydrogen atom of different levels is also derived from this model.

E = K + Uelectric

1)
<math> E = {\frac{mv^2}{2}} - {\frac{{\frac{1}{2}}*{\frac{1}{4π ε0}}*{\frac{me^2}{h*}}}{N^2}}</math>

2)
<math> E = {\frac{13.6 eV}{N^2}}</math> where N = 1,2,3
====Wavelengths====
The Debroglie relationship can tell us about the wavelength associated with the electron:
<math>λ = \frac{h}{mv}</math>

This equation is used to help derive the equation for the angular momentum of an electron in orbit
<math> L = \frac{nh}{2π} </math>

===A Computational Model===

[[File:bohr2.gif|upright]]
[[File:bohr3.gif]]

To interact with this glowscript visualization of the Bohr Model, click [http://www.glowscript.org/#/user/matterandinteractions/folder/matterandinteractions/program/08-Bohr-levels here]. You will also find a graph of Energy (eV), Kinetic Energy, and Potential Energy. This visualization shows how the electrons jump from level to level according to the Bohr Model. There is also an energy vs distance graph shown which varies according to these levels. Notice that the further away the electron is, the more energy it has.

==Examples==

===Simple===
Find the magnitude of the translational angular momentum of an electron when a hydrogen atom is in its 2nd excited state above the ground state.

--

We know that the only possible states of the hydrogen atom are those when the electron's translational angular momentum is an integer multiple of ħ.
<math>|\vec L_{trans,c}| = Nħ </math>
For the 2nd excited state, N = 3
Now just plug the numbers in
<math>|\vec L_{trans,c}| = (3)(1.05*10^{-34} J*s) = 3.15*10^{-34} J*s</math>

===Middling===
[6]A hydrogen atom is in state N = 3, where N = 1 is the lowest energy state. What is K+U in electron volts for this atomic hydrogen energy state?

1)
<math>E(3) = {\frac{-13.6 eV}{3^2}}</math> = -1.51 Joules

2)
<math>E(1) = {\frac{-13.6 eV}{1^2}}</math> = -13.6 Joules

3)
K+U = energy of photon = <math>E(1) - E(3) = {\frac{-13.6 eV}{3^2}} - {\frac{-13.6 eV}{1^2}}</math> = 12.09 Joules

Below is the graph of E as the hydrogen atom goes from N = 3 to N = 1.

[[Image:Graphproblem.png]]

===Difficult===
Hydrogen has been detected transitioning from the 101st to the 100th energy levels. What is the wavelength of the radiation? Where in the electromagnetic spectrum is this emission?

To solve this problem, we first need to use formulas derived from Bohr Model of hydrogen atom. It is <math>E = {\frac{-13.6 eV}{N^2}}</math>

Then solve for the wavelength using formula from Electromagnetic Wave Theory.

[[Image:Screen Shot 2015-12-01 at 8.21.56 PM.png|200x250px|]]

This wavelength falls in the microwave portion of the electromagnetic spectrum.

==Connectedness==
I chose this topic because in our chemistry class, we learned a brief introduction to quantum mechanics, and this was one of the things that really interested me. I also really enjoyed the lab report we did involving this topic, and we got to use spectrometers to observe the emission spectrum of hydrogen and other elements. Bohr's discovery of this model was extremely important in the physics and chemistry world, because it laid down the framework for other theories in quantum mechanics. This model has applications to almost everything in our everyday lives - Bohr's notion that atoms emit light with very specific and quantized energies can explain how lasers work, and aid in figuring out what elements galaxies are made of, since every element has its own emission spectrum. Bohr's model has numerous amounts of applications to chemical engineering especially, because it explains why atoms have certain properties, such as conductivity. Every chemical process involves bonds between atoms being broken or formed.

== See also ==

===Further reading===

Matter and Interactions I Modern Mechanics 4th Edition Chapter 11.10

===External links===

https://en.wikipedia.org/wiki/Bohr_model

https://en.wikipedia.org/wiki/Quantization_(physics)

Videos:

https://www.khanacademy.org/science/chemistry/electronic-structure-of-atoms/bohr-model-hydrogen/v/bohr-model-energy-levels

https://www.youtube.com/watch?v=nVW1zDPPZGM

Simulation of Bohr Model:

https://phet.colorado.edu/en/simulation/legacy/hydrogen-atom

Why Bohr's model explains everything around us:

http://scitech.au.dk/en/roemer/apr13/bohrs-model-of-the-atom-explains-science-in-everyday-life/

==References==

This section contains the the references you used while writing this page

[http://pachamamatrust.org/f2/1_K/SP_physics/H4_bohr_model_KSP.htm]
[http://chemistry.about.com/od/atomicstructure/a/bohr-model.htm]
[http://scitech.au.dk/en/roemer/apr13/bohrs-model-of-the-atom-explains-science-in-everyday-life/]
[http://www.encyclopedia.com/topic/Bohr_model.aspx]
[http://ib-physics-ii-6b-e.aspen.high.schoolfusion.us/modules/locker/files/get_group_file.phtml?gid=4397333&fid=18592200]
[https://files.t-square.gatech.edu/access/content/group/gtc-80c7-d8ef-5c05-873d-52adf5b9ce5c/Lecture%20Notes/Fenton/Phys2211_C_2015_fall_week_16_xMonday.pdf]
[http://hyperphysics.phy-astr.gsu.edu/hbase/bohr.html]
[http://web.ihep.su/dbserv/compas/src/bohr13/eng.pdf]
[http://chemed.chem.purdue.edu/genchem/history/bohr.html]
Note: All images on this page are either free for commercial use (with no attribution required) or made by myself.
[[Category:Energy]]

File:RutherfordModel2.png

2017-03-28T18:33:33Z

Sguo70:

2017-03-26T18:55:37Z

Sguo70: /* History */

Sherry Guo (Spring 2017)

This page gives basic information about the Bohr model and quantization. It also includes examples using the Bohr model.
==Main Idea==

[[File: niels-bohr-model-of-the-hydrogen-atom.png|right|300x300px|thumb|Bohr Model Diagram, [2]]] The Bohr model of the atom was proposed by Niels Bohr in 1913, and it was radically different from any other models that previously existed. His model depicted atoms as having negatively charged electrons which orbited as small, positively charged nucleus. Bohr's incorporation of quantum theory was the most groundbreaking part; in this model, electrons can only exist in discrete energy levels, which are quantized. This model is very simplistic and is useful in introducing students to quantum mechanics. Although this model successfully predicts energy levels of the hydrogen atom, it has its shortcomings.
===Bohr's Assumptions===
* Electrons travel in a circular orbit around the nucleus, similar to how planets orbit around the sun.
* The energy that electrons have is related to their distance from the nucleus, or which energy level they occupy. The further away the electron, the more energy it has.
* Electrons can gain or lose energy by jumping from one orbit to another. The energy is quantized - the orbitals have discreet radii.
* When an excited electron returns back to its ground state, then it releases the energy that it absorbed, in the form of light. This quantized energy is equal to the difference between the energies of the orbits.

==History==
[[Image:200px-Niels_Bohr_Date_Unverified_LOC.jpg |right|frame|Niels Bohr]] Ernest Rutherford proposed that the basic structure of an atom is a cloud of electrons which surround a small nucleus. With his experiments, he developed a model that depicted an atom like a solar system, with electrons orbiting around the nucleus. He proposed that the emission spectrum of hydrogen would look more like a smear rather than being made up of distinct lines. However, this was flawed, as it suggested that atoms can emit energy that isn't quantized. [8]

Niels Bohr was able to explain the emission spectra of hydrogen by improving on the Rutherford model of the atom. Bohr noticed that in the atomic emissions spectrum of hydrogen, only certain colors could be seen. Thus, he theorized that electrons need to be in energy levels that are quantized. He related the energies of the colors he saw to the differences of hydrogen's energy levels. Although this model is not entirely correct, as it only applies to systems where two charged particles orbit each other, it still has many features that are very applicable to physics today. [9]

== Shortcomings of the Bohr Model ==
While the Bohr model is an important predecessor to the current quantum mechanical models of the atom, it doesn't correctly describe some aspects of electron orbitals. It does not provide any reasoning as to why certain spectral lines are brighter than others. Its main shortcoming is that it violates the uncertainty principle, as it considers electrons to have a definite radius and momentum. This model is very basic, and in order to know more specific details about spectra and charge distribution, more calculations must be done. Many of the failures of the Bohr model can be corrected by the [http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hydsch.html#c2 Schrodinger equation]for the hydrogen atom.

==Application==
===A Mathematical Model===
==== The Angular Momentum Quantum ====
<math>ħ = h/2π =1.05*10^{-34} J*s </math>
h is Planck's constant, which is a physical constant that is essential in quantum mechanics.

==== Angular Momentum Is Quantized ====
<math> |\vec L_{trans,C}| = rp = Nħ</math>

p is the electron's momentum
r is the radius of circular orbit
N is an integer (1,2,3, ...)

We can derive the equation for r, the allowed Bohr radii for electron orbits.

1) Electric force the proton exerts on the electron:
<math> F_{el} = {\frac{e^2}{4π ε0 r^2}} </math>

2) Using the momentum principle and curving motion:
<math>|F_{perpendicular}| = {\frac{|p| |v|}{r}} = {\frac{e^2}{4π ε0 r^2}} </math>

3)Substituting in Bohr's conditions:
<math> {\frac{N^2 h*^2}{mr^3}} = {\frac{e^2}{4π ε0 r^2}} </math>

4)Solving for the allowed radii
<math> r = {\frac{4π ε0 h*^2}{me^2}} *N^2</math>
where N = 1,2,3,...

Additionally, the formula for energy of hydrogen atom of different levels is also derived from this model.

E = K + Uelectric

1)
<math> E = {\frac{mv^2}{2}} - {\frac{{\frac{1}{2}}*{\frac{1}{4π ε0}}*{\frac{me^2}{h*}}}{N^2}}</math>

2)
<math> E = {\frac{13.6 eV}{N^2}}</math> where N = 1,2,3
====Wavelengths====
The Debroglie relationship can tell us about the wavelength associated with the electron:
<math>λ = \frac{h}{mv}</math>

This equation is used to help derive the equation for the angular momentum of an electron in orbit
<math> L = \frac{nh}{2π} </math>

===A Computational Model===

[[File:bohr2.gif|upright]]
[[File:bohr3.gif]]

To interact with this glowscript visualization of the Bohr Model, click [http://www.glowscript.org/#/user/matterandinteractions/folder/matterandinteractions/program/08-Bohr-levels here]. You will also find a graph of Energy (eV), Kinetic Energy, and Potential Energy. This visualization shows how the electrons jump from level to level according to the Bohr Model. There is also an energy vs distance graph shown which varies according to these levels. Notice that the further away the electron is, the more energy it has.

==Examples==

===Simple===
Find the magnitude of the translational angular momentum of an electron when a hydrogen atom is in its 2nd excited state above the ground state.

--

We know that the only possible states of the hydrogen atom are those when the electron's translational angular momentum is an integer multiple of ħ.
<math>|\vec L_{trans,c}| = Nħ </math>
For the 2nd excited state, N = 3
Now just plug the numbers in
<math>|\vec L_{trans,c}| = (3)(1.05*10^{-34} J*s) = 3.15*10^{-34} J*s</math>

===Middling===
[6]A hydrogen atom is in state N = 3, where N = 1 is the lowest energy state. What is K+U in electron volts for this atomic hydrogen energy state?

1)
<math>E(3) = {\frac{-13.6 eV}{3^2}}</math> = -1.51 Joules

2)
<math>E(1) = {\frac{-13.6 eV}{1^2}}</math> = -13.6 Joules

3)
K+U = energy of photon = <math>E(1) - E(3) = {\frac{-13.6 eV}{3^2}} - {\frac{-13.6 eV}{1^2}}</math> = 12.09 Joules

Below is the graph of E as the hydrogen atom goes from N = 3 to N = 1.

[[Image:Graphproblem.png]]

===Difficult===
Hydrogen has been detected transitioning from the 101st to the 100th energy levels. What is the wavelength of the radiation? Where in the electromagnetic spectrum is this emission?

To solve this problem, we first need to use formulas derived from Bohr Model of hydrogen atom. It is <math>E = {\frac{-13.6 eV}{N^2}}</math>

Then solve for the wavelength using formula from Electromagnetic Wave Theory.

[[Image:Screen Shot 2015-12-01 at 8.21.56 PM.png|200x250px|]]

This wavelength falls in the microwave portion of the electromagnetic spectrum.

==Connectedness==
This topic is quite interesting as it is the initial introduction to quantum physics, which I find particularly intriguing. Although, this model has had shortcomings, it is one of the most successful models of its time. It has many features that are used in the actual model for quantum physics. This model also involves coding to show the visualization, which is how it somewhat related to Computer Science. Additionally, there are quite a few applications of the Bohr's Model. Bohr's discovery of the quantum leap is, in many ways, the most shining example of the consequences which Bohr's model of the atom has had for modern society. Bohr's notion that atoms emit light quanta with very specific energies is behind many of the technologies on which we depend our daily lives. Laser technology, which is becoming very popular in today's world, depends entirely on principles behind Bohr's model of atom because laser light is produced by quantum leaps. The quantum leaps between the specific energy levels show that light has specific frequency and wavelength which measures time and length precisely. However, due to shortcomings in Bohr's Model, we can't completely utilize these calculations, but the framework is used.

== See also ==

===Further reading===

Matter and Interactions I Modern Mechanics 4th Edition Chapter 11.10

===External links===

https://en.wikipedia.org/wiki/Bohr_model

https://en.wikipedia.org/wiki/Quantization_(physics)

Videos:

https://www.khanacademy.org/science/chemistry/electronic-structure-of-atoms/bohr-model-hydrogen/v/bohr-model-energy-levels

https://www.youtube.com/watch?v=nVW1zDPPZGM

Simulation of Bohr Model:

https://phet.colorado.edu/en/simulation/legacy/hydrogen-atom

==References==

This section contains the the references you used while writing this page

[http://pachamamatrust.org/f2/1_K/SP_physics/H4_bohr_model_KSP.htm]
[http://chemistry.about.com/od/atomicstructure/a/bohr-model.htm]
[http://scitech.au.dk/en/roemer/apr13/bohrs-model-of-the-atom-explains-science-in-everyday-life/]
[http://www.encyclopedia.com/topic/Bohr_model.aspx]
[http://ib-physics-ii-6b-e.aspen.high.schoolfusion.us/modules/locker/files/get_group_file.phtml?gid=4397333&fid=18592200]
[https://files.t-square.gatech.edu/access/content/group/gtc-80c7-d8ef-5c05-873d-52adf5b9ce5c/Lecture%20Notes/Fenton/Phys2211_C_2015_fall_week_16_xMonday.pdf]
[http://hyperphysics.phy-astr.gsu.edu/hbase/bohr.html]
[http://web.ihep.su/dbserv/compas/src/bohr13/eng.pdf]
[http://chemed.chem.purdue.edu/genchem/history/bohr.html]
[[Category:Energy]]

Bohr Model

2017-03-26T18:55:05Z

Sguo70: /* Middling */

Sherry Guo (Spring 2017)

This page gives basic information about the Bohr model and quantization. It also includes examples using the Bohr model.
==Main Idea==

[[File: niels-bohr-model-of-the-hydrogen-atom.png|right|300x300px|thumb|Bohr Model Diagram, [2]]] The Bohr model of the atom was proposed by Niels Bohr in 1913, and it was radically different from any other models that previously existed. His model depicted atoms as having negatively charged electrons which orbited as small, positively charged nucleus. Bohr's incorporation of quantum theory was the most groundbreaking part; in this model, electrons can only exist in discrete energy levels, which are quantized. This model is very simplistic and is useful in introducing students to quantum mechanics. Although this model successfully predicts energy levels of the hydrogen atom, it has its shortcomings.
===Bohr's Assumptions===
* Electrons travel in a circular orbit around the nucleus, similar to how planets orbit around the sun.
* The energy that electrons have is related to their distance from the nucleus, or which energy level they occupy. The further away the electron, the more energy it has.
* Electrons can gain or lose energy by jumping from one orbit to another. The energy is quantized - the orbitals have discreet radii.
* When an excited electron returns back to its ground state, then it releases the energy that it absorbed, in the form of light. This quantized energy is equal to the difference between the energies of the orbits.

==History==
[[Image:200px-Niels_Bohr_Date_Unverified_LOC.jpg |right| alt=caption|Niels Bohr]] Ernest Rutherford proposed that the basic structure of an atom is a cloud of electrons which surround a small nucleus. With his experiments, he developed a model that depicted an atom like a solar system, with electrons orbiting around the nucleus. He proposed that the emission spectrum of hydrogen would look more like a smear rather than being made up of distinct lines. However, this was flawed, as it suggested that atoms can emit energy that isn't quantized. [8]

Niels Bohr was able to explain the emission spectra of hydrogen by improving on the Rutherford model of the atom. Bohr noticed that in the atomic emissions spectrum of hydrogen, only certain colors could be seen. Thus, he theorized that electrons need to be in energy levels that are quantized. He related the energies of the colors he saw to the differences of hydrogen's energy levels. Although this model is not entirely correct, as it only applies to systems where two charged particles orbit each other, it still has many features that are very applicable to physics today. [9]

== Shortcomings of the Bohr Model ==
While the Bohr model is an important predecessor to the current quantum mechanical models of the atom, it doesn't correctly describe some aspects of electron orbitals. It does not provide any reasoning as to why certain spectral lines are brighter than others. Its main shortcoming is that it violates the uncertainty principle, as it considers electrons to have a definite radius and momentum. This model is very basic, and in order to know more specific details about spectra and charge distribution, more calculations must be done. Many of the failures of the Bohr model can be corrected by the [http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hydsch.html#c2 Schrodinger equation]for the hydrogen atom.

==Application==
===A Mathematical Model===
==== The Angular Momentum Quantum ====
<math>ħ = h/2π =1.05*10^{-34} J*s </math>
h is Planck's constant, which is a physical constant that is essential in quantum mechanics.

==== Angular Momentum Is Quantized ====
<math> |\vec L_{trans,C}| = rp = Nħ</math>

p is the electron's momentum
r is the radius of circular orbit
N is an integer (1,2,3, ...)

We can derive the equation for r, the allowed Bohr radii for electron orbits.

1) Electric force the proton exerts on the electron:
<math> F_{el} = {\frac{e^2}{4π ε0 r^2}} </math>

2) Using the momentum principle and curving motion:
<math>|F_{perpendicular}| = {\frac{|p| |v|}{r}} = {\frac{e^2}{4π ε0 r^2}} </math>

3)Substituting in Bohr's conditions:
<math> {\frac{N^2 h*^2}{mr^3}} = {\frac{e^2}{4π ε0 r^2}} </math>

4)Solving for the allowed radii
<math> r = {\frac{4π ε0 h*^2}{me^2}} *N^2</math>
where N = 1,2,3,...

Additionally, the formula for energy of hydrogen atom of different levels is also derived from this model.

E = K + Uelectric

1)
<math> E = {\frac{mv^2}{2}} - {\frac{{\frac{1}{2}}*{\frac{1}{4π ε0}}*{\frac{me^2}{h*}}}{N^2}}</math>

2)
<math> E = {\frac{13.6 eV}{N^2}}</math> where N = 1,2,3
====Wavelengths====
The Debroglie relationship can tell us about the wavelength associated with the electron:
<math>λ = \frac{h}{mv}</math>

This equation is used to help derive the equation for the angular momentum of an electron in orbit
<math> L = \frac{nh}{2π} </math>

===A Computational Model===

[[File:bohr2.gif|upright]]
[[File:bohr3.gif]]

To interact with this glowscript visualization of the Bohr Model, click [http://www.glowscript.org/#/user/matterandinteractions/folder/matterandinteractions/program/08-Bohr-levels here]. You will also find a graph of Energy (eV), Kinetic Energy, and Potential Energy. This visualization shows how the electrons jump from level to level according to the Bohr Model. There is also an energy vs distance graph shown which varies according to these levels. Notice that the further away the electron is, the more energy it has.

==Examples==

===Simple===
Find the magnitude of the translational angular momentum of an electron when a hydrogen atom is in its 2nd excited state above the ground state.

--

We know that the only possible states of the hydrogen atom are those when the electron's translational angular momentum is an integer multiple of ħ.
<math>|\vec L_{trans,c}| = Nħ </math>
For the 2nd excited state, N = 3
Now just plug the numbers in
<math>|\vec L_{trans,c}| = (3)(1.05*10^{-34} J*s) = 3.15*10^{-34} J*s</math>

===Middling===
[6]A hydrogen atom is in state N = 3, where N = 1 is the lowest energy state. What is K+U in electron volts for this atomic hydrogen energy state?

1)
<math>E(3) = {\frac{-13.6 eV}{3^2}}</math> = -1.51 Joules

2)
<math>E(1) = {\frac{-13.6 eV}{1^2}}</math> = -13.6 Joules

3)
K+U = energy of photon = <math>E(1) - E(3) = {\frac{-13.6 eV}{3^2}} - {\frac{-13.6 eV}{1^2}}</math> = 12.09 Joules

Below is the graph of E as the hydrogen atom goes from N = 3 to N = 1.

[[Image:Graphproblem.png]]

===Difficult===
Hydrogen has been detected transitioning from the 101st to the 100th energy levels. What is the wavelength of the radiation? Where in the electromagnetic spectrum is this emission?

To solve this problem, we first need to use formulas derived from Bohr Model of hydrogen atom. It is <math>E = {\frac{-13.6 eV}{N^2}}</math>

Then solve for the wavelength using formula from Electromagnetic Wave Theory.

[[Image:Screen Shot 2015-12-01 at 8.21.56 PM.png|200x250px|]]

This wavelength falls in the microwave portion of the electromagnetic spectrum.

==Connectedness==
This topic is quite interesting as it is the initial introduction to quantum physics, which I find particularly intriguing. Although, this model has had shortcomings, it is one of the most successful models of its time. It has many features that are used in the actual model for quantum physics. This model also involves coding to show the visualization, which is how it somewhat related to Computer Science. Additionally, there are quite a few applications of the Bohr's Model. Bohr's discovery of the quantum leap is, in many ways, the most shining example of the consequences which Bohr's model of the atom has had for modern society. Bohr's notion that atoms emit light quanta with very specific energies is behind many of the technologies on which we depend our daily lives. Laser technology, which is becoming very popular in today's world, depends entirely on principles behind Bohr's model of atom because laser light is produced by quantum leaps. The quantum leaps between the specific energy levels show that light has specific frequency and wavelength which measures time and length precisely. However, due to shortcomings in Bohr's Model, we can't completely utilize these calculations, but the framework is used.

== See also ==

===Further reading===

Matter and Interactions I Modern Mechanics 4th Edition Chapter 11.10

===External links===

https://en.wikipedia.org/wiki/Bohr_model

https://en.wikipedia.org/wiki/Quantization_(physics)

Videos:

https://www.khanacademy.org/science/chemistry/electronic-structure-of-atoms/bohr-model-hydrogen/v/bohr-model-energy-levels

https://www.youtube.com/watch?v=nVW1zDPPZGM

Simulation of Bohr Model:

https://phet.colorado.edu/en/simulation/legacy/hydrogen-atom

==References==

This section contains the the references you used while writing this page

[http://pachamamatrust.org/f2/1_K/SP_physics/H4_bohr_model_KSP.htm]
[http://chemistry.about.com/od/atomicstructure/a/bohr-model.htm]
[http://scitech.au.dk/en/roemer/apr13/bohrs-model-of-the-atom-explains-science-in-everyday-life/]
[http://www.encyclopedia.com/topic/Bohr_model.aspx]
[http://ib-physics-ii-6b-e.aspen.high.schoolfusion.us/modules/locker/files/get_group_file.phtml?gid=4397333&fid=18592200]
[https://files.t-square.gatech.edu/access/content/group/gtc-80c7-d8ef-5c05-873d-52adf5b9ce5c/Lecture%20Notes/Fenton/Phys2211_C_2015_fall_week_16_xMonday.pdf]
[http://hyperphysics.phy-astr.gsu.edu/hbase/bohr.html]
[http://web.ihep.su/dbserv/compas/src/bohr13/eng.pdf]
[http://chemed.chem.purdue.edu/genchem/history/bohr.html]
[[Category:Energy]]

Bohr Model

2017-03-26T18:48:35Z

Sguo70: /* Middling */

Sherry Guo (Spring 2017)

This page gives basic information about the Bohr model and quantization. It also includes examples using the Bohr model.
==Main Idea==

[[File: niels-bohr-model-of-the-hydrogen-atom.png|right|300x300px|thumb|Bohr Model Diagram, [2]]] The Bohr model of the atom was proposed by Niels Bohr in 1913, and it was radically different from any other models that previously existed. His model depicted atoms as having negatively charged electrons which orbited as small, positively charged nucleus. Bohr's incorporation of quantum theory was the most groundbreaking part; in this model, electrons can only exist in discrete energy levels, which are quantized. This model is very simplistic and is useful in introducing students to quantum mechanics. Although this model successfully predicts energy levels of the hydrogen atom, it has its shortcomings.
===Bohr's Assumptions===
* Electrons travel in a circular orbit around the nucleus, similar to how planets orbit around the sun.
* The energy that electrons have is related to their distance from the nucleus, or which energy level they occupy. The further away the electron, the more energy it has.
* Electrons can gain or lose energy by jumping from one orbit to another. The energy is quantized - the orbitals have discreet radii.
* When an excited electron returns back to its ground state, then it releases the energy that it absorbed, in the form of light. This quantized energy is equal to the difference between the energies of the orbits.

==History==
[[Image:200px-Niels_Bohr_Date_Unverified_LOC.jpg |right| alt=caption|Niels Bohr]] Ernest Rutherford proposed that the basic structure of an atom is a cloud of electrons which surround a small nucleus. With his experiments, he developed a model that depicted an atom like a solar system, with electrons orbiting around the nucleus. He proposed that the emission spectrum of hydrogen would look more like a smear rather than being made up of distinct lines. However, this was flawed, as it suggested that atoms can emit energy that isn't quantized. [8]

Niels Bohr was able to explain the emission spectra of hydrogen by improving on the Rutherford model of the atom. Bohr noticed that in the atomic emissions spectrum of hydrogen, only certain colors could be seen. Thus, he theorized that electrons need to be in energy levels that are quantized. He related the energies of the colors he saw to the differences of hydrogen's energy levels. Although this model is not entirely correct, as it only applies to systems where two charged particles orbit each other, it still has many features that are very applicable to physics today. [9]

== Shortcomings of the Bohr Model ==
While the Bohr model is an important predecessor to the current quantum mechanical models of the atom, it doesn't correctly describe some aspects of electron orbitals. It does not provide any reasoning as to why certain spectral lines are brighter than others. Its main shortcoming is that it violates the uncertainty principle, as it considers electrons to have a definite radius and momentum. This model is very basic, and in order to know more specific details about spectra and charge distribution, more calculations must be done. Many of the failures of the Bohr model can be corrected by the [http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hydsch.html#c2 Schrodinger equation]for the hydrogen atom.

==Application==
===A Mathematical Model===
==== The Angular Momentum Quantum ====
<math>ħ = h/2π =1.05*10^{-34} J*s </math>
h is Planck's constant, which is a physical constant that is essential in quantum mechanics.

==== Angular Momentum Is Quantized ====
<math> |\vec L_{trans,C}| = rp = Nħ</math>

p is the electron's momentum
r is the radius of circular orbit
N is an integer (1,2,3, ...)

We can derive the equation for r, the allowed Bohr radii for electron orbits.

1) Electric force the proton exerts on the electron:
<math> F_{el} = {\frac{e^2}{4π ε0 r^2}} </math>

2) Using the momentum principle and curving motion:
<math>|F_{perpendicular}| = {\frac{|p| |v|}{r}} = {\frac{e^2}{4π ε0 r^2}} </math>

3)Substituting in Bohr's conditions:
<math> {\frac{N^2 h*^2}{mr^3}} = {\frac{e^2}{4π ε0 r^2}} </math>

4)Solving for the allowed radii
<math> r = {\frac{4π ε0 h*^2}{me^2}} *N^2</math>
where N = 1,2,3,...

Additionally, the formula for energy of hydrogen atom of different levels is also derived from this model.

E = K + Uelectric

1)
<math> E = {\frac{mv^2}{2}} - {\frac{{\frac{1}{2}}*{\frac{1}{4π ε0}}*{\frac{me^2}{h*}}}{N^2}}</math>

2)
<math> E = {\frac{13.6 eV}{N^2}}</math> where N = 1,2,3
====Wavelengths====
The Debroglie relationship can tell us about the wavelength associated with the electron:
<math>λ = \frac{h}{mv}</math>

This equation is used to help derive the equation for the angular momentum of an electron in orbit
<math> L = \frac{nh}{2π} </math>

===A Computational Model===

[[File:bohr2.gif|upright]]
[[File:bohr3.gif]]

To interact with this glowscript visualization of the Bohr Model, click [http://www.glowscript.org/#/user/matterandinteractions/folder/matterandinteractions/program/08-Bohr-levels here]. You will also find a graph of Energy (eV), Kinetic Energy, and Potential Energy. This visualization shows how the electrons jump from level to level according to the Bohr Model. There is also an energy vs distance graph shown which varies according to these levels. Notice that the further away the electron is, the more energy it has.

==Examples==

===Simple===
Find the magnitude of the translational angular momentum of an electron when a hydrogen atom is in its 2nd excited state above the ground state.

--

We know that the only possible states of the hydrogen atom are those when the electron's translational angular momentum is an integer multiple of ħ.
<math>|\vec L_{trans,c}| = Nħ </math>
For the 2nd excited state, N = 3
Now just plug the numbers in
<math>|\vec L_{trans,c}| = (3)(1.05*10^{-34} J*s) = 3.15*10^{-34} J*s</math>

===Middling===
[6]A hydrogen atom is in state N = 3, where N = 1 is the lowest energy state. What is K+U in electron volts for this atomic hydrogen energy state?
[[Image:Graphproblem.png|right|250x250px, [6]]]

1)
<math>E(3) = {\frac{-13.6 eV}{3^2}}</math> = -1.51 Joules

2)
<math>E(1) = {\frac{-13.6 eV}{1^2}}</math> = -13.6 Joules

3)
K+U = energy of photon = <math>E(1) - E(3) = {\frac{-13.6 eV}{3^2}} - {\frac{-13.6 eV}{1^2}}</math> = 12.09 Joules

To the right is the graph of E as the hydrogen atom goes from N = 3 to N = 1.

===Difficult===
Hydrogen has been detected transitioning from the 101st to the 100th energy levels. What is the wavelength of the radiation? Where in the electromagnetic spectrum is this emission?

To solve this problem, we first need to use formulas derived from Bohr Model of hydrogen atom. It is <math>E = {\frac{-13.6 eV}{N^2}}</math>

Then solve for the wavelength using formula from Electromagnetic Wave Theory.

[[Image:Screen Shot 2015-12-01 at 8.21.56 PM.png|200x250px|]]

This wavelength falls in the microwave portion of the electromagnetic spectrum.

==Connectedness==
This topic is quite interesting as it is the initial introduction to quantum physics, which I find particularly intriguing. Although, this model has had shortcomings, it is one of the most successful models of its time. It has many features that are used in the actual model for quantum physics. This model also involves coding to show the visualization, which is how it somewhat related to Computer Science. Additionally, there are quite a few applications of the Bohr's Model. Bohr's discovery of the quantum leap is, in many ways, the most shining example of the consequences which Bohr's model of the atom has had for modern society. Bohr's notion that atoms emit light quanta with very specific energies is behind many of the technologies on which we depend our daily lives. Laser technology, which is becoming very popular in today's world, depends entirely on principles behind Bohr's model of atom because laser light is produced by quantum leaps. The quantum leaps between the specific energy levels show that light has specific frequency and wavelength which measures time and length precisely. However, due to shortcomings in Bohr's Model, we can't completely utilize these calculations, but the framework is used.

== See also ==

===Further reading===

Matter and Interactions I Modern Mechanics 4th Edition Chapter 11.10

===External links===

https://en.wikipedia.org/wiki/Bohr_model

https://en.wikipedia.org/wiki/Quantization_(physics)

Videos:

https://www.khanacademy.org/science/chemistry/electronic-structure-of-atoms/bohr-model-hydrogen/v/bohr-model-energy-levels

https://www.youtube.com/watch?v=nVW1zDPPZGM

Simulation of Bohr Model:

https://phet.colorado.edu/en/simulation/legacy/hydrogen-atom

==References==

This section contains the the references you used while writing this page

[http://pachamamatrust.org/f2/1_K/SP_physics/H4_bohr_model_KSP.htm]
[http://chemistry.about.com/od/atomicstructure/a/bohr-model.htm]
[http://scitech.au.dk/en/roemer/apr13/bohrs-model-of-the-atom-explains-science-in-everyday-life/]
[http://www.encyclopedia.com/topic/Bohr_model.aspx]
[http://ib-physics-ii-6b-e.aspen.high.schoolfusion.us/modules/locker/files/get_group_file.phtml?gid=4397333&fid=18592200]
[https://files.t-square.gatech.edu/access/content/group/gtc-80c7-d8ef-5c05-873d-52adf5b9ce5c/Lecture%20Notes/Fenton/Phys2211_C_2015_fall_week_16_xMonday.pdf]
[http://hyperphysics.phy-astr.gsu.edu/hbase/bohr.html]
[http://web.ihep.su/dbserv/compas/src/bohr13/eng.pdf]
[http://chemed.chem.purdue.edu/genchem/history/bohr.html]
[[Category:Energy]]

2017-03-24T22:09:15Z

Sguo70: /* Main Idea */

Sherry Guo (Spring 2017)

This page gives basic information about the Bohr Model and Quantization. It also includes examples using Bohr Model.
==Main Idea==

[[File: niels-bohr-model-of-the-hydrogen-atom.png|right|300x300px|thumb|Bohr Model Diagram, [2]]] The Bohr model of the atom was proposed by Niels Bohr in 1913, and it was radically different from any other models that previously existed. His model depicted atoms as having negatively charged electrons which orbited as small, positively charged nucleus. Bohr's incorporation of quantum theory was the most groundbreaking part; in this model, electrons can only exist in discrete energy levels, which are quantized. This model is very simplistic and is useful in introducing students to quantum mechanics. Although this model successfully predicts energy levels of the hydrogen atom, it has its shortcomings.
===A Mathematical Model===
This model gives us the formula for the radius derived from translational angular momentum.

|Ltrans,nucleus| = Nh* where N = 1,2,3

Note: h* is used when is not the actual notation for it.

h* = h/2π = 1.05 * 10^-34 J*s

1)
|Ltrans,nucleus| = Nh*,rp = Nh*

2)
From curving motion:
<math>|Fperpendicular| = {\frac{|p| |v|}{r}} = {\frac{e^2}{4π ε0 r^2}} </math>

3)
Substitute in Bohr's Condition:
<math> {\frac{N^2 h*^2}{mr^3}} = {\frac{e^2}{4π ε0 r^2}} </math>

4)
Solve for the Radius
<math> r = {\frac{4π ε0 h*^2}{me^2}} *N^2</math> where N = 1,2,3

Thus Bohr's Model derives equation for the radius.

Additionally, the formula for energy of hydrogen atom of different levels is also derived from this model.

E = K + Uelectric

1)
<math> E = {\frac{mv^2}{2}} - {\frac{{\frac{1}{2}}*{\frac{1}{4π ε0}}*{\frac{me^2}{h*}}}{N^2}}</math>

2)
<math> E = {\frac{13.6 eV}{N^2}}</math> where N = 1,2,3

===A Computational Model===

[[File:bohr2.gif|upright]]
[[File:bohr3.gif]]

To interact with this glowscript visualization of the Bohr Model, click [http://www.glowscript.org/#/user/matterandinteractions/folder/matterandinteractions/program/08-Bohr-levels here]. You will also find a graph of Energy (eV), Kinetic Energy, and Potential Energy. This visualization shows how the electrons jump from level to level according to the Bohr Model. There is also an energy vs distance graph shown which varies according to these levels. Notice that the further away the electron is, the more energy it has.

==Examples==

===Simple===
How much energy in electron volts is required to ionize a hydrogen atom, if initially the atom is in the state N = 3?
Here we can use the formula for the hydrogen atom which is

1)
<math>E = {\frac{-13.6 eV}{N^2}}</math>

2)
<math>={\frac{-13.6 eV}{3^2}}</math>
= -1.51 Joules

===Middling===
[6]A hydrogen atom is in state N = 3, where N = 1 is the lowest energy state. What is K+U in electron volts for this atomic hydrogen energy state?
[[Image:Screen Shot 2015-12-03 at 9.37.48 PM.png|right|250x250px, [6]]]

1)
<math>E(3) = {\frac{-13.6 eV}{3^2}}</math> = -1.51 Joules

2)
<math>E(1) = {\frac{-13.6 eV}{1^2}}</math> = -13.6 Joules

3)
K+U = energy of photon = <math>E(1) - E(3) = {\frac{-13.6 eV}{3^2}} - {\frac{-13.6 eV}{1^2}}</math> = 12.09 Joules

To the right is the graph of E as the hydrogen atom goes from N = 3 to N = 1.

===Difficult===
Hydrogen has been detected transitioning from the 101st to the 100th energy levels. What is the wavelength of the radiation? Where in the electromagnetic spectrum is this emission?

To solve this problem, we first need to use formulas derived from Bohr Model of hydrogen atom. It is <math>E = {\frac{-13.6 eV}{N^2}}</math>

Then solve for the wavelength using formula from Electromagnetic Wave Theory.

[[Image:Screen Shot 2015-12-01 at 8.21.56 PM.png|left|350x350px|]]

This wavelength falls in the microwave portion of the electromagnetic spectrum.

==Connectedness==
This topic is quite interesting as it is the initial introduction to quantum physics, which I find particularly intriguing. Although, this model has had shortcomings, it is one of the most successful models of its time. It has many features that are used in the actual model for quantum physics. This model also involves coding to show the visualization, which is how it somewhat related to Computer Science. Additionally, there are quite a few applications of the Bohr's Model. Bohr's discovery of the quantum leap is, in many ways, the most shining example of the consequences which Bohr's model of the atom has had for modern society. Bohr's notion that atoms emit light quanta with very specific energies is behind many of the technologies on which we depend our daily lives. Laser technology, which is becoming very popular in today's world, depends entirely on principles behind Bohr's model of atom because laser light is produced by quantum leaps. The quantum leaps between the specific energy levels show that light has specific frequency and wavelength which measures time and length precisely. However, due to shortcomings in Bohr's Model, we can't completely utilize these calculations, but the framework is used.

==History==

[[Image:200px-Niels_Bohr_Date_Unverified_LOC.jpg|right|100x100px, [5]]] Ernest Rutherford proposed that the basic structure of an atom is a cloud of electrons which surround a small nucleus. With his experiments, he developed a model that depicted an atom like a solar system, with electrons orbiting around the nucleus. However, his proposal was flawed, as it suggested all atoms are unstable and can emit energy that isn't quantized. [8]

Niels Bohr was able to explain the emission spectra of hydrogen by improving on the Rutherford model of the atom. Bohr’s model improved the classical atomic models of physicists J. J. Thomson and Ernest Rutherford by incorporating quantum theory. While working on his doctoral dissertation at Copenhagen University, Bohr studied physicist Max Planck’s quantum theory of radiation. Then after graduating, Bohr worked in England with Thomson and subsequently with Rutherford to come up with this model. During this time, Bohr developed his model of atomic structure. Initially, Rutherford's planetary model predicted a continuous spectrum of light from hydrogen. However, Bohr corrected for this by proposing that the translational angular momentum of the electron can be quantized. Although this model is not entirely correct, as it only applies to systems where two charged particles orbit each other, it still has many features that are very applicable to physics today.

== Shortcomings of the Bohr Model ==
The Bohr Model is an important predecessor to the current quantum mechanical models of the atom. However, there are some characteristics of the Bohr model that are not entirely correct. The actual quantization rules in a hydrogen atom are much more complex than those assumed in the Bohr Model. The translational angular momentum in ground state (N = 1), is zero, not h, and for the next higher state of N = 2, the z component of translational angular momentum can either be zero or h. Other issues with the Bohr Model include that it violates the Heisenberg Uncertainty Principle because it considers electrons to have both a known radius and orbit. It also makes poor predictions regarding spectra of larger atoms, and does not predict the relative intensities of spectral lines.

== See also ==

===Further reading===

Matter and Interactions I Modern Mechanics 4th Edition Chapter 11.10

===External links===

https://en.wikipedia.org/wiki/Bohr_model

https://en.wikipedia.org/wiki/Quantization_(physics)

Videos:

https://www.khanacademy.org/science/chemistry/electronic-structure-of-atoms/bohr-model-hydrogen/v/bohr-model-energy-levels

https://www.youtube.com/watch?v=nVW1zDPPZGM

Simulation of Bohr Model:

https://phet.colorado.edu/en/simulation/legacy/hydrogen-atom

==References==

This section contains the the references you used while writing this page

[http://pachamamatrust.org/f2/1_K/SP_physics/H4_bohr_model_KSP.htm]
[http://chemistry.about.com/od/atomicstructure/a/bohr-model.htm]
[http://scitech.au.dk/en/roemer/apr13/bohrs-model-of-the-atom-explains-science-in-everyday-life/]
[http://www.encyclopedia.com/topic/Bohr_model.aspx]
[http://ib-physics-ii-6b-e.aspen.high.schoolfusion.us/modules/locker/files/get_group_file.phtml?gid=4397333&fid=18592200]
[https://files.t-square.gatech.edu/access/content/group/gtc-80c7-d8ef-5c05-873d-52adf5b9ce5c/Lecture%20Notes/Fenton/Phys2211_C_2015_fall_week_16_xMonday.pdf]
[http://hyperphysics.phy-astr.gsu.edu/hbase/bohr.html]
[http://web.ihep.su/dbserv/compas/src/bohr13/eng.pdf]

[[Category:Energy]]

Bohr Model

2017-03-24T21:24:22Z

Sguo70: /* A Computational Model */

Sherry Guo (Spring 2017)

This page gives basic information about the Bohr Model and Quantization. It also includes examples using Bohr Model.
==Main Idea==

[[File: niels-bohr-model-of-the-hydrogen-atom.png|right|300x300px|thumb|Bohr Model Diagram, [2]]] In atomic physics, the Bohr model depicts the atom as a small, positively charged nucleus surrounded by electrons in orbit similar in structure to the solar system. In this model, the neutrons and protons occupy a dense central region (the nucleus), while the electrons orbit the nucleus, like the planets orbit the Sun. This is why the Bohr Model is commonly referred to as the "planetary model" [2].It is taught as an introduction to quantum physics. In the Bohr Model, electrons can only be at certain, different, distances from the proton to which it is bound. Energy is quantized as explained by the Bohr Model. This means that only orbits with certain radii are allowed, while orbits in between simply don't exist. These levels are knows an quantized energy levels and are labeled with integer N known as quantum number. The lowest energy state is generally termed the ground state. The states with successively more energy than the ground state are called the first excited state, the second excited state, and so on. As the electrons become further away from the nucleus, they become larger and have higher energy. Beyond an energy called the ionization potential the single electron of the hydrogen atom is no longer bound to the atom. The Bohr model works well for very simple atoms such as hydrogen (which has 1 electron) but not for more complex atoms. Although the Bohr model is still used today, especially in elementary textbooks, a more complex model known as the quantum mechanical model is the more accurate version of the Bohr Model and used universally.

===A Mathematical Model===
This model gives us the formula for the radius derived from translational angular momentum.

|Ltrans,nucleus| = Nh* where N = 1,2,3

Note: h* is used when is not the actual notation for it.

h* = h/2π = 1.05 * 10^-34 J*s

1)
|Ltrans,nucleus| = Nh*,rp = Nh*

2)
From curving motion:
<math>|Fperpendicular| = {\frac{|p| |v|}{r}} = {\frac{e^2}{4π ε0 r^2}} </math>

3)
Substitute in Bohr's Condition:
<math> {\frac{N^2 h*^2}{mr^3}} = {\frac{e^2}{4π ε0 r^2}} </math>

4)
Solve for the Radius
<math> r = {\frac{4π ε0 h*^2}{me^2}} *N^2</math> where N = 1,2,3

Thus Bohr's Model derives equation for the radius.

Additionally, the formula for energy of hydrogen atom of different levels is also derived from this model.

E = K + Uelectric

1)
<math> E = {\frac{mv^2}{2}} - {\frac{{\frac{1}{2}}*{\frac{1}{4π ε0}}*{\frac{me^2}{h*}}}{N^2}}</math>

2)
<math> E = {\frac{13.6 eV}{N^2}}</math> where N = 1,2,3

===A Computational Model===

[[File:bohr2.gif|upright]]
[[File:bohr3.gif]]

To interact with this glowscript visualization of the Bohr Model, click [http://www.glowscript.org/#/user/matterandinteractions/folder/matterandinteractions/program/08-Bohr-levels here]. You will also find a graph of Energy (eV), Kinetic Energy, and Potential Energy. This visualization shows how the electrons jump from level to level according to the Bohr Model. There is also an energy vs distance graph shown which varies according to these levels. Notice that the further away the electron is, the more energy it has.

==Examples==

===Simple===
How much energy in electron volts is required to ionize a hydrogen atom, if initially the atom is in the state N = 3?
Here we can use the formula for the hydrogen atom which is

1)
<math>E = {\frac{-13.6 eV}{N^2}}</math>

2)
<math>={\frac{-13.6 eV}{3^2}}</math>
= -1.51 Joules

===Middling===
[6]A hydrogen atom is in state N = 3, where N = 1 is the lowest energy state. What is K+U in electron volts for this atomic hydrogen energy state?
[[Image:Screen Shot 2015-12-03 at 9.37.48 PM.png|right|250x250px, [6]]]

1)
<math>E(3) = {\frac{-13.6 eV}{3^2}}</math> = -1.51 Joules

2)
<math>E(1) = {\frac{-13.6 eV}{1^2}}</math> = -13.6 Joules

3)
K+U = energy of photon = <math>E(1) - E(3) = {\frac{-13.6 eV}{3^2}} - {\frac{-13.6 eV}{1^2}}</math> = 12.09 Joules

To the right is the graph of E as the hydrogen atom goes from N = 3 to N = 1.

===Difficult===
Hydrogen has been detected transitioning from the 101st to the 100th energy levels. What is the wavelength of the radiation? Where in the electromagnetic spectrum is this emission?

To solve this problem, we first need to use formulas derived from Bohr Model of hydrogen atom. It is <math>E = {\frac{-13.6 eV}{N^2}}</math>

Then solve for the wavelength using formula from Electromagnetic Wave Theory.

[[Image:Screen Shot 2015-12-01 at 8.21.56 PM.png|left|350x350px|]]

This wavelength falls in the microwave portion of the electromagnetic spectrum.

==Connectedness==
This topic is quite interesting as it is the initial introduction to quantum physics, which I find particularly intriguing. Although, this model has had shortcomings, it is one of the most successful models of its time. It has many features that are used in the actual model for quantum physics. This model also involves coding to show the visualization, which is how it somewhat related to Computer Science. Additionally, there are quite a few applications of the Bohr's Model. Bohr's discovery of the quantum leap is, in many ways, the most shining example of the consequences which Bohr's model of the atom has had for modern society. Bohr's notion that atoms emit light quanta with very specific energies is behind many of the technologies on which we depend our daily lives. Laser technology, which is becoming very popular in today's world, depends entirely on principles behind Bohr's model of atom because laser light is produced by quantum leaps. The quantum leaps between the specific energy levels show that light has specific frequency and wavelength which measures time and length precisely. However, due to shortcomings in Bohr's Model, we can't completely utilize these calculations, but the framework is used.

==History==

[[Image:200px-Niels_Bohr_Date_Unverified_LOC.jpg|right|100x100px, [5]]] Ernest Rutherford proposed that the basic structure of an atom is a cloud of electrons which surround a small nucleus. With his experiments, he developed a model that depicted an atom like a solar system, with electrons orbiting around the nucleus. However, his proposal was flawed, as it suggested all atoms are unstable and can emit energy that isn't quantized. [8]

Niels Bohr was able to explain the emission spectra of hydrogen by improving on the Rutherford model of the atom. Bohr’s model improved the classical atomic models of physicists J. J. Thomson and Ernest Rutherford by incorporating quantum theory. While working on his doctoral dissertation at Copenhagen University, Bohr studied physicist Max Planck’s quantum theory of radiation. Then after graduating, Bohr worked in England with Thomson and subsequently with Rutherford to come up with this model. During this time, Bohr developed his model of atomic structure. Initially, Rutherford's planetary model predicted a continuous spectrum of light from hydrogen. However, Bohr corrected for this by proposing that the translational angular momentum of the electron can be quantized. Although this model is not entirely correct, as it only applies to systems where two charged particles orbit each other, it still has many features that are very applicable to physics today.

== Shortcomings of the Bohr Model ==
The Bohr Model is an important predecessor to the current quantum mechanical models of the atom. However, there are some characteristics of the Bohr model that are not entirely correct. The actual quantization rules in a hydrogen atom are much more complex than those assumed in the Bohr Model. The translational angular momentum in ground state (N = 1), is zero, not h, and for the next higher state of N = 2, the z component of translational angular momentum can either be zero or h. Other issues with the Bohr Model include that it violates the Heisenberg Uncertainty Principle because it considers electrons to have both a known radius and orbit. It also makes poor predictions regarding spectra of larger atoms, and does not predict the relative intensities of spectral lines.

== See also ==

===Further reading===

Matter and Interactions I Modern Mechanics 4th Edition Chapter 11.10

===External links===

https://en.wikipedia.org/wiki/Bohr_model

https://en.wikipedia.org/wiki/Quantization_(physics)

Videos:

https://www.khanacademy.org/science/chemistry/electronic-structure-of-atoms/bohr-model-hydrogen/v/bohr-model-energy-levels

https://www.youtube.com/watch?v=nVW1zDPPZGM

Simulation of Bohr Model:

https://phet.colorado.edu/en/simulation/legacy/hydrogen-atom

==References==

This section contains the the references you used while writing this page

[http://pachamamatrust.org/f2/1_K/SP_physics/H4_bohr_model_KSP.htm]
[http://chemistry.about.com/od/atomicstructure/a/bohr-model.htm]
[http://scitech.au.dk/en/roemer/apr13/bohrs-model-of-the-atom-explains-science-in-everyday-life/]
[http://www.encyclopedia.com/topic/Bohr_model.aspx]
[http://ib-physics-ii-6b-e.aspen.high.schoolfusion.us/modules/locker/files/get_group_file.phtml?gid=4397333&fid=18592200]
[https://files.t-square.gatech.edu/access/content/group/gtc-80c7-d8ef-5c05-873d-52adf5b9ce5c/Lecture%20Notes/Fenton/Phys2211_C_2015_fall_week_16_xMonday.pdf]
[http://hyperphysics.phy-astr.gsu.edu/hbase/bohr.html]
[http://web.ihep.su/dbserv/compas/src/bohr13/eng.pdf]

[[Category:Energy]]

Bohr Model

2017-03-24T21:18:13Z

Sguo70: /* A Computational Model */

Sherry Guo (Spring 2017)

This page gives basic information about the Bohr Model and Quantization. It also includes examples using Bohr Model.
==Main Idea==

[[File: niels-bohr-model-of-the-hydrogen-atom.png|right|300x300px|thumb|Bohr Model Diagram, [2]]] In atomic physics, the Bohr model depicts the atom as a small, positively charged nucleus surrounded by electrons in orbit similar in structure to the solar system. In this model, the neutrons and protons occupy a dense central region (the nucleus), while the electrons orbit the nucleus, like the planets orbit the Sun. This is why the Bohr Model is commonly referred to as the "planetary model" [2].It is taught as an introduction to quantum physics. In the Bohr Model, electrons can only be at certain, different, distances from the proton to which it is bound. Energy is quantized as explained by the Bohr Model. This means that only orbits with certain radii are allowed, while orbits in between simply don't exist. These levels are knows an quantized energy levels and are labeled with integer N known as quantum number. The lowest energy state is generally termed the ground state. The states with successively more energy than the ground state are called the first excited state, the second excited state, and so on. As the electrons become further away from the nucleus, they become larger and have higher energy. Beyond an energy called the ionization potential the single electron of the hydrogen atom is no longer bound to the atom. The Bohr model works well for very simple atoms such as hydrogen (which has 1 electron) but not for more complex atoms. Although the Bohr model is still used today, especially in elementary textbooks, a more complex model known as the quantum mechanical model is the more accurate version of the Bohr Model and used universally.

===A Mathematical Model===
This model gives us the formula for the radius derived from translational angular momentum.

|Ltrans,nucleus| = Nh* where N = 1,2,3

Note: h* is used when is not the actual notation for it.

h* = h/2π = 1.05 * 10^-34 J*s

1)
|Ltrans,nucleus| = Nh*,rp = Nh*

2)
From curving motion:
<math>|Fperpendicular| = {\frac{|p| |v|}{r}} = {\frac{e^2}{4π ε0 r^2}} </math>

3)
Substitute in Bohr's Condition:
<math> {\frac{N^2 h*^2}{mr^3}} = {\frac{e^2}{4π ε0 r^2}} </math>

4)
Solve for the Radius
<math> r = {\frac{4π ε0 h*^2}{me^2}} *N^2</math> where N = 1,2,3

Thus Bohr's Model derives equation for the radius.

Additionally, the formula for energy of hydrogen atom of different levels is also derived from this model.

E = K + Uelectric

1)
<math> E = {\frac{mv^2}{2}} - {\frac{{\frac{1}{2}}*{\frac{1}{4π ε0}}*{\frac{me^2}{h*}}}{N^2}}</math>

2)
<math> E = {\frac{13.6 eV}{N^2}}</math> where N = 1,2,3

===A Computational Model===

[[File:bohr2.gif|upright]]
[[File:bohr3.gif]]

Here is a [http://www.glowscript.org/#/user/matterandinteractions/folder/matterandinteractions/program/08-Bohr-levels glowscript visualization] of the Bohr Model, and its graph of Energy (eV), Kinetic Energy, and Potential Energy. This visualization shows how the electrons jump from level to level according to the Bohr Model. There is also an energy vs distance graph shown which varies according to these levels.

==Examples==

===Simple===
How much energy in electron volts is required to ionize a hydrogen atom, if initially the atom is in the state N = 3?
Here we can use the formula for the hydrogen atom which is

1)
<math>E = {\frac{-13.6 eV}{N^2}}</math>

2)
<math>={\frac{-13.6 eV}{3^2}}</math>
= -1.51 Joules

===Middling===
[6]A hydrogen atom is in state N = 3, where N = 1 is the lowest energy state. What is K+U in electron volts for this atomic hydrogen energy state?
[[Image:Screen Shot 2015-12-03 at 9.37.48 PM.png|right|250x250px, [6]]]

1)
<math>E(3) = {\frac{-13.6 eV}{3^2}}</math> = -1.51 Joules

2)
<math>E(1) = {\frac{-13.6 eV}{1^2}}</math> = -13.6 Joules

3)
K+U = energy of photon = <math>E(1) - E(3) = {\frac{-13.6 eV}{3^2}} - {\frac{-13.6 eV}{1^2}}</math> = 12.09 Joules

To the right is the graph of E as the hydrogen atom goes from N = 3 to N = 1.

===Difficult===
Hydrogen has been detected transitioning from the 101st to the 100th energy levels. What is the wavelength of the radiation? Where in the electromagnetic spectrum is this emission?

To solve this problem, we first need to use formulas derived from Bohr Model of hydrogen atom. It is <math>E = {\frac{-13.6 eV}{N^2}}</math>

Then solve for the wavelength using formula from Electromagnetic Wave Theory.

[[Image:Screen Shot 2015-12-01 at 8.21.56 PM.png|left|350x350px|]]

This wavelength falls in the microwave portion of the electromagnetic spectrum.

==Connectedness==
This topic is quite interesting as it is the initial introduction to quantum physics, which I find particularly intriguing. Although, this model has had shortcomings, it is one of the most successful models of its time. It has many features that are used in the actual model for quantum physics. This model also involves coding to show the visualization, which is how it somewhat related to Computer Science. Additionally, there are quite a few applications of the Bohr's Model. Bohr's discovery of the quantum leap is, in many ways, the most shining example of the consequences which Bohr's model of the atom has had for modern society. Bohr's notion that atoms emit light quanta with very specific energies is behind many of the technologies on which we depend our daily lives. Laser technology, which is becoming very popular in today's world, depends entirely on principles behind Bohr's model of atom because laser light is produced by quantum leaps. The quantum leaps between the specific energy levels show that light has specific frequency and wavelength which measures time and length precisely. However, due to shortcomings in Bohr's Model, we can't completely utilize these calculations, but the framework is used.

==History==

[[Image:200px-Niels_Bohr_Date_Unverified_LOC.jpg|right|100x100px, [5]]] Ernest Rutherford proposed that the basic structure of an atom is a cloud of electrons which surround a small nucleus. With his experiments, he developed a model that depicted an atom like a solar system, with electrons orbiting around the nucleus. However, his proposal was flawed, as it suggested all atoms are unstable and can emit energy that isn't quantized. [8]

Niels Bohr was able to explain the emission spectra of hydrogen by improving on the Rutherford model of the atom. Bohr’s model improved the classical atomic models of physicists J. J. Thomson and Ernest Rutherford by incorporating quantum theory. While working on his doctoral dissertation at Copenhagen University, Bohr studied physicist Max Planck’s quantum theory of radiation. Then after graduating, Bohr worked in England with Thomson and subsequently with Rutherford to come up with this model. During this time, Bohr developed his model of atomic structure. Initially, Rutherford's planetary model predicted a continuous spectrum of light from hydrogen. However, Bohr corrected for this by proposing that the translational angular momentum of the electron can be quantized. Although this model is not entirely correct, as it only applies to systems where two charged particles orbit each other, it still has many features that are very applicable to physics today.

== Shortcomings of the Bohr Model ==
The Bohr Model is an important predecessor to the current quantum mechanical models of the atom. However, there are some characteristics of the Bohr model that are not entirely correct. The actual quantization rules in a hydrogen atom are much more complex than those assumed in the Bohr Model. The translational angular momentum in ground state (N = 1), is zero, not h, and for the next higher state of N = 2, the z component of translational angular momentum can either be zero or h. Other issues with the Bohr Model include that it violates the Heisenberg Uncertainty Principle because it considers electrons to have both a known radius and orbit. It also makes poor predictions regarding spectra of larger atoms, and does not predict the relative intensities of spectral lines.

== See also ==

===Further reading===

Matter and Interactions I Modern Mechanics 4th Edition Chapter 11.10

===External links===

https://en.wikipedia.org/wiki/Bohr_model

https://en.wikipedia.org/wiki/Quantization_(physics)

Videos:

https://www.khanacademy.org/science/chemistry/electronic-structure-of-atoms/bohr-model-hydrogen/v/bohr-model-energy-levels

https://www.youtube.com/watch?v=nVW1zDPPZGM

Simulation of Bohr Model:

https://phet.colorado.edu/en/simulation/legacy/hydrogen-atom

==References==

This section contains the the references you used while writing this page

[http://pachamamatrust.org/f2/1_K/SP_physics/H4_bohr_model_KSP.htm]
[http://chemistry.about.com/od/atomicstructure/a/bohr-model.htm]
[http://scitech.au.dk/en/roemer/apr13/bohrs-model-of-the-atom-explains-science-in-everyday-life/]
[http://www.encyclopedia.com/topic/Bohr_model.aspx]
[http://ib-physics-ii-6b-e.aspen.high.schoolfusion.us/modules/locker/files/get_group_file.phtml?gid=4397333&fid=18592200]
[https://files.t-square.gatech.edu/access/content/group/gtc-80c7-d8ef-5c05-873d-52adf5b9ce5c/Lecture%20Notes/Fenton/Phys2211_C_2015_fall_week_16_xMonday.pdf]
[http://hyperphysics.phy-astr.gsu.edu/hbase/bohr.html]
[http://web.ihep.su/dbserv/compas/src/bohr13/eng.pdf]

[[Category:Energy]]