Density

This topic covers Density -- Claimed by Ritvik Khanna, M08 Dr. Greco Spring 2016

The Main Idea

Density can be simply defined as the relative compactness of a substance. The units for density are quantified by mass per unit volume. The density, also called the volumetric mass density, of a substance is denoted by the greek symbol ρ (the lowercase Greek letter rho), but sometimes it can also be represented by the Latin letter D. The density of materials varies, different materials will have different densities. For instance, osmium and iridium are different elements and therefore have different densities. Osmium has a density of 22.59 g/cm^3 while iridium has a density of 22.42 g/cm^3. Also the density of a material can be related to the materials buoyancy, packaging, and purity.

A Mathematical Model

Mathematically, density is defined as mass divided by volume.

[math]\displaystyle{ \rho = \frac{m}{V}, }[/math]

Density = Mass / Volume

For dealing with mass distributions in the more general sense, density refers to the mass distributed in a certain volume, often denoted with differentials:

Density = dm/dV

In this equation ρ is the density, m is the mass, and V is the volume.The mass can be expressed in grams [g] or kilograms [kg], and the volume is measured in liters [L], cubic centimeters [cm^3], or milliliters [mL]. In this equation mass is divided by volume and in the US oil and gas industry sometimes density is referred to as weight per unit volume. This is technically incorrect as this quantity is actually called specific weight.

A Computational Model

The density of a solution is the sum of the mass of the components of that solution. Mass concentrations of each component ρi in a solution sums to density of the solution. Therefore this can be modeled by the following equation.

[math]\displaystyle{ \rho = \sum_i \varrho_i \ }[/math]

This expressed as a function of the densities of pure components of the mixture allows for the computation of the excess molar volumes through the following summation.

[math]\displaystyle{ \rho = \sum_i \rho_i \frac{V_i}{V}\,= \sum_i \rho_i \varphi_i =\sum_i \rho_i \frac{V_i}{\sum_i V_i + \sum_i V_i^{E}} }[/math]

From this point by having the excess volumes and activity coefficients it is possible to determine the activity coefficients by using the equation below.

[math]\displaystyle{ \bar{V^E}_i= RT \frac{\partial (ln(\gamma_i))}{\partial P} }[/math]

Examples

Simple

What is the weight of the isopropyl alcohol that exactly fills a 200.0 mL container? The density of isopropyl alcohol is 0.786 g/mL.

[math]\displaystyle{ D = \frac{m}{V} }[/math]

Isolating for mass (m): [math]\displaystyle{ m = D \cdot V }[/math]

[math]\displaystyle{ m = 0.789 \frac{g}{mL} \cdot 200.0 }[/math] mL [math]\displaystyle{ = 157.2 }[/math] g

Thus, the isopropyl alcohol weighs [math]\displaystyle{ 157.2 }[/math] grams.

Middling

A copper ingot has a mass of 2.15 kg. If the copper is drawn into a wire whose diameter is 2.27 mm, how many inches of copper wire can be obtained from the ingot?

It is easier and best practice to break a problem into several smaller steps.

a) First, determine volume of copper:

[math]\displaystyle{ 8.94 \frac{g}{cm^3} = 2150 \frac{g}{V} }[/math]

[math]\displaystyle{ V = 240.49217 }[/math] cm^3

Note: this is the volume of the wire.

b) Determine h in the volume of a cylinder (the wire)

This "height" translates to the length of the wire, since the wire can essentially be treated as a long cylinder.

[math]\displaystyle{ V = \pi \cdot r^2 \cdot h }[/math]

Isolating for height (h):

[math]\displaystyle{ h = \frac{V}{\pi \cdot r^2} }[/math]

where r = radius

[math]\displaystyle{ h = \frac{240.49217 cm^3}{\pi \cdot (0.1135 cm)^2} }[/math]

{h} = [math]\displaystyle{ 5942.37 }[/math] cm

Note: 0.1135 cm is the radius

c) Convert cm to inch:

5942.37 cm divided by 2.54 cm/in = 2340 inches

Difficult

Copper can be drawn into thin wires. How many meters of 34-gauge wire (diameter = [math]\displaystyle{ 6.304 \cdot 10^{-3} }[/math] inches) can be produced from the copper that is in [math]\displaystyle{ 5.88 }[/math] lbs of covellite, an ore of copper that is [math]\displaystyle{ 66% }[/math] copper by mass? (Hint: treat the wire as a cylinder. The density of copper is [math]\displaystyle{ 8.94 }[/math] [math]\displaystyle{ \frac{g}{cm^3} }[/math]; one kg weighs [math]\displaystyle{ 2.2046 }[/math] lbs. The volume of a cylinder is [math]\displaystyle{ V_{cyl} = \pi \cdot r^2 \cdot h }[/math].

a) Determine pounds of pure copper in 5.88 lbs of covellite:

[math]\displaystyle{ 5.88 }[/math] lbs [math]\displaystyle{ \cdot 0.66 = 3.8808 }[/math] lbs

b) Convert pounds to kilograms:

[math]\displaystyle{ 3.8808 }[/math] lbs / [math]\displaystyle{ 2.6046 }[/math] lbs kg^-1 = 1.489979 kg (keep a few guard digits)

1.489979 kg = 1489.979 g

c) Determine the volume that this mass of copper occupies:

[math]\displaystyle{ 8.94 }[/math] g/cm^3 = [math]\displaystyle{ 1489.979 }[/math] g / V

V = 166.664 cm^3

Note: this is the volume of the cylinder.

d) Convert the diameter in inches to a radius in centimeters:

diameter = [math]\displaystyle{ 6.304 \cdot 10^{-3} }[/math] in; radius = [math]\displaystyle{ 3.152 \cdot 10^{-3} }[/math] in

[math]\displaystyle{ 3.152 \cdot 10^{-3} }[/math] in [math]\displaystyle{ \cdot }[/math] [math]\displaystyle{ 2.54 }[/math] cm/in = [math]\displaystyle{ 8.00 \cdot 10^{-3} cm }[/math]

e) Determine h in the volume of a cylinder:

[math]\displaystyle{ 166.664 }[/math] cm^3 = [math]\displaystyle{ \pi \cdot (8.00 \cdot 10^{-3})^{2} }[/math] cm [math]\displaystyle{ \cdot h }[/math]

[math]\displaystyle{ h = 5.63 \cdot 10^{3} }[/math] m

Connectedness

Importance of Density

Density is a very important physical characteristic of matter. Density can increase or decrease as the result of actions taken on an object. Overall the characteristics of density are very important in the the way the universe works and in our daily lives.

Density is often a critical component in analyzing certain charge and mass distributions; for instance, in electricity in magnetism, the charge density affects the electric field and is often used to find quantities like total charge.

How Density Changes

Density can increase either due to increasing mass or decreasing volume. For instance, if you have two objects that have the same mass and you compress one of the objects to a smaller size, the smaller sized object will have a higher density than the non compressed object. On the other hand, if you have two objets of equal volume but different mass, the object with the greater mass will have a high density. It is for this reason that a bowling ball has a higher density than a volleyball, even though both are similar in volume.

Helium Balloons

Helium balloons that we see at most birthday parties work in the way they do because of helium's density. Since the helium air inside the balloon is less dense than the air around the balloon, the balloon is able to float. This phenomenon is true for weather balloons and many other things.

Ice and Icebergs

Icebergs often cause many problems and can be hazardous to ships. The reason that icebergs stay afloat is because the density of water changes as the temperature drops and ice floats. In addition, icebergs are made of frozen freshwater which has a lower density than the surrounding cold salty seawater which is more dense. Therefore icebergs remain afloat and pose quite the danger for ships.

Density and Chemical Engineers

It is very important for chemical engineers to understand the microscopic structure and macroscopic properties of matter. Chemical engineers and bio chemists also use the concept of density functional theory (DFT) in their work. DFT is a computational quantum mechanical modeling method used to investigate the electronic structure of particular atoms, molecules, and the condensed phases. Recently DFT advances have expanded on the knowledge of condensed fluids, polymer solutions, liquid crystal, and colloids all of which help with the fabrication of different types of materials.

History

Archimedes in 250 B.C. "discovered" density. He suspected that he was being cheated on by the metal craftsmen who had constructed his golden crown. Archimedes suspected that the craftsman was using a sizeable amount of silver with his gold crown, which is cheaper, and ultimately an insult to the king. Archimedes put the crown that the craftsmen had created as well as a crown of pure gold both into a bathtub full of water. He measured the amount of water that was in excess and confirmed his suspicions. He then proceeds to chant "Eureka! Eureka!" in the streets.

Density

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