Maxwell Relations: Difference between revisions
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The Maxwell Relations are a set of partial derivative relations derived using [https://en.wikipedia.org/wiki/Symmetry_of_second_derivatives Clairaut's Theorem] that enable the expression of physical quantities such as [https://en.wikipedia.org/wiki/Gibbs_free_energy Gibbs Free Energy] and [https://en.wikipedia.org/wiki/Enthalpy Enthalpy] as infinitesimal changes in pressure ('''P'''), volume ('''V'''), temperature ('''T'''), and entropy ('''S'''). They are named after [[James Maxwell | James Maxwell]] and build upon the work done by [https://en.wikipedia.org/wiki/Ludwig_Boltzmann Ludwig Boltzmann] in [[Temperature & Entropy | Statistical Mechanics]]. | The Maxwell Relations are a set of partial derivative relations derived using [https://en.wikipedia.org/wiki/Symmetry_of_second_derivatives Clairaut's Theorem] that enable the expression of physical quantities such as [https://en.wikipedia.org/wiki/Gibbs_free_energy Gibbs Free Energy] and [https://en.wikipedia.org/wiki/Enthalpy Enthalpy] as infinitesimal changes in pressure ('''P'''), volume ('''V'''), temperature ('''T'''), and entropy ('''S'''). They are named after [[James Maxwell | James Maxwell]] and build upon the work done by [https://en.wikipedia.org/wiki/Ludwig_Boltzmann Ludwig Boltzmann] in [[Temperature & Entropy | Statistical Mechanics]]. | ||
==Basic Thermodynamic Quantities== | |||
===Internal Energy=== | |||
The [https://en.wikipedia.org/wiki/First_law_of_thermodynamics first law of thermodynamics] states that <math>{\Delta U = \Delta Q - \Delta W}</math>. We can re-express Q (heat) and W (work) with the [https://en.wikipedia.org/wiki/State_variable state variables] '''P''','''V''','''T''','''S''' using the substitutions <math>{dQ_{rev} = TdS}</math> (see [https://en.wikipedia.org/wiki/Clausius_theorem Clausius Theorem]) and <math>{dW = -PdV}</math> (see [https://en.wikipedia.org/wiki/Work_(thermodynamics) Pressure-Volume Work]). We thus arrive at the thermodynamic definition for internal energy <math>{dU = T dS − P dV}</math> | The [https://en.wikipedia.org/wiki/First_law_of_thermodynamics first law of thermodynamics] states that <math>{\Delta U = \Delta Q - \Delta W}</math>. We can re-express Q (heat) and W (work) with the [https://en.wikipedia.org/wiki/State_variable state variables] '''P''','''V''','''T''','''S''' using the substitutions <math>{dQ_{rev} = TdS}</math> (see [https://en.wikipedia.org/wiki/Clausius_theorem Clausius Theorem]) and <math>{dW = -PdV}</math> (see [https://en.wikipedia.org/wiki/Work_(thermodynamics) Pressure-Volume Work]). We thus arrive at the thermodynamic definition for internal energy <math>{dU = T dS − P dV}</math> | ||
==Utility of Maxwell Relations== | |||
What are the mathematical equations that allow us to model this topic. For example <math>{\frac{d\vec{p}}{dt}}_{system} = \vec{F}_{net}</math> where '''p''' is the momentum of the system and '''F''' is the net force from the surroundings. | What are the mathematical equations that allow us to model this topic. For example <math>{\frac{d\vec{p}}{dt}}_{system} = \vec{F}_{net}</math> where '''p''' is the momentum of the system and '''F''' is the net force from the surroundings. | ||
Revision as of 11:33, 24 November 2024
Claimed by Ram Vempati (Fall 2024)
The Maxwell Relations are a set of partial derivative relations derived using Clairaut's Theorem that enable the expression of physical quantities such as Gibbs Free Energy and Enthalpy as infinitesimal changes in pressure (P), volume (V), temperature (T), and entropy (S). They are named after James Maxwell and build upon the work done by Ludwig Boltzmann in Statistical Mechanics.
Basic Thermodynamic Quantities
Internal Energy
The first law of thermodynamics states that [math]\displaystyle{ {\Delta U = \Delta Q - \Delta W} }[/math]. We can re-express Q (heat) and W (work) with the state variables P,V,T,S using the substitutions [math]\displaystyle{ {dQ_{rev} = TdS} }[/math] (see Clausius Theorem) and [math]\displaystyle{ {dW = -PdV} }[/math] (see Pressure-Volume Work). We thus arrive at the thermodynamic definition for internal energy [math]\displaystyle{ {dU = T dS − P dV} }[/math]
Utility of Maxwell Relations
What are the mathematical equations that allow us to model this topic. For example [math]\displaystyle{ {\frac{d\vec{p}}{dt}}_{system} = \vec{F}_{net} }[/math] where p is the momentum of the system and F is the net force from the surroundings.
All Maxwell Relations
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