Maxwell Relations: Difference between revisions
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===Internal Energy=== | ===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> where <math>{\Delta Q}</math> is the heat ''added to the system'' and <math>{\Delta W}</math> is the work done ''by the system''. We can re-express differentials Q and W 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> where <math>{\Delta Q}</math> is the heat ''added to the system'' and <math>{\Delta W}</math> is the work done ''by the system''. We can re-express differentials Q and W 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>, which suggests U is a function of '''S''' and '''V'''. | ||
The Maxwell Relation for Internal Energy can be found by applying [https://en.wikipedia.org/wiki/Symmetry_of_second_derivatives Clairaut's Theorem] to the system with the fundamental assumption that the variables can be expressed as [https://en.wikipedia.org/wiki/Exact_differential Exact Differentials]. | The Maxwell Relation for Internal Energy can be found by applying [https://en.wikipedia.org/wiki/Symmetry_of_second_derivatives Clairaut's Theorem] to the system with the fundamental assumption that the variables can be expressed as [https://en.wikipedia.org/wiki/Exact_differential Exact Differentials]. The exact differential form of <math>{dU = T dS − P dV}</math> is <math>{dU = \left(\frac{\partial{U}}{\partial{S}}\right)_V dS + \left(\frac{\partial{U}}{\partial{V}}\right)_S dV}</math> which enables the redefining <math>T=\left(\frac{\partial{U}}{\partial{S}}\right)_V</math> and <math>P=-\left(\frac{\partial{U}}{\partial{V}}\right)_S</math>. Note that the "opposite" variable is kept constant i.e for <math>{dS}</math>, '''V''' is kept constant and for <math>{dV}</math>, '''S''' is kept constant. This is very similar to performing integration in multivariable calculus where either x or y is kept constant while the other function is varied to obtain the area under a "slice" for a 3D surface. We can then apply Clairaut's Theorem to obtain the relation <math>\left[ \frac{\partial}{\partial V} \left( \frac{\partial U}{\partial S} \right)_V \right]_S = \left[ \frac{\partial}{\partial S} \left( \frac{\partial U}{\partial V} \right)_S \right]_V</math> which can be simplified to <math>\left[ \frac{\partial}{\partial V} \left( T \right)_V \right]_S = \left[ \frac{\partial}{\partial S} \left( -P \right)_S \right]_V</math>. The ''Maxwell Relation'' for internal energy follows from this result as <math>{dU = T dS − P dV}</math> is <math>{\left(\frac{\partial{T}}{\partial{V}}\right)_S = -\left(\frac{\partial{P}}{\partial{S}}\right)_V}</math> which includes ''only'' the state variables P,V,T,S as we wanted | ||
==Utility of Maxwell Relations== | ==Utility of Maxwell Relations== | ||
Revision as of 17:03, 25 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 Thermodynamics and Statistical Mechanics
Derivations
Internal Energy
The first law of thermodynamics states that [math]\displaystyle{ {\Delta U = \Delta Q - \Delta W} }[/math] where [math]\displaystyle{ {\Delta Q} }[/math] is the heat added to the system and [math]\displaystyle{ {\Delta W} }[/math] is the work done by the system. We can re-express differentials Q and W 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], which suggests U is a function of S and V.
The Maxwell Relation for Internal Energy can be found by applying Clairaut's Theorem to the system with the fundamental assumption that the variables can be expressed as Exact Differentials. The exact differential form of [math]\displaystyle{ {dU = T dS − P dV} }[/math] is [math]\displaystyle{ {dU = \left(\frac{\partial{U}}{\partial{S}}\right)_V dS + \left(\frac{\partial{U}}{\partial{V}}\right)_S dV} }[/math] which enables the redefining [math]\displaystyle{ T=\left(\frac{\partial{U}}{\partial{S}}\right)_V }[/math] and [math]\displaystyle{ P=-\left(\frac{\partial{U}}{\partial{V}}\right)_S }[/math]. Note that the "opposite" variable is kept constant i.e for [math]\displaystyle{ {dS} }[/math], V is kept constant and for [math]\displaystyle{ {dV} }[/math], S is kept constant. This is very similar to performing integration in multivariable calculus where either x or y is kept constant while the other function is varied to obtain the area under a "slice" for a 3D surface. We can then apply Clairaut's Theorem to obtain the relation [math]\displaystyle{ \left[ \frac{\partial}{\partial V} \left( \frac{\partial U}{\partial S} \right)_V \right]_S = \left[ \frac{\partial}{\partial S} \left( \frac{\partial U}{\partial V} \right)_S \right]_V }[/math] which can be simplified to [math]\displaystyle{ \left[ \frac{\partial}{\partial V} \left( T \right)_V \right]_S = \left[ \frac{\partial}{\partial S} \left( -P \right)_S \right]_V }[/math]. The Maxwell Relation for internal energy follows from this result as [math]\displaystyle{ {dU = T dS − P dV} }[/math] is [math]\displaystyle{ {\left(\frac{\partial{T}}{\partial{V}}\right)_S = -\left(\frac{\partial{P}}{\partial{S}}\right)_V} }[/math] which includes only the state variables P,V,T,S as we wanted
Utility of Maxwell Relations
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All Maxwell Relations
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