Maxwell Relations: Difference between revisions

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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} = T\,dS} }[/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 is consistent with the fact that 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{ {\frac{\partial{U}^2}{\partial{S}\partial{V}} = \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], the right side of 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{ {\frac{\partial{U}^2}{\partial{S}\partial{V}} = \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 desired.

The other common Maxwell Relations

The other Maxwell relations can be derived from their Thermodynamic Potentials in a similar fashion to internal energy.

Enthalpy: The thermodynamic potential for Enthalpy is [math]\displaystyle{ H = U + PV }[/math] and in differential form is [math]\displaystyle{ dH = T, dS + V, dP }[/math]. Its Maxwell relation is [math]\displaystyle{ \frac{\partial^2 H}{\partial S \partial P} = \left( \frac{\partial T}{\partial P} \right)_S = \left( \frac{\partial V}{\partial S} \right)_P }[/math].

Gibbs Free Energy: The thermodynamic potential for Gibbs Free Energy is [math]\displaystyle{ G = H - TS }[/math] and in differential form is [math]\displaystyle{ dG = -S\, dT + V\, dP }[/math]. Its Maxwell relation is [math]\displaystyle{ \frac{\partial^2 G}{\partial T \partial P} = \left( \frac{\partial S}{\partial P} \right)_T = -\left( \frac{\partial V}{\partial T} \right)_P }[/math]

Helmholtz Free Energy: The thermodynamic potential for Helmholtz Free Energy is [math]\displaystyle{ F = U - TS }[/math] and in differential form is [math]\displaystyle{ dF = -S, dT - P, dV }[/math]. Its Maxwell relation is [math]\displaystyle{ \frac{\partial^2 F}{\partial T \partial V} = \left( \frac{\partial S}{\partial V} \right)_T = \left( \frac{\partial P}{\partial T} \right)_V }[/math].

While these relations are immediately useful to describe the behavior of the thermodynamic potential as a "surface" which ca nbe traversed by changing state variables, quantities like [math]\displaystyle{ {\left(\frac{\partial{T}}{\partial{V}}\right)_S} }[/math] are difficult to measure empirically because constant-entropy systems are extremely difficult to set up. Luckily, as we will see in the next part of the discussion, Maxwell relations can aid us in re-expressing difficult to measure quantities with ones that are well established.

Experimental Quantities

Heat Capacity: From Chemistry, we know that the molar heat capacity (at constant pressure) relates the addition of heat to a material to the resulting temperature via [math]\displaystyle{ {\Delta Q_{rev} = C_p \Delta T} }[/math] i.e [math]\displaystyle{ {C_p = \frac{\Delta Q_{rev}}{\Delta T}} }[/math]. In derivative form, [math]\displaystyle{ C_p =\left(\frac{\partial{Q_{rev}}}{\partial{T}}\right)_P }[/math]. Earlier, we established that any reversible change in heat is related to a change in entropy i.e [math]\displaystyle{ {\partial{Q_{rev}} = TdS} }[/math], so [math]\displaystyle{ {C_p =T\left(\frac{\partial{S}}{\partial{T}}\right)_P } }[/math] which is in terms of canonical state variables as with Maxwell relations. Thus, this allows Heat Capacity and the other following quantities to be used as a substitutes for difficult to measure quantities within Maxwell relations.

Thermal Expansion Coefficient: The amount a material expands when heated (at constant pressure) is expressed by the equation [math]\displaystyle{ \Delta V_T = \alpha V_0 \Delta T }[/math] where [math]\displaystyle{ {\alpha} }[/math] is the thermal expansion coefficient of a material. It can be expressed in derivative form (by rearranging the preceding equation): [math]\displaystyle{ {\alpha = \frac{1}{V} \left(\frac{\partial{V}}{\partial{T}}\right)_P} }[/math], which is already in terms of our state variables and thus no further work is needed.

Bulk Modulus: The amount of pressure required to obtain a fractional change in volume (at constant temperature). [math]\displaystyle{ {\frac{\Delta V_p}{V} = \frac{-P}{\Beta}} }[/math] where [math]\displaystyle{ {\beta} }[/math] is the Bulk Modulus of the material. Its derivative can be expressed (by rearranging the preceding equation) as [math]\displaystyle{ {\beta =V\left(\frac{\partial{P}}{\partial{V}}\right)_T} }[/math]. This experimental quantity is also already in terms of state variables, so no further work is needed to make it "compatible" with Maxwell relations.

Utility of Maxwell Relations

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All Maxwell Relations

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Examples

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Connectedness

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@@ Line 7: / Line 7: @@
 ===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} = T\,dS}</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 is consistent with the fact that 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 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>{\frac{\partial{U}^2}{\partial{S}\partial{V}} =  \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>, the right side of 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>{\frac{\partial{U}^2}{\partial{S}\partial{V}} = \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 desired.