Maxwell Relations
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. There are more Maxwell Relations than listed here; For a full list explore the linked materials.
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 can be 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.
Example - Bringing It All Together
Now we will see how to express complicated thermodynamic quantities like The Heat Capacity at Constant Volume using Maxwell Relations and Substitutions. Using the previous result for the Heat Capacity at Constant Pressure, we can write the heat Capacity at constant volume as [math]\displaystyle{ {C_p =T\left(\frac{\partial{S}}{\partial{T}}\right)_V } }[/math]. However, this quantity is difficult to measure experimentally as quantifying changes in entropy of a macroscopic system is no easy task. Therefore, Maxwell Relations can help us re-express this quantity in terms of ones that are experimentally measurable. Using Clairaut's Theorem on the Entropy with its natural variables (P,T), we obtain the relation [math]\displaystyle{ {dS = \left(\frac{\partial{S}}{\partial{T}}\right)_P dT + \left(\frac{\partial{S}}{\partial{P}}\right)_T dP} }[/math]. Dividing by [math]\displaystyle{ dT }[/math] and multiplying by T (while keeping volume constant) allows us to substitute in the definition for heat capacity as follows: [math]\displaystyle{ {\left(\frac{\partial{S}}{\partial{T}}\right)_V = \left(\frac{\partial{S}}{\partial{T}}\right)_P + T\left(\frac{\partial{S}}{\partial{P}}\right)_T \left(\frac{\partial{P}}{\partial{T}}\right)_V} }[/math] which becomes [math]\displaystyle{ {C_V = C_P + T\left(\frac{\partial{S}}{\partial{P}}\right)_T \left(\frac{\partial{P}}{\partial{T}}\right)_V} }[/math]. The right side of the equation looks daunting but can also be re-expressed in terms of the experimental quantities defined previously, specifically [math]\displaystyle{ {T\left(\frac{\partial{S}}{\partial{P}}\right)_T \left(\frac{\partial{P}}{\partial{T}}\right)_V = -\alpha^2 \beta VT} }[/math] (feel free to confirm this for yourself), which gives a definition for [math]\displaystyle{ C_V }[/math] as [math]\displaystyle{ {C_V = C_P -\alpha^2 \beta VT} }[/math] which is significantly easier to measure experimentally since [math]\displaystyle{ \alpha }[/math] and [math]\displaystyle{ \beta }[/math] are well known or easily measured for a large number of materials and compounds. Hopefully, this example illustrates the utility and power of Maxwell Relations in enabling the measurement of all sorts of (differential) thermodynamic and physical quantities using measured constants and accessible state variables.