4.3.12. First Derivative and Maxwell Relations#
4.3.12.1. Learning goals:#
After this section, student’s should be able to:
Identify the two first partial derivative relationships of each Thermodynamic energy function
Define a Maxwell relation
Identify the Maxwell relation in each Thermodynamic energy function (of two variables)
4.3.12.2. First Partial Derivatives of Thermodynamic Energy Functions#
We have already seen the differential forms of each of the four Thermodynamic energy function and how these can be transformed from one to the other using Legendre transformations. As a reminder, here are the four differential forms:
Recall that for a differntiable function of two variables, e.g. \(f(x,y)\), the exact or total differential is given as
Let’s consider each Thermodynamic energy function as a function of its natural variables and compare the two forms of its differential.
4.3.12.2.1. Internal Energy#
For internal energy, \(U(S,V)\), we have the following two forms for its differential:
where the first one comes from the standard form of the internal energy and the second is just the mathematical relationship between a differential and the partials of the independent variables. Notice what these equalities imply:
4.3.12.2.2. Enthalpy#
For enthalpy, \(H(S,P)\), we have the following two forms for its differential:
where the first one comes from the standard form of the enthalpy and the second is just the mathematical relationship between a differential and the partials of the independent variables. Notice what these equalities imply:
4.3.12.2.3. Helmholtz Free Energy#
For Helmholtz free energy, \(A(T,V)\), we have the following two forms for its differential:
where the first one comes from the standard form of the Helmholtz free energy and the second is just the mathematical relationship between a differential and the partials of the independent variables. Notice what these equalities imply:
4.3.12.2.4. Gibbs Free Energy#
For Gibbs free energy, \(G(T,P)\), we have the following two forms for its differential:
where the first one comes from the standard form of the Helmholtz free energy and the second is just the mathematical relationship between a differential and the partials of the independent variables. Notice what these equalities imply:
4.3.12.2.5. Summary of first derivative relationships#
Energy Function |
Differential Form \( \) \( \) \( \) \( \) |
First Derivative Relationships |
|---|---|---|
U - Internal Energy |
\( dU = TdS - PdV \) |
\(T = \left( \frac{\partial U}{\partial S}\right)_V\) and \(-P =\left( \frac{\partial U}{\partial V}\right)_S\) |
H - Enthalpy |
\(dH = TdS + VdP\) |
\(T = \left( \frac{\partial H}{\partial S}\right)_P\) and \(V =\left( \frac{\partial H}{\partial P}\right)_S\) |
A - Helmholtz Free Energy |
\(dA = -SdT - PdV\) |
\(-S = \left( \frac{\partial A}{\partial T}\right)_V\) and \(-P =\left( \frac{\partial A}{\partial V}\right)_T\) |
G - Gibbs Free Energy |
\(dG = -SdT + VdP\) |
\(-S = \left( \frac{\partial G}{\partial T}\right)_P\) and \(V =\left( \frac{\partial G}{\partial P}\right)_T\) |
4.3.12.3. Maxwell Relations#
The Maxwell relations are the equivalencies of the second cross partials of the Thermodynamic energy functions
The Maxwell relations rely on an additional mathematical property of exact differentials. Namely, that for an exact differential of (at least) two independent variables, the second cross paritial derivatives are equivalent.
What is a second cross partial derivative? Let’s again consider a function, \(f(x,y)\), of independent variables \(x\) and \(y\). The differential, \(df\), is expressed in terms of first partial derivatives of each independent variable:
where \(\left( \frac{\partial f}{\partial x}\right)\) is the first partial derivative of \(f\) with respect to (wrt) \(x\) and \(\left( \frac{\partial f}{\partial y}\right)\) is the first partial derivative of \(f\) wrt \(y\). A second cross paritial derivative is the derivative of a first partial derivative wrt the other independent variable. In this case, there are two cross partials:
This is as opposed to the second partial derivatives wrt to the same variables
The second cross paritial derivatives of an exact differential are equivalent, meaning:
4.3.12.3.1. Example: Second Cross Partials are Equivalent#
Consider the function \(f(x,y) = 2xy + x^2 + \frac{y^3}{x}\). Show that the second cross partials are equivalent.
Strategy: Determine each cross partial by first determining the first partial derivatives and subsequently differentiating those wrt the other variable.
Now compute the second cross partials
We see that these two are equivalent.
4.3.12.3.2. Thermodynamic Energy Functions#
So what does this mean for the Thermodynamic energy functions? We will consider the internal energy and extrapolate to the rest
The real power of this comes from plugging in for the first derivatives of the energy functions (namely \(T = \left( \frac{\partial U}{\partial S}\right)_V\) and \(-P =\left( \frac{\partial U}{\partial V}\right)_S\) in this case)
Similar relationship can be derived from the differential forms of \(H\), \(A\), \(G\). These are summmarized in following table.
4.3.12.3.3. Summary of Maxwell Relations#
Energy Function |
Differential Form $\( \)$ |
Maxwell Relations $\( \)$ |
|---|---|---|
U - Internal Energy |
\( dU = TdS - PdV \) |
\(\left(\frac{\partial T}{\partial V}\right)_S = -\left(\frac{\partial P}{\partial S}\right)_V\) |
H - Enthalpy |
\(dH = TdS + VdP\) |
\(\left(\frac{\partial T}{\partial P}\right)_S = \left(\frac{\partial V}{\partial S}\right)_P\) |
A - Helmholtz Free Energy |
\(dA = -SdT - PdV\) |
\(\left(\frac{\partial S}{\partial V}\right)_T = \left(\frac{\partial P}{\partial T}\right)_V\) |
G - Gibbs Free Energy |
\(dG = -SdT + VdP\) |
\(\left(\frac{\partial S}{\partial P}\right)_T = -\left(\frac{\partial V}{\partial T}\right)_P\) |