2.3 Manipulating partial derivatives

We sometimes need to find derivatives we don’t obtain easily from one of these potentials. For this, we use some calculus rules.

Inversion rule¶

The inversion rule states that the partial derivative of each variable with respect to each other are reciprocals if the two variables are related through an inverse function.

To demonstrate use of this rule, consider two ways of calculating the partial derivative $(\partial x/\partial y)_z$ for the function

and the inverse function

Evaluating the partial derivative of x with respect to y directly gives

while the partial derivative of y with respect to x is:

Now, use the inversion rule and substitute $x(y,z)$ back in:

The results are the same!

Chain rule¶

When working with multivariable functions, we often want to compute the derivative of one variable with respect to another, even when the relationship between them is indirect. In such cases, the chain rule allows us to break down the derivative into intermediate steps.

To demonstrate use of this rule, let $w = yz$, which means that

We will calculate the partial derivative of x with respect to y using the chain rule, then show it gives the same as if we differentiated directly. First, we need partial derivatives of x with respect to w and w with respect to y:

Therefore, by the chain rule:

This matches our previous result for the derivative!

Triple product rule¶

Sometimes, the variables that are held constant in a derivative are inconvenient. The “triple product rule” (sometimes called the cyclic or xyz-1 rule) lets us replace such a derivative by two other derivatives of involving that variable.

To demonstrate use of this rule, use $x = y^2/z$, which has the inverse functions $y = \sqrt{xz}$ and $z = y^2/x$. Compute the relevant partial derivatives:

Now plug into the triple product rule, substituting the definition of x:

This confirms the rule!

We can use these rules to manipulate thermodynamic derivatives. Choosing the right rules to use is like solving a puzzle, and you’ll get better at it with practice.

Example: Constant-volume heat capacity¶

Express $(\partial S/\partial T)_V$ for a pure substance using the constant-volume heat capacity

where S is the molar entropy, T is the temperature, V is the molar volume, and U is the molar internal energy, given that

Using the chain rule, we can introduce U, then invert the S derivative:

The last line follows from the definition of the heat capacity.

Derivatives as functions¶

The first law of thermodynamics for a pure substance is:

where U is the molar internal energy, T is the temperature, S is the molar entropy, P is the pressure, and V is the molar volume. This is an exact differential for $U(S,V)$. Mathematically, we also have the total differential:

By comparing the two, we see that

This shows that T and P are functions of S and V! Their derivatives can be computed and manipulated using rules of multivariable calculus in order to relate measurable quantities like T and P to the derivatives of an unmeasurable quantities like U! For example, the change in internal energy for an adiabatic process (constant S) is:

We can also relate quantities as mixed derivatives. For example, the entropy derivative of the pressure cannot be measured easily, but it is related to the temperature change during adiabatic compression:

Swapping variables and derivatives¶

We say U has S and V as “natural” variables because they are what appears in the differential first law. But, we do not like S as a variable because we cannot measure it. We would love to use T instead. Can we swap the two?

Yes, if we define the Helmholtz free energy:

where S is now a function of T and V and so is U as a result. The total differential for A confirms this:

As a result,

Other quantities can be defined to use different sets of natural variables.

• Enthalpy H

  $$\begin{align} H &= U + P V \\ \d{H} &= T \d{S} + V \d{P} \end{align}$$

  (26)

• Gibbs free energy G

  $$\begin{align} G &= A + P V = U - T S + P V \\ \d{G} &= - S \d{T} + V \d{P} \end{align}$$

  (27)

The reasons for making these definitions are based on a concept called a Legendre transformation and this has important implications in thermodynamics (e.g., why $\Delta G < 0$ for a spontaneous process at constant T and P).

Example: Change in internal energy¶

We want to compute the change in molar internal energy $\Delta U$ of a substance as we vary the temperature T and pressure P in terms of quantities we can measure. In addition to T and P, these quantities are the molar volume V, the thermal expansion coefficient $\alpha_V$, the isothermal compressibility $\kappa_T$, and the constant-pressure heat capacity $c_P$:

where $H$ is the molar enthalpy. The following total differentials are known from thermodynamics:

where S is the molar entropy and $G$ is the molar Gibbs free energy.

First, we express the total differential for U as a function of T and P:

Next, we form the derivatives using the given total differential for U:

Then, we go about replacing what we don’t like because we can’t measure it with things that we can. For $(\partial S/ \partial T)_P$, use the chain rule followed by the inversion rule:

where the last step used the total differential for $H$ to replace the derivative in the denominator. Additionally using the definition of $\alpha_V$ gives

For $(\partial S/\partial P)_T$, use the total differential for $G$ and equate its mixed second derivatives:

Additionally using the definition of $\kappa_T$ gives

Putting it all together:

This total differential is now suitable for integration with respect to T and P using only measurable quantities!

Skill builder problems¶

Given

Evaluate the following for $f=\cos(4x+y^2)+x^2y$