Manipulating partial derivatives

2.3. Manipulating partial derivatives#

2.3.1. Derivatives as functions#

The first law of thermodynamics for a pure substance is:

(2.14)#\[\begin{equation} \d{U} = T\d{S} - P \d{V} \end{equation}\]

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:

(2.15)#\[\begin{equation} \d{U} = \td{}{U}{S}{V} \d{S} + \td{}{U}{V}{S} \d{V} \end{equation}\]

By comparing the two, we see that

(2.16)#\[\begin{equation} T = \td{}{U}{S}{V} \qquad P = -\td{}{U}{V}{S} \end{equation}\]

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:

(2.17)#\[\begin{equation} \Delta U = \int_{V_1}^{V_2} \td{}{U}{V}{S} \d{V} = \int_{V_1}^{V_2} -P \d{V} \end{equation}\]

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:

(2.18)#\[\begin{equation} -\td{}{P}{S}{V} = \frac{\partial ^2 U}{\partial S \partial V} = \frac{\partial^2 U}{\partial V \partial S} = \td{}{T}{V}{S} \end{equation}\]

2.3.2. 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:

(2.19)#\[\begin{equation} A = U - T S \end{equation}\]

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

(2.20)#\[\begin{align} \d{A} &= \d{U} - (T \d{S} + S \d{T}) \\ &= (T \d{S} -P \d{V}) - T \d{S} - S \d{T} \\ &= -S \d{T} -P \d{V} \end{align}\]

As a result,

(2.21)#\[\begin{align} S &= -\td{}{A}{T}{V} \\ P &= - \td{}{A}{V}{T} \\ \td{}{S}{V}{T} &= \td{}{P}{T}{V} \end{align}\]

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

  • Enthalpy H

    (2.22)#\[\begin{align} H &= U + P V \\ \d{H} &= T \d{S} + V \d{P} \end{align}\]

  • Gibbs free energy G

    (2.23)#\[\begin{align} G &= A + P V = U - T S + P V \\ \d{G} &= - S \d{T} + V \d{P} \end{align}\]

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 \(c_P\):

(2.24)#\[\begin{align} \alpha_V &= \frac{1}{V} \td{}{V}{T}{P} \\ \kappa_T &= -\frac{1}{V} \td{}{V}{P}{T} \\ c_P &= \td{}{H}{T}{P} \end{align}\]

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

(2.25)#\[\begin{align} \d{U} &= T \d{S} - P \d{V} \\ \d{H} &= T \d{S} + V \d{P} \\ \d{G} &= -S \d{T} + V \d{P} \end{align}\]

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:

(2.26)#\[\begin{equation} \d{U} = \td{}{U}{T}{P} \d{T} + \td{}{U}{P}{T} \d{P} \end{equation}\]

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

(2.27)#\[\begin{align} \td{}{U}{T}{P} &= T \td{}{S}{T}{P} - P \td{}{V}{T}{P} \\ \td{}{U}{P}{T} &= T \td{}{S}{P}{T} - P \td{}{V}{P}{T} \end{align}\]

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:

(2.28)#\[\begin{align} \td{}{S}{T}{P} &= \td{}{S}{H}{P} \td{}{H}{T}{P} \\ &= \frac{ (\partial H/\partial T)_P }{ (\partial H/\partial S)_P } \\ &= \frac{c_P}{T} \end{align}\]

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

(2.29)#\[\begin{align} \td{}{U}{T}{P} &= T \left( \frac{c_P}{T} \right) - P V \alpha_V \\ &= c_P - P V \alpha_V \end{align}\]

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

(2.30)#\[\begin{equation} -\td{}{S}{P}{T} = \td{}{V}{T}{P} = V\alpha_V \end{equation}\]

Additionally using the definition of \(\kappa_T\) gives

(2.31)#\[\begin{align} \td{}{U}{P}{T} &= T (-V \alpha_V) - P(-V\kappa_T) \\ &= P V \kappa_T - T V \alpha_V \end{align}\]

Putting it all together:

(2.32)#\[\begin{equation} \d{U} = (c_P - PV\alpha_V)\d{T} + V(P\kappa_T - T\alpha_V)\d{P} \end{equation}\]

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

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