2.6: Heat capacity and the partial derivative

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Let’s review the ideas within this chapter, and restate some principles a new way as we go.

All of the internal energy change in a system comes from one of two sources: heat transfer into or out of the system, and work done by or on the system.

$Δ U = q + w$

If there’s no volume change in the system, w = 0, and the internal energy change equates to heat transfer. This is the reason why we can equate the internal energy change to constant-volume heat capacity times temperature difference:

$Δ U = C_{V} Δ T$

This is true not merely for discrete changes that we represent by deltas, but for infinitesimal changes as well, so we can write this expression differently:

$d U = C_{V} d T$

This implies a derivative - but we have to be careful about how we write this derivative. The internal energy U is capable of changing when volume changes as well as when the temperature changes. When we write most simple derivatives in calculus, it’s either directly stated or strongly implied that only one variable is changing as a function of the other. In thermodynamics, we’re rarely constrained to a single variable changing, and we have to ensure that other variables are held constant.

Therefore it’s necessary to express the derivative that’s equivalent to C_V in a new way:

$C_{V} = {(\frac{\partial U}{\partial T})}_{V}$

As we've referred to before, the quantity in parentheses is referred to as the partial derivative of internal energy with respect to temperature. The V outside of the parentheses means that, when taking this derivative of internal energy, volume is held constant and not allowed to change the internal energy; the derivative is based on the changes in temperature alone. In every other sense, taking this derivative is no different than any other derivative of a single variable we might take; but because internal energy is not merely based on possible changes in temperature, we have to constrain the derivative.

As you've seen in previous sections, though, we don't usually study the constant-volume heat capacity; we're more likely to study the constant-volume specific heat. So it's better to express this partial derivative in terms of molar properties:

$\bar{C_{V}} = {(\frac{\partial \bar{U}}{\partial T})}_{V}$

Internal energy is equivalent to heat transfer at constant volume; we built the concept of enthalpy to be equivalent to heat transfer at constant pressure. So we can write an equivalent expression to equate to the constant-pressure specific heat:

$\bar{C_{P}} = {(\frac{\partial \bar{H}}{\partial T})}_{P}$

(The difference between the quantity $H$ and the quantity $\bar{H}$ is the difference in an energy presented in Joules or kiloJoules and an energy presented in Joules per mole or kiloJoules per mole. This has been implied previously; I'm making sure I say it explicitly here.)

There are several implications of this equivalence of the specific heats to partial derivatives of internal energy and enthalpy, respectively. One of these implications is that the specific heats are state variables, just like internal energy and enthalpy are. It turns out that we can apply Hess' Law to specific heats in the exact same way as we apply it to enthalpy itself:

$Δ \bar{C_{P}} = \sum n {\bar{C_{P}}}_{, p r o d u c t s} - \sum n {\bar{C_{P}}}_{, r e a c t a n t s}$

The idea that the specific heat of substances change when substances themselves chemically change should not be totally surprising, but it may build confidence to know that there's a clear pattern to how that specific heat changes.

There's an additional power to knowing how specific heat changes in a chemical reaction, and that's found in the connection between the enthalpy and the constant-pressure specific heat. We can undo the partial derivative just as easily as we can do it:

$d \bar{H} = \bar{C_{P}} d T$

We can integrate both sides of this equation and arrive at a change in enthalpy as a function of temperature discretely:

$integral Subscript upper H 1 Superscript upper H 2 Baseline d upper H overbar equals integral Subscript upper T 1 Superscript upper T 2 Baseline upper C Subscript upper P Baseline overbar d upper T$

${\bar{H}}_{T_{2}} - {\bar{H}}_{T_{1}} = \int_{T_{1}}^{T_{2}} \bar{C_{P}} d T$

Now, we could just rewrite the left side as $Δ \bar{H}$ . However, that's not a typical enthalpy change chemically; we prefer to write enthalpy changes not for simply transitioning the temperature of a single known stuff, but for the chemical transformation of one group of substances to a new group of substances. This corresponds to $Δ \bar{C_{P}}$ that we just defined above. It's customary to rewrite this new equation in terms of the changes in reaction of these substances:

$Δ {\bar{H}}_{T_{2}} - Δ {\bar{H}}_{T_{1}} = \int_{T_{1}}^{T_{2}} Δ \bar{C_{P}} d T$

This new equation, relating enthalpy changes for a reaction at different temperatures to the integrated change in specific heat between those temperatures, is known as Kirchhoff's Law. It's typically simplified two ways. The first is assuming T₁ is the reference enthalpy change, at standard ambient temperature and pressure. We can therefore rewrite $normal upper Delta upper H overbar Subscript upper T 1$ as $Δ {\bar{H}}^{\circ}$ , assuming that the standard enthalpy change is always associated with a value of T₁ = 298 K. (T₂ becomes T_new in this arrangement.) The second is the usually-satisfactory assumption that constant-pressure specific heats don't change substantially with temperature - indeed, many of the equations we've derived previously assume that $\bar{C_{P}}$ values don't change with temperature. This brings us to the form of Kirchoff's Law we will normally use:

$Δ {\bar{H}}_{T_{n e w}} = Δ {\bar{H}}^{\circ} + Δ \bar{C_{P}} (T_{n e w} - 298 K)$

This brings us to a new, practical relationship. If we know the standard enthalpy change for a reaction and we know how the specific heats of the substances in that reaction change, we can know the enthalpy change for that reaction at any temperature T_new.