11.12 The New Variables

The new kid on the block is the entropy

But a lot of systems of interest are not well described as being
adiabatic. A typical alternative case might be a system in a rigid
box in an environment that is big enough, and conducts heat well
enough, that it can at all times be taken to be at the same
temperature

A slightly different version occurs even more often in real
applications. In these the system is not in a rigid box, but instead
its surface is at all times exposed to ambient atmospheric pressure.
Energy conservation now says that the heat added

There are a number of differential expressions that are very useful
in doing thermodynamics. The primary one is obtained by combining the
differential first law (11.11) with the differential second
law (11.19) for reversible processes:

The differentials of the Helmholtz and Gibbs free energies are, after cleaning
up with the two expressions immediately above:

Expression (11.25) shows that the work obtainable in an isothermal reversible process is given by the decrease in Helmholtz free energy. That is why Helmholtz called it “free energy” in the first place. The Gibbs free energy is applicable to steady flow devices such as compressors and turbines; the first law for these devices must be corrected for the “flow work” done by the pressure forces on the substance entering and leaving the device. The effect is to turn

Maxwell noted that, according to the total differential of calculus,
the coefficients of the differentials in the right hand sides of
(11.23) through (11.26) must be the partial
derivatives of the quantity in the left hand side:

The final equation in each line can be verified by substituting in the previous two and noting that the order of differentiation does not make a difference. Those are called the “Maxwell relations.” They have a lot of practical uses. For example, either of the final equations in the last two lines allows the entropy to be found if the relationship between the

normalvariables

There are two even more remarkable relations along these lines. They
follow from dividing (11.23) and (11.24) by

What is so remarkable is the final equation in each case: they do not involve entropy in any way, just the

normalvariables

As an example, consider an ideal gas, more precisely, any substance
that satisfies the ideal gas law

(The final relation is because

There is a sleeper among the Maxwell equations; the very first one, in
(11.27). Turned on its head, it says that

Of course, this new definition of temperature is completely consistent
with the ideal gas one; it was derived from it. However, the new
definition also works fine for negative temperatures. Assume a system

And now the big question: what is that “chemical potential” you hear so much about? Nothing new,
really. For a pure substance with a single constituent like this
chapter is supposed to discuss, the chemical potential is just the
specific Gibbs free energy on a molar basis,

then the chemical potential

The partial derivatives described by the chemical potentials are
important for figuring out the stable equilibrium state a system
will achieve in an isothermal, isobaric, environment, i.e. in an
environment that is at constant temperature and pressure. As noted
earlier in this section, the Gibbs free energy must be as small as it
can be in equilibrium at a given temperature and pressure. Now
according to calculus, the full differential for a change in Gibbs
free energy is

The first two partial derivatives, which keep the number of particles fixed, were identified in the discussion of the Maxwell equations as

This generalizes (11.26) to the case that the numbers of constituents change. At equilibrium at given temperature and pressure, the Gibbs energy must be minimal. It means that

Note that there are typically constraints on the changes

The equality of chemical potentials suffices to derive the famous
Clausius-Clapeyron equation relating pressure changes under two-phase,
or “saturated,” conditions to the corresponding temperature
changes. For, the changes in chemical potentials must be equal too,

and rearranging gives the Clausius-Clapeyron equation:

Note that since the right-hand side is a ratio, it does not make a difference whether you take the entropies and volumes on a molar basis or on a mass basis. The mass basis is shown since that is how you will typically find the entropy and volume tabulated. Typical engineering thermodynamic textbooks will also tabulate

In case your tables do not have the entropies of the liquid and vapor
phases, they often still have the “latent heat of vaporization,” also known as “enthalpy of vaporization” or similar, and in engineering
thermodynamics books typically indicated by

For chemical reactions, like maybe

the changes in the amounts of the constituents are related as

where

However, chemical reactions are often posed in a context of constant
volume rather than constant pressure, for one because it simplifies
the reaction kinematics. For constant volume, the Helmholtz free
energy must be used instead of the Gibbs one. Does that mean that a
second set of chemical potentials is needed to deal with those
problems? Fortunately, the answer is no, the same chemical potentials
will do for Helmholtz problems. To see why, note that by definition

Does this mean that the chemical potentials are also specific
Helmholtz free energies, just like they are specific Gibbs free
energies? Of course the answer is no, and the reason is that the
partial derivatives of