11.12 The New Variables

The new kid on the block is the entropy . For an adiabatic system the entropy is always increasing. That is highly useful information, if you want to know what thermodynamically stable final state an adiabatic system will settle down into. No need to try to figure out the complicated time evolution leading to the final state. Just find the state that has the highest possible entropy , that will be the stable final state.

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 . Also assume that initially the
system itself is in some state 1 at the ambient temperature
, and that it ends up in a state 2 again at that
temperature. In the evolution from 1 to 2, however, the system
temperature could be be different from the surroundings, or even
undefined, no thermal equilibrium is assumed. The first law, energy
conservation, says that the heat added to the system from the
surroundings equals the change in internal energy of the
system. Also, the entropy change in the isothermal environment will
be , so the system entropy change
must be at least in order for the
net entropy in the universe not to decrease. From that it can be seen
by simply writing it out that the “Helmholtz free energy”

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 equals the
change in internal energy plus the work done
expanding against the atmospheric pressure, which is
. Assuming that both the initial state
1 and final state 2 are at ambient atmospheric pressure, as well as at
ambient temperature as before, then it is seen that the quantity that
decreases is the “Gibbs free energy”

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 into as the differential for the actual work obtainable from the device. (This assumes that the kinetic and/or potential energy that the substance picks up while going through the device is a not a factor.)

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 , , and is known, assuming that at least one data point at every temperature is already available. Even more important from an applied point of view, the Maxwell relations allow whatever data you find about a substance in literature to be stretched thin. Approximate the derivatives above with difference quotients, and you can compute a host of information not initially in your table or graph.

There are two even more remarkable relations along these lines. They
follow from dividing (11.23) and (11.24) by
and rearranging so that becomes the quantity differentiated. That
produces

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

normalvariables , , , , and . Merely because entropy exists, there must be relationships between these variables which seemingly have absolutely nothing to do with the second law.

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

(The final relation is because with and .) Ideal gas tables can therefore be tabulated by temperature only, there is no need to include a second independent variable. You might think that entropy should be tabulated against both varying temperature and varying pressure, because it does depend on both pressure and temperature. However, the Maxwell equation (11.30) may be used to find the entropy at any pressure as long as it is listed for just one pressure, say for one bar.

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 has a negative temperature according to he definition above. Then its messiness (entropy) increases if it gives up heat. That is in stark contrast to normal substances at positive temperatures that increase in messiness if they take in heat. So assume that system is brought into thermal contact with a normal system at a positive temperature. Then will give off heat to , and both systems increase their messiness, so everyone is happy. It follows that will give off heat however hot is the normal system it is brought into contact with. While the temperature of may be negative, it is hotter than any substance with a normal positive temperature!

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,
. More generally, if there is more than one
constituent the chemical potential of each constituent
is best defined as

then the chemical potential of each constituent is simply the molar specific Gibbs free energy of that constituent,

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 and ; also the partial derivatives with respect to the numbers of particles of the constituent have been defined as the chemical potentials . Therefore more shortly,

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 must be zero whenever 0, regardless of any infinitesimal changes in the amounts of the constituents. That gives a condition on the fractions of the constituents present.

Note that there are typically constraints on the changes in the amounts of the constituents. For example, in a liquid-vapor “phase equilibrium,” any additional amount of particles that condenses to liquid must equal the amount of particles that disappears from the vapor phase. (The subscripts follow the unfortunate convention liquid=fluid=f and vapor=gas=g. Don’t ask.) Putting this relation in (11.37) it can be seen that the liquid and vapor phase must have the same chemical potential, . Otherwise the Gibbs free energy would get smaller when more particles enter whatever is the phase of lowest chemical potential and the system would collapse completely into that phase alone.

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 substituting in
the differential (11.26) for the Gibbs free energy, taking
it on a molar basis since ,

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 and , making the formula above very convenient.

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 . That
is the difference between the enthalpy of the saturated liquid and
vapor phases, . If
saturated liquid is turned into saturated vapor by adding heat under
conditions of constant pressure and temperature, (11.24)
shows that the change in enthalpy equals
. So the Clausius-Clapeyron equation
can be rewritten as

For chemical reactions, like maybe

the changes in the amounts of the constituents are related as

where is the additional number of times the forward reaction takes place from the starting state. The constants 2, 1, and 2 are called the “stoichiometric coefficients.” They can be used when applying the condition that at equilibrium, the change in Gibbs energy due to an infinitesimal amount of further reactions must be zero.

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
, so
, and substituting for from
(11.37), that gives

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 represented by the chemical potentials keep extensive volume , instead of intensive molar specific volume constant. A single-constituent molar specific Helmholtz energy can be considered to be a function of temperature and molar specific volume, two intensive variables, and then , but does not simply produce , even if produces .