11.7 The Basic Thermodynamic Variables

This section introduces the most important basic players in thermodynamics.

The primary thermodynamic property introduced so far is the temperature. Recall that temperature is a measure of the hotness of the substance, a measure of how eager it is to dump energy onto other systems. Temperature is called an “intensive variable;“ it is the same for two systems that differ only in size.

The total number of particles

Note that under equilibrium conditions, it suffices to know the temperature and particle density to fully fix the state that a given system is in. More generally, the rule is that:

(To be precise, in a two-phase equilibrium like a liquid-vapor mixture, pressure and temperature are related, and would not be sufficient to determine something like net specific volume. They do still suffice to determine the specific volumes of the liquid and vapor parts individually, in any case.) If the amount of substance is also desired, knowledge of at least one extensive variable is required, making three variables that must be known in total.Two intensive variables must be known to fully determine the intensive properties of a simple substance in thermal equilibrium.

Since the number of particles will have very large values, for
macroscopic work the particle density is often not very convenient,
and somewhat differently defined, but completely equivalent variables
are used. The most common are the (mass) “density”

Alternatively, to keep the values for the number of particles in
check, they may be expressed in “moles,” multiples of Avogadro’s number

That produces the “molar density”

So what else is there? Well, there is the energy of the system. In
view of the uncertainty in energy, the appropriate system energy is
defined as the expectation value,

As a demonstration of the importance of the partition function
mentioned in the previous section, if the partition function
(11.5) is differentiated with respect to temperature, you get

(The volume of the system should be held constant in order that the energy eigenfunctions do not change.) Dividing both sides by

Next there is the “pressure”

(The force is pressure times area and is normal to the area; the work is force times displacement in the direction of the force; combining the two, area times displacement normal to that area gives change in volume. The minus sign is because the displacement must be inwards for the pressure force on the substance to do positive work.) So for the system in a single eigenstate, the pressure equals

It may be verified by simple substitution that this, too may be obtained
from the partition function, now by differentiating with respect to
volume keeping temperature constant:

While the final quantum mechanical definition of the pressure
is quite sound, it should be pointed out that the original definition
in terms of force was very artificial. And not just because force is
a poor quantum variable. Even if a system in a single eigenfunction
could be created, the walls of the system would have to be idealized
to assume that the energy change equals the work

Often a particular combination of the variables defined above is very
convenient; the“enthalpy”

Assuming that the system evolves while staying at least approximately
in thermal equilibrium, the “first law of thermodynamics” can be stated macroscopically as
follows:

Note the use of a straight

Just two more variables. The “specific heat at constant volume”

it means that

Note that in thermodynamics the quantity being held constant while taking the partial derivative is shown as a subscript to parentheses enclosing the derivative. You did not see that in calculus, but that is because in mathematics, they tend to choose a couple of independent variables and stick with them. In thermodynamics, two independent variables are needed, (assuming the amount of substance is a given), but the choice of which two changes all the time. Therefore, listing what is held constant in the derivatives is crucial.

The specific heat at constant pressure