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 or the total volume of their box are not intensive variables; they are “extensive variables,“ variables that increase in value proportional to the system size. Often, however, you are only interested in the properties of your substance, not the amount. In that case, intensive variables can be created by taking ratios of the extensive ones; in particular, is an intensive variable called the “particle density.” It is the number of particles per unit volume. If you restrict your attention to only one half of your box with particles, the particle density is still the same, with half the particles in half the volume.
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:
Two intensive variables must be known to fully determine the intensive properties of a simple substance in thermal equilibrium.(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.
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” , found by multiplying the particle density with the single-particle mass , , or its reciprocal, the “specific volume” . The density is the system mass per unit system volume, and the specific volume is the system volume per unit system mass.
Alternatively, to keep the values for the number of particles in
check, they may be expressed in “moles,” multiples of Avogadro’s number
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
Next there is the “pressure” , being the force with which the substance
pushes on the surfaces of the box it is in per unit surface area. To
identify quantum mechanically, first consider a system in a single
energy eigenfunction for certain. If the volume of the box is
slightly changed, there will be a corresponding slight change in the
energy eigenfunction , (the boundary conditions of the
Hamiltonian eigenvalue problem will change), and in particular its
energy will slightly change. Energy conservation requires that the
change in energy is offset by the work done by the
containing walls on the substance. Now the work done by the wall
pressure on the substance 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 . For example, if the walls of the box would consist of molecules that were hotter than the particles inside, the walls too would add energy to the system, and take it out of its single energy eigenstate to boot. And even macroscopically, for pressure times area to be the force requires that the system is in thermal equilibrium. It would not be true for a system evolving in a violent way.
Often a particular combination of the variables defined above is very
convenient; the“enthalpy” is defined as
Assuming that the system evolves while staying at least approximately
in thermal equilibrium, the “first law of thermodynamics” can be stated macroscopically as
Note the use of a straight for the changes in internal energy and volume , but a for the heat energy added. It reflects that and are changes in properties of the system, but is not; is a small amount of energy exchanged between systems, not a property of any system. Also note that while popularly you might talk about the heat within a system, it is standard in thermodynamics to refer to the thermal energy within a system as internal energy, and reserve the term “heat” for exchanged thermal energy.
Just two more variables. The “specific heat at constant volume”
is defined as the heat that must be added to the substance for each
degree temperature change, per unit mass and keeping the volume
constant. In terms of the first law on a unit mass basis,
The specific heat at constant pressure is defined similarly as
, except that pressure, instead of volume, is being held
constant. According to the first law above, the heat added is now
and that is the change in enthalpy .
There is the first practical application of the enthalpy already! It