Consider a system of N particles in a cubic box of volume .
The particles are assumed not to interact with each other. Thus, the Hamiltonian in Cartesian coordinates may be taken to be
where we are assuming that all particles are of the same type.
The microcanonical partition function is
Since the Hamiltonian is independent of the coordinates, the 3N coordinate integrations can be done straightforwardly. The range of each one is 0 to L. Thus, these integrations give an overall factor :
To do the momentum integrals, we first change variables to
Substitution into the partition function gives
The 3N dimensional integral can now be seen to an integration over the surface of a sphere defined by the equation
Therefore, it proves useful to transform to 3N dimensional spherical coordinates, , where
where is the 3N-1 solid angle integral over the 3N-1 angles. The partition function now becomes:
At this point, we use an identity of -functions:
We will also make use of the fact that N is large, so that we may take . Using the general formula for an n dimensional solid angle integral:
where is the Gamma function:
Thus, the solid angle integral is
and the partition function finally becomes
The entropy S(N,V,E) is given by
Note that the term since , which is negligibly small compared to the term proportional to N and . Thus, we can neglect it. Now, we can simplify using Stirling's approximation
which is valid for N very large. Also, note that
Substituting these approximations into the expression for the entropy, we obtain
We could also simplify the using Stirling's approximation, however, let us keep it as it is for now, since, as we remember from our past treatment of the microcanonical ensemble, this factor was included in the partition function in an ad hoc manner, in order to account for the indistinguishability of the particles. We will want to explore the effect of removing this term. Without it, the entropy is the purely classical entropy
Other thermodynamic quantities can be easily obtained. For example, the temperature is
which is the result we obtained from our analysis of the classical virial theorem. The pressure is given by
which is the famous ideal gas law. This is actually the equation of state of the ideal gas, as it expresses the pressure as a function of the volume and temperature. It can also be written as
where is the constant density of the gas.
One often expresses the equation of state graphically. For the ideal gas, if we plot P vs. V, for different values of T, we obtain the following plot:
The different curves shown are called the isotherms, since they represent P vs. V for fixed temperature.
The heat capacity of the ideal gas follows from the expression for the energy
From our previous analysis of the virial theorem, we can conclude that each kinetic mode contributes k/2 to the heat capacity.