The Ideal Gas: From Kinetic Theory and Boltzmann's Law
The MB distribution is a specific case of a more general law that states that the distribution of states with different energies is given by
which is also called the Boltzmann distribution. Since above for the ideal gas, this reduces to the MB distribution in that case. This distribution can be derived from the idea that the ratio of probabilities between two states of energy E1 and E2, P(E1)/P(E2) = function of (E1 - E2) only = f(E1 - E2). (See Chapter 17.1 in McQuarrie.) This last assumption simply states that only relative energies matter--the absolute zero of energy can have no physical meaning.
From this, we can also have a simple interpretation of the fundamental law of statistical mechanics, namely Boltzmann's formula
where is the number of states corresponding to a given energy. The connection can be made by using the thermodynamic expression for the free energy A = E - TS, which generalizes energy to include thermal effects. Now this shows that TS plays the role of an energy, or more precisely, a free energy. Using this in P(E), we find
P(state) = exp(-(E-TS)/kBT) = exp(-E/kBT) exp(ln) = exp(-E/kBT)
In other words, when we have states corresponding to a given energy, the probability is multiplied by that number.
Entropy of an Ideal Gas
Classically, calculation of the entropy of the ideal gas is much simpler than using a full quantum description. When a gas particle which was originally in volume V0 is reduced to a volume V, the number of possible states (positions) it can be in is reduced by a factor of (V/V0). If we have N particles, this factor must occur N times, we find the change in entropy in going from V0 to V is
Since the pressure is given by P = - (dA/dV) = (T dS/dV) = T kB N (1/V), we recover the ideal gas law! :
P V = N kb T
Notice that the idea of pressure, which mechanically is a force per area, arises without any mention of force, momentum, kinetic energy, or velocity! It arises solely in consideration of possible positions of the particles. Do you find it puzzling that a calculation of the entropy can give an expression for the pressure which is the same as that arising from a detailed analysis of the force imparted on container walls due to particle collisions?
Question: In the above, we assumed that the particle volumes are negligible. At higher densities, however, the molecular or particle volume, b, can be a noticeable fraction of the total volume. One approximate way to take this into account is to say that the total available volume at any time is reduced by N b. Use this to arrive at a modified expression for the change in entropy, .
Furthermore, at higher densities, the energetic interactions between particles also become significant. We can approximate the total interaction energy by assuming it is proportional to the number of particle-particle contacts. This in turn is proportional to the density squared, and is proportional to the total volume:
E = a V (N/V)2 = a N2 / V
where a is the van der Waals interaction parameter. Now using the relations P = - (dA/dV)N,T , A = E - TS, and the expressions above for E and , calculate an expression for the pressure. Does this look familiar? (As above, we rely on d()/dV = dS/dV. Why is this true?)