1132 27 Equilibrium Statistical Mechanics. III. Ensemblessteeply and the small factor is falling so rapidly that there is a small range of values
ofEwhere the product of the two factors is much larger than for other values ofE,
as shown in Figure 27.2c. For a system ofNmolecules it turns out that the width of
the canonical energy probability distributionpEis proportional to√
N, whereas the
energy itself is proportional toN. For large values ofNthe width of the distribution
inEbecomes very small compared to the value ofE. The energy probability distribu-
tion of the microcanonical ensemble is nonzero for only a single energy, as depicted
Figure 27.2d. The canonical distribution is so narrow that this distribution is virtually
indistinguishable from it. This is related to the reason that the results of the canonical
and microcanonical ensembles coincide.
The narrowness of the canonical energy probability distribution is also related to
the very interesting question: Why is a macroscopic variable such asUgiven as a
smooth mathematical function of other thermodynamic variables such asT,V, andN?
We are not prepared to give a complete answer to this question, but comment that the
thermodynamic functions are assumed to be equal to ensemble average values. Average
quantities in very large populations tend to behave regularly and predictably, even if
the properties of each member of the population do not. This regular behavior is related
to the fact that the ensemble average energy is a smooth single-valued mathematical
function ofT,V, andN.
Applications of the canonical ensemble to quantum mechanical systems other than
dilute gases are beyond the scope of this book, and we omit them. These applications
are discussed in some of the statistical mechanics books listed at the end of this volume.
We will make some comments on the application of the canonical ensemble to systems
obeying classical mechanics in the next section.PROBLEMS
Section 27.3: The Dilute Gas in the Canonical Ensemble
27.7 Show that the canonical ensemble formulas lead toU3
2nRTfor a monatomic dilute gas without electronic
excitation.
27.8 Show that in the canonical ensemble the energy of a
dilute gas would be the same if the molecules were
distinguishable from each other.
27.9 a.Find the ensemble average energy of 1.000 mol of
argon gas at 298.15 K and 1.000 bar.
b.Find the value of the canonical partition function for
this system.
c.Find the probability of a single microstate of the
system with energy equal to the ensemble average
energy. Comment on the magnitude of your answer.27.10a.The experimental value of the standard-state (1.000
bar) third-law molar entropy of O 2 gas at 298.15 K is
205.146 J K−^1 mol−^1. Using this value and the
statistical mechanical value for the energy, calculate
the experimental value of the Helmholtz energy of
1.000 mol of oxygen gas at 298.15 K and 1.000 bar.
Take the energy of supercooled oxygen gas at 0 K to
equal zero.b.Find the value of the canonical partition function for
this system from the result of part a.c.Find the value of the molecular partition function
from the result of part b.27.11The experimental value of H◦m(298.15 K)−H◦m(0 K) for
neon is 6.197 kJ mol−^1 and the experimental value of
S◦m(298.15 K) is 146.327 J K−^1 mol−^1.a.Find the value of G◦mand A◦mfor neon at 298.15 K.b.Find the value of the canonical partition function of
neon at 298.15 K and 1.000 bar from the value of A◦m.
27.12Calculate the value of the molecular partition function of
oxygen gas at 298.15 K if 1.000 mol of the gas is confined