Page 267 - Instant notes

Statistical thermodynamics 253

The partition function may be used for direct calculation of the
values of thermodynamic parameters, the most significant of
N
which is the entropy: S=k B lnq +U/T. The specific case of a
perfect monatomic gas yields the Sackur-Tetrode equation:
5/2
2 3/2
S=nRln[e (2πmk B T/h ) (k B T/p)].
Because the thermal component of the energy of a system may be
calculated from the partition function, the heat capacity may be

calculated from the differential of this value with respect to time. The maximum molar
heat capacity is equal to R/2 for each degree of freedom, that is, each independent
mode of motion. Thus, a gaseous diatomic molecule may have three translational
degrees of freedom (one for each orthogonal direction of motion), two rotational
degrees of freedom (from rotation about each of two equatorial axes). One vibrational
degree of freedom contributes R to the molar heat capacity—R/2 from each of the
potential and kinetic energy components.
Related The first law (B1) Entropy and change (B5)
topics
Entropy (B4) Free energy (B6)

Statistical thermodynamics

Classical thermodynamics neither requires, nor takes account of, the molecular nature of
matter, whereas chemists are interested in the molecular nature of matter and its
properties. Statistical thermodynamics has been a highly successful approach to bridging
the gap between the quantized, molecular properties of a system and its macroscopic
thermodynamic properties. It is a fundamental premise of statistical thermodynamics that
the microscopic properties of a system directly influence those properties which are
observable and measurable at the macroscopic level (heat capacity or entropy, for
example). Statistical thermodynamics operates effectively because the microscopic
properties of a system can be described by focusing only on the most probable molecular
state.
Furthermore, since nature places no weighting on any particular one of a set of states
of equal energy, the most probable states are those which can be generated in the greatest
number of ways. Once the statistical properties of the most probable state have been
ascertained, it is then possible to use this information to describe the macroscopic
thermodynamic properties of the system in terms of experimentally measurable
quantities.

The Boltzmann distribution law

The Boltzmann distribution is a statistical description of the manner in which the
molecules in a system are distributed over the available states of that system.