Physical chemists reconcile the empirical theory of classical
thermodynamics
with the quantum nature of matter and energy when they recover thermodynamics
from a statistical mechanical treatment of the individual particles’
quantized eigenspectrum. The conclusion is that, when systems are
very large collections of particles, interactions between adjacent
systems are comparatively negligible, resulting in an additive thermodynamic
framework where the energy of a composite systemABmay be expressed as the sum of the individual energies of subsystemsAandB. This powerful theory is consistent
with quantum theory, and it accurately describes the macroscopic properties
of sufficiently large systems subject to comparatively short-ranged
interactions. Nevertheless, classical thermodynamics has its limitations.
Its main drawback is the theory’s failure to accurately describe
systems not sufficiently large for the aforementioned interaction
to be neglected. This shortcoming was addressed by the celebrated
chemist Terrell L. Hill in the 1960s when he generalized classical
thermodynamics by adding a phenomenological energy term to describe
systems not captured by the additivity ansatz (i.e.,AB≠A+B) of classical thermodynamics.
Despite its elegance and success, Hill’s generalization mostly
remained a specialist tool rather than becoming part of the standard
chemical thermodynamics corpus. A probable reason is that, in contrast
to the classical large-system case, Hill’s small-system framework
does not reconcile with a thermostatistical treatment of quantum mechanical
eigenenergies. In this work we show that, by introducing a temperature-dependent
perturbation in the particles’ energy spectrum, Hill’s
generalized framework is in fact recovered with a simple thermostatistical
analysis accessible to physical chemists.