A microscopic, mechanical derivation of the adiabatic gas relation

The simplest form of the adiabatic gas relation is the observation that the temperature of a thermally insulated gas increases when it is compressed and decreases when it is expanded. According to the historical review of this subject by Kuhn, the first publication documenting this behavior was by the Scottish physician William Cullen in the mid17th century. Experimental observations were summarized by the relation PV5constant, where the exponent g was determined to exceed unity. The deviation of g from unity is what allowed Sadi Carnot to develop his eponymous cycle. ~Apparently Carnot did not have the correct value of g, but the use of an incorrect value did not affect his fundamental result that heat engine efficiency depends only on inlet and outlet temperatures.! Serious attempts to develop a theoretical explanation for the adiabatic gas relation were undertaken by Laplace, Poisson, and others in the early 19th century, but no single individual has been identified as being the first to provide the correct theoretical explanation. Since the mid-19th century development of thermodynamics, the adiabatic gas relation has been established from first principles using thermodynamic arguments. The standard thermodynamic derivation is based on considering the temperature change of the gas in a cylinder at constant pressure or constant volume while taking into account specific heats at constant pressure and constant volume. The purpose of this paper is to show that the adiabatic gas relation PV5constant is a direct consequence of an important property of periodic mechanical motion, namely adiabatic invariance. Although the word adiabatic is used in both the mechanical and thermodynamic contexts, its meaning in the mechanical context differs from its meaning in the thermodynamic context because the concept of heat does not exist in the mechanical context. The derivation presented here provides insight into the fundamental microscopic dynamics underlying adiabaticity. The derivation will first be presented for molecules with no internal degrees of freedom and then extended to molecules with internal degrees of freedom. Two standard properties of an ideal gas will be invoked repeatedly, namely, the properties of an ideal gas occupying a volume V do not depend on the shape of the volume, and collisions cause all properties of an ideal gas to become isotropic.