Positive and Negative Temperatures in a Two-Level System: Thermodynamic and Statistical-Mechanical Perspectives

Transient negative temperature states have been reported for a range of systems having a finite number of energy levels. While such systems are rare and seem to contradict the common notion that temperature is always positive, they provide an effective platform for illustrating the relationship between the thermodynamic and statistical-mechanical formulations of temperature. In this article we present a set of calculations for a two-level system containing N particles (1 ≤ N ≤ ∞) that graphically illustrates the statistical nature of temperature as well as the fundamental equivalence of its thermodynamic and statistical-mechanical formulations. These calculations, which we have applied in our undergraduate- and graduate-level physical chemistry courses, provide pedagogically useful insights into the meaning of a variety of thermodynamic and statistical mechanical concepts that students frequently have difficulty grasping