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The Relation of Temperature to Energy

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Christopher King

Department of Chemistry, Troy University, Troy, AL 36082

J. Chem. Educ., 2005, 82 (6), p 861

DOI: 10.1021/ed082p861

Publication Date (Web): June 1, 2005

Section:

History, Education, and Documentation

Abstract

The object of this article is to present known information about the relation between temperature and energy in a manner accessible to physical chemistry students. Temperature, T, can be defined as

or .

Combining this with the Boltzmann expression gives an operational definition of temperature:

The temperature of a collection of N harmonic oscillators, representing vibrations of diatomic molecules, containing n quanta of energy is calculated using

The validity of this definition of temperature can be demonstrated by calculating both the temperature and the vibrational contribution to heat capacity for N₂ and showing that they agree well with experiment. The calculated heat capacity is not constant with temperature, demonstrating that, contrary to the ideal gas case, temperature is not generally proportional to energy. Temperature can be calculated exactly even for systems containing many moles of oscillators. The lowest possible nonzero vibrational temperature of a mole of hydrogen molecules is calculated to be 109 K. A two-level system is used to illustrate negative Kelvin temperatures. That system also provides an analogy to account for the observation that systems with a limited numbers of energy levels, such as F₂, (i) have heat capacities that reach a maximum, then decrease with temperature; and (ii) have a maximum heat capacity that is not a multiple of R/2.