12.1: Temperature Scales

A heat engine is a machine that extracts heat from the environment and produces work, typically in mechanical or electrical form. As we will see, for a heat engine to function there need to be two different environments available. The formulas below place restrictions on the efficiency of energy conversion, in terms of the different values of \(\beta\) of the two environments. We will derive these restrictions.

First, however, it is useful to start to deal with the reciprocal of \(\beta\) rather than \(\beta\) itself. Recall that \(\beta\) is an intensive property: if two systems with different values of \(\beta\) are brought into contact, they will end up with a common value of \(\beta\), somewhere between the original two values, and the overall entropy will rise. The same is true of 1/\(\beta\), and indeed of any constant times 1/\(\beta\). (Actually this statement is not true if one of the two values of \(\beta\) is positive and the other is negative; in this case the resulting value of \(\beta\) is intermediate but the resulting value of 1/\(\beta\) is not.) Note that 1/\(\beta\) can, by using the formulas in Chapter 11, be interpreted as a small change in energy divided by the change in entropy that causes it, to within the scale factor \(k_B\).

Let us define the “absolute temperature” as

\(T = \dfrac{1}{k_B\beta} \tag{12.1}\)

where \(k_B = 1.381 × 10^{−23}\) Joules per Kelvin is Boltzmann’s constant. The probability distribution that comes from the use of the Principle of Maximum Entropy is, when written in terms of \(T\),

\(\begin{align*} p_i &= e^{−\alpha}e^{−\beta E_i} \tag{12.2} \\ &= e^{−\alpha}e^{−E_i/k_BT} \tag{12.3} \end{align*}\)

The interpretation of \(\beta\) in terms of temperature is consistent with the everyday properties of temperature, namely that two bodies at the same temperature do not exchange heat, and if two bodies at different temperatures come into contact one heats up and the other cools down so that their temperatures approach each other. In ordinary experience absolute temperature is positive, and the corresponding value of \(\beta\) is also. Because temperature is a more familiar concept than dual variables or Lagrange multipliers, from now on we will express our results in terms of temperature.

Absolute temperature \(T\) is measured in Kelvins (sometimes incorrectly called degrees Kelvin), in honor of William Thomson (1824–1907), who proposed an absolute temperature scale in 1848.\(^1\) The Celsius scale, which is commonly used by the general public in most countries of the world, differs from the Kelvin scale by an additive constant, and the Fahrenheit scale, which is in common use in the United States, differs by both an additive constant and a multiplicative factor. Finally, to complete the roster of scales, William Rankine (1820–1872) proposed a scale which had 0 the same as the Kelvin scale, but the size of the degrees was the same as in the Fahrenheit scale.

More than one temperature scale is needed because temperature is used for both scientific purposes (for which the Kelvin scale is well suited) and everyday experience. Naturally, the early scales were designed for use by the general public. Gabriel Fahrenheit (1686–1736) wanted a scale where the hottest and coldest weather in Europe would lie between 0 and 100. He realized that most people can deal most easily with numbers in that range. In 1742 Anders Celsius (1701–1744) decided that temperatures between 0 and 100 should cover the range where water is a liquid. In his initial Centigrade Scale, he represented the boiling point of water as 0 degrees and the freezing point as 100 degrees. Two years later it was suggested that these points be reversed.\(^2\) The result, named after Celsius in 1948, is now used throughout the world.

For general interest, Table 12.1 shows a few temperatures of interest on the four scales, along with \(\beta\).

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\(^1\)Thomson was a prolific scientist/engineer at Glasgow University in Scotland, with major contributions to electromagnetism, thermodynamics, and their industrial applications. He invented the name “Maxwell’s Demon.” In 1892 he was created Baron Kelvin of Largs for his work on the transatlantic cable. Kelvin is the name of the river that flows through the University.

\(^2\)According to some accounts the suggestion was made by Carolus Linnaeus (1707–1778), a colleague on the faculty of Uppsala University and a protege of Celsius’ uncle. Linnaeus is best known as the inventor of the scientific notation for plants and animals that is used to this day by botanists and zoologists.