Before introducing the temperature scale now known as the

Chapter 5, Problem 5.16

(choose chapter or problem)

Before introducing the temperature scale now known as the Kelvin scale, Kelvin suggested a logarithmic scale in which the function \(\psi\) of Sec. 5.8.1 takes the form

\(\psi=\exp \theta_{\mathrm{C}} / \exp \theta_{\mathrm{H}}\)

where \(\theta_{\mathrm{H}}\) and \(\theta_{\mathrm{C}}\) denote, respectively, the temperatures of the hot and cold reservoirs on this scale.

(a) Show that the relation between the Kelvin temperature T and the temperature \(\theta\) on the logarithmic scale is

\(\theta=\ln T+C\)

where C is a constant.

(b) On the Kelvin scale, temperatures vary from 0 to \(+\infty\). Determine the range of temperature values on the logarithmic scale.

(c) Obtain an expression for the thermal efficiency of any system undergoing a reversible power cycle while operating between reservoirs at temperatures \(\theta_{\mathrm{H}}\) and \(\theta_{\mathrm{C}}\) on the logarithmic scale.