Carnot’s Function

To continue with his analysis, Carnot had to deduce what he could concerning
the physical and mathematical nature of ideal engine operation. Here he seems
to have exploited further his idea that heat engines do workby dropping heat
from a higher to a lower temperature. It seemed that the ability of heat to do
workin a heat engine depended on its thermal level expressed by the temperature
t, just as the ability of water to do workin a water engine depends on its
gravitational level.
Carnot emphasized a function F(t) that expressed the ideal heat engine’s operating
efficiency at the temperature t. He made three remarkable calculations of
numerical values for his function F(t). These calculations were based on three different heat engine designs that used air, boiling water, and boiling alcohol as
the working materials. Carnot’s theory required that ideal heat engine behavior
be entirely independent of the nature of the working material and other special
design features: values obtained for F(t) in the three cases had to be dependent
only on the temperature t. Although the primitive data available to Carnot for
the calculation limited the accuracy, his results for F(t) seemed to satisfy this
requirement. No doubt this success helped convince Carnot that his heat engine
theory was fundamentally correct.
To complete his theory, Carnot had to find not just numbers but a mathematical
expression for his function F(t). In this effort, he was unsuccessful; he could see
only that F(t) decreased with increasing temperature. Many of Carnot’s successors
also became fascinated with this problem. Although in the end Carnot’s
function was found to be nothing more complicated than the reciprocal of the
temperature expressed on an absolute scale, it tookno fewer than eight thermodynamicists,
spanning two generations, to establish this conclusion unequivocally;
five of them (Carnot, Clausius, Joule, Helmholtz, and Thomson) were major
figures in nineteenth-century physics.