4.3 The Absolute Thermodynamic Temperature
A corollary of Carnot’s Principle is as follows:
The motive power of heat is independent of the agents employed to realize it; its quantity is
fixed solely by the temperatures of the bodies between which is effected, finally, the transfer
of the caloric [1:20].
Expressing the corollary mathematically, it is
W ¼ W Q A ; t A ; t B
ð
Þ
ð 44Þ
The first decisive step toward the formulation of the second law was taken by
Kelvin [4:100–106 (1848)] by introducing the concept of the absolute temperature.
What follows is the modern treatment of the topic (e.g., Fermi [5]), which starts
with the premise of the MEH firmly in place while, originally in his 1848 treatment,
Kelvin had not yet accepted the premise.
4
Consider a reversible Carnot cycle engine receives Q A from reservoir at t A
operating in a cyclic process. Work produced in the cyclic process equals to the net
heat or Q A − Q B (see Eq. [24] in Chap. 3), where Q B is heat rejected by the engine
to reservoir at t B . Introduce the efficiency of the Carnot cycle to be the ratio of the
work performed by the cycle to the heat absorbed from the heat source at temperature t A .
g
W
Q A
¼
Q A À Q B
Q A
¼ 1 À
Q B
Q A
ð45Þ
Accordingly, the corollary may be restated
The efficiency of the reversible work derived from a heat source depends solely on the
temperature of the heat source, t A , and the temperature of the heat sink, t B , independent of
working fluids used in the production of work.
That is, the mathematical expression of Carnot’s principle becomes
W ¼ W Q A ; t A ; t B
ð
Þ¼Q A g t A ; t B
ð
Þ
ð44AÞ
We shall prove that the function g t A ; t B
ð
Þ has the following property. Imagine a
third body of heat reservoir at a temperature t C (t A > t B > t C ). We now consider two
reversible Carnot heat engines in series: the first, engine A, operates between
reservoirs A and B and a second engine, engine B, operates between reservoirs
B and C. We assume, for sake of simplicity, that the two engines operate in such
manner that the amount of heat, Q
0 (at the temperature t B ) rejected by engine A to
4
In the 1848 paper, he had this to say: “the conversion of heat (or caloric) into mechanical effect is
probably impossible*, certainly undiscovered.” In the *footnote, however, Kelvin acknowledged
the contrary opinion advocated by Joule, and signaled the move to accept the MEH that he would
take in a very short time (1851).
4.3 The Absolute Thermodynamic Temperature
67
A corollary of Carnot’s Principle is as follows:
The motive power of heat is independent of the agents employed to realize it; its quantity is
fixed solely by the temperatures of the bodies between which is effected, finally, the transfer
of the caloric [1:20].
Expressing the corollary mathematically, it is
W ¼ W Q A ; t A ; t B
ð
Þ
ð 44Þ
The first decisive step toward the formulation of the second law was taken by
Kelvin [4:100–106 (1848)] by introducing the concept of the absolute temperature.
What follows is the modern treatment of the topic (e.g., Fermi [5]), which starts
with the premise of the MEH firmly in place while, originally in his 1848 treatment,
Kelvin had not yet accepted the premise.
4
Consider a reversible Carnot cycle engine receives Q A from reservoir at t A
operating in a cyclic process. Work produced in the cyclic process equals to the net
heat or Q A − Q B (see Eq. [24] in Chap. 3), where Q B is heat rejected by the engine
to reservoir at t B . Introduce the efficiency of the Carnot cycle to be the ratio of the
work performed by the cycle to the heat absorbed from the heat source at temperature t A .
g
W
Q A
¼
Q A À Q B
Q A
¼ 1 À
Q B
Q A
ð45Þ
Accordingly, the corollary may be restated
The efficiency of the reversible work derived from a heat source depends solely on the
temperature of the heat source, t A , and the temperature of the heat sink, t B , independent of
working fluids used in the production of work.
That is, the mathematical expression of Carnot’s principle becomes
W ¼ W Q A ; t A ; t B
ð
Þ¼Q A g t A ; t B
ð
Þ
ð44AÞ
We shall prove that the function g t A ; t B
ð
Þ has the following property. Imagine a
third body of heat reservoir at a temperature t C (t A > t B > t C ). We now consider two
reversible Carnot heat engines in series: the first, engine A, operates between
reservoirs A and B and a second engine, engine B, operates between reservoirs
B and C. We assume, for sake of simplicity, that the two engines operate in such
manner that the amount of heat, Q
0 (at the temperature t B ) rejected by engine A to
4
In the 1848 paper, he had this to say: “the conversion of heat (or caloric) into mechanical effect is
probably impossible*, certainly undiscovered.” In the *footnote, however, Kelvin acknowledged
the contrary opinion advocated by Joule, and signaled the move to accept the MEH that he would
take in a very short time (1851).
4.3 The Absolute Thermodynamic Temperature
67