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The Carnot function and the second law
Brief Communications The Car-notFunction by
JOHN
and the Second Law
S. TJZOMSEN
The Johns Hopkins University, Baltimore, Ma yibnd The Carnot Function, recently discussed by Potter (l), is equivalent to a function used by Thomsen (2) in a reformulation of the Second Law of Thermodynamics. In this note we demonstrate the equivalence and the role of this function in the reformulation. The Carnot Function, as defined by Potter is
CT=
Aw
(1)
ij&j
where AW and Q are the work done and heat added in a Carnot (reversible) cycle with a small temperature difference AO. He commented that this function has suffered from neglect. Hence, it is interesting to note that essentially the same function has previously been used by the writer in a reformulation of the Second Law of Thermodynamics. In the latter treatment the Second Law was presented in the form of three postulates, the first being aF = g(t)[F -
at
VU],
(2)
where F is a generalized n-component force, t is the temperature on any empirical scale, and U is the internal energy. This assumes n + 1 independent variables, the temperature t and n displacement coordinates; the latter occur in the gradient term, VU. The sign convention is taken such that F is positive for a compressive force. (For a simple fluid, n = 1 and F and VU reduce to p and W/au, respectively.) The essence of the Law lies in g(t), which is postulated to be a universal (the same for all systems), positive (nonzero, except possibly for isolated points) function of temperature alone; the form of the function will vary for the particular empirical temperature scale employed. We readily show that the quantity g(t) is the Carnot Function discussed by Potter. If, as in his Eq. 14, empirical temperature is denoted by t rather than 8, it becomes
250
The Carnot Function and the Second Law
Comparison with Eq. 15 of (2) indicates immediately that CT and g(t) are indeed identical. AS both papers show, absolute temperature is then defined in terms of this function through the equation
T =
‘lbexp([~dt)dt)
(4)
The treatment of the writer essentially makes the Carnot Function (defined in terms of an arbitrary empirical temperature scale) the starting point for the statement of the Second Law rather than a derived concept. It avoids on the one hand use of hypothetical engines and heat reservoirs in developing the theory, and on the other hand the more abstract approach of Caratheodory (3), based on adiabatic inaccessibility. Whether or not this course is preferable is largely a matter of preference. At any rate, it deserves mention as another interesting aspect of the Carnot Function. Bibliography (1) J. H. Potter, “A Note on the Carnot Function,” Jour. Frank. Inst., Vol. 276, No. 6,
pp. 507-518, Dec. 1963. (2) J. S. Thomeen, Am. Jour. of Physics, Vol. 29, No. 5, pp. 300-307, May 1961. (3) C. Caratheodory, Mathemiische Annalen, Vol. 67, pp[355-386<1909.
Vol. 281. No. 3, March 1966
251
JOHN
and the Second Law
S. TJZOMSEN
The Johns Hopkins University, Baltimore, Ma yibnd The Carnot Function, recently discussed by Potter (l), is equivalent to a function used by Thomsen (2) in a reformulation of the Second Law of Thermodynamics. In this note we demonstrate the equivalence and the role of this function in the reformulation. The Carnot Function, as defined by Potter is
CT=
Aw
(1)
ij&j
where AW and Q are the work done and heat added in a Carnot (reversible) cycle with a small temperature difference AO. He commented that this function has suffered from neglect. Hence, it is interesting to note that essentially the same function has previously been used by the writer in a reformulation of the Second Law of Thermodynamics. In the latter treatment the Second Law was presented in the form of three postulates, the first being aF = g(t)[F -
at
VU],
(2)
where F is a generalized n-component force, t is the temperature on any empirical scale, and U is the internal energy. This assumes n + 1 independent variables, the temperature t and n displacement coordinates; the latter occur in the gradient term, VU. The sign convention is taken such that F is positive for a compressive force. (For a simple fluid, n = 1 and F and VU reduce to p and W/au, respectively.) The essence of the Law lies in g(t), which is postulated to be a universal (the same for all systems), positive (nonzero, except possibly for isolated points) function of temperature alone; the form of the function will vary for the particular empirical temperature scale employed. We readily show that the quantity g(t) is the Carnot Function discussed by Potter. If, as in his Eq. 14, empirical temperature is denoted by t rather than 8, it becomes
250
The Carnot Function and the Second Law
Comparison with Eq. 15 of (2) indicates immediately that CT and g(t) are indeed identical. AS both papers show, absolute temperature is then defined in terms of this function through the equation
T =
‘lbexp([~dt)dt)
(4)
The treatment of the writer essentially makes the Carnot Function (defined in terms of an arbitrary empirical temperature scale) the starting point for the statement of the Second Law rather than a derived concept. It avoids on the one hand use of hypothetical engines and heat reservoirs in developing the theory, and on the other hand the more abstract approach of Caratheodory (3), based on adiabatic inaccessibility. Whether or not this course is preferable is largely a matter of preference. At any rate, it deserves mention as another interesting aspect of the Carnot Function. Bibliography (1) J. H. Potter, “A Note on the Carnot Function,” Jour. Frank. Inst., Vol. 276, No. 6,
pp. 507-518, Dec. 1963. (2) J. S. Thomeen, Am. Jour. of Physics, Vol. 29, No. 5, pp. 300-307, May 1961. (3) C. Caratheodory, Mathemiische Annalen, Vol. 67, pp[355-386<1909.
Vol. 281. No. 3, March 1966
251