THE SECOND L A W AND ENTROPY R. C. CANTEW, UNIVERSITY OX CINCINNATI. CINCINNATI, OHIO
The pliewpoint is taken that in a first course in chemical thermodynamics, the second law should be a fundamental postulate, and that the afiproach to the thennodynamic functions should be physical. The second law i s ezpressed in the form, heat cannot be converted into work without compensation; and with this as a fundnmental postulate, a method is ginen for the development of the concept of entrofiy.
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I n a recent paper, Hazlehurst ( I ) , speaking of eutropy says, "It is bound up in some obscure way with an equally mysterious second law, which has the reputation of saying a great deal but which is actually as meaniugless to the tyro as a magic alchemical recipe." That such a vagueness exists in the mind of the student of thermodynamics as to the meaning of the second law, is due, in the present writer's opinion, to the student's inability t o interpret the natural processes which are a part of his every-day experiences. For this reason, the writer feels that in a first course in chemical thermodynamics, the second law should be made a fundamental postulate and that the approach to the thermodynamic functions should be physical. If a fundamental statement of the second law is given, it is quite possible to have this physical approach, for example, to the concept of eutropy without sacrificing entirely the elegance of a purely mathematical treatment. Tolman (2) says: "Moreover, the truth of the two fundamental laws, a t least as far as the macroscopic behavior of physical-chemical systems is concerned, can hardly be questioned; deviations from the laws have never been found.. .. We must regard the theorems of thermodynamics as among the most certain possessions of science." Surely, in the light of these facts we are justified in taking the second law as one of the fundamental laws of chemical thermodynamics. The present paper will be limited t o the physical approach, through the second law to the thermodynamic function known as entropy. The most useful statement of the second law is, heat cannot be converted into work without compensation. It is in this respect that heat energy differs from mechanical energy. There are two possible cases t o be considered both of which have come within the student's experience. The first case is that in which heat is converted into work a t constant temperature, that is, an isothermal procesS by which heat is converted into work. What is the compensation? A few examples of such processes as the isothermal expansion of a compressed gas, the isothermal evaporation of a liquid a t its boiling point, and the isothermal production of electricity in a voltaic cell show a t once that an isothermal process can convert heat into work only if that isothermal process be accompanied by a change in state. The change in the state is the compensation. This
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experimental fact enables us to draw several deductions; for example, an isothermal cycle obviously cannot convert heat into work. A cycle involves no change in state. Perpetual motion obtained from an isothermal cycle has been called "perpetual motion of the second kind." We see therefore, that "perpetual motion of the second kind is impossible." From the impossibility of perpetual motion of the second kind arises the concept of a reversible isothermal process. The isothermal expansion of a perfect gas from state A to state B can be carried out in such a way that a definite maximum amount of work can be obtained from it, a t the expense of the heat of the surroundings. Since work cannot be obtained from heat by an isothermal cycle, it will require as a minimum of work, to restore the gas from state B to state A, a quantity of work equal to the maximum obtained by the direct process. I n the reverse process an equivalent quantity of heat will be given to the surroundings. We can define, therefore. an isothermal reversible process as one in which the maximum amount of work is obtained or the minimum amount of work is expended. The usual analysis can then be used to show that the criterion of a reversible process is not only that the work of the process is the maximum possible, but also that an infinitesimal change in the force against which the system is acting will actually cause the process to take place in the reverse direction. The reversible process consists of a series of equilibrium states infinitely close together, and when the process is reversed the system goes through the same series of equilibrium states but in the reverse direction. It becomes obvious, then, that a reversible process does not have to be, necessarily, isothermal. The second possible case by which heat energy is converted into work is seen to be a cyclic process, and such a cyclic process cannot be an isothermal one. What then is the compensation? Experience shows us that in this case always a certain quantity of heat must be absorbed from a "source" a t a given temperature, that a part of this heat can be converted into work, but the remainder must be rejected to a "sink" a t a lower temperature. This rejection a t the lower temperature of a part of the heat taken up a t the higher is the compensation in this case. If the quantity of heat Q1-be absorbed from a source a t the temperaby the heat engine, then a quantity of heat Q2+must. be rejected ture TI, the quantity Q1- - Qz+ being conto a sink a t a lower temperature Tz, verted into the work W+. The efficiency of the reversible cyclic process can then be obtained in the usual way. The expression obtained is
3OURNAL OF CHEMICAL EDUCATION
NOYB~ER, 1931
If TI = Tz, Wi = 0, which is in harmony with the conclusion already drawn. Equation (1) may be put in the form,
and if we write Q1- for the quantity of heat absorbed a t T I , and the quantity of heat absorbed a t T2.we have finally,
Qz - for
Without difficulty, we can extend our treatment to any reversible cycle. We imagine that a t various stages of a reversible cycle, the heat elements 6 , 6qz, . . .6qn are absorbed from reservoirs a t the temperatures T I , Tz, . . . T", and the heat elements 6qlf, 6qz1, . . .6q.' are rejected a t the temperatures T I f ,Tz', . . . T,'. We have in addition a source a t a temperature T , where T > T I ,Tz . . . T , and a sink a t a temperature T' where T' < TI', T%',. . . T,,'. Wbeu the cycle has been completed the reservoir a t T1has lost the heat element 6 9 . We can restore this quantity of heat to the reservoir a t T I , by a reversible cycle operating between the temperatures T and TI. If the reservoir a t TIgains the heat 6q1, the source loses the heat X,- and
The source a t T loses in all the heat quantity Q-.
Similarly the gains 6q1', 6@', . . ,647,' of the reservoirs a t T,', Tz1,. . . T.' can he used in reversible cycles between the respective temperatures T I f , TZr,. . . and T'. The sink a t T' gains, in all, the heat quantity Q+.
The original cycle with all its stages is equivalent to a reversible cycle in which the quantity of heat Q- is absorbed by the work mechanism from
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the source a t T,and the quantity of heat to the sink a t T'. Hence from (3) and (4) we have
As m -+
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Q+ is rejected by the mechanism
and 6p +DQ,equation (5) becomes
DQ is an inexact differential which for a reversible path of change in state is given by the first law equation for such a path, DQ = dE
+ pdV
(7)
is a line integral which is independent of the path which connects the points
. an exact differential of some funcA and B. Otherwise expressed, DQ is T tiou S of the variables which determine the state of the system. In this respect S is in the same class with the other thermodynamic functions U,A, H, F and is seen to he as much a property of a given body or system as is the familiar physical quantity V. S is the property of the body or system that is known as its entropy. For some entirely unknown reason, the student "shies" a t S, but endures F (the free energy function) and embraces V. The interesting fact has arisen in our treatment that although DQ is an
. exact and equal to dS. inexact differential, DQ - IS 7. If we take, therefore, the equation DQ = TdS
as the expression for the second law (for a reversible path) we can combine the two laws, for a system of constant mass, in a single equation, dE = T d S - @ P
(9)
where all the differentials are exact. Literature cited (1) HAZLEHURST, ''Exo~cisinpa Spectre: Entropy," J. Cmw. Eonc., 8, 498 (March, 1931). (2) T~LMAN, "Statistical Mechanics," The Chemical Catalog Company. New Ymk City, 1927, p. 20.
