Entropy and the Second Law of Thermodynamics ARTHUR E. MARTELL Clark University, Worcester, Massachusetts
T
HE concept of entropy and its use in the understandtng of the second law of thermodynamics has always caused difficulty for many students of elementary physical chemistry. The development of thermodynamics usually begins by explaining the impossibility of perpetual motion with the aid of a number of imaginary machines. The conditions necessary for the operation of real machines are then deduced and used to derive the second law. This completely negative approach to the science of thermodynamics often serves only to increase the confusion of the student. A more logical approach to the second law was developed in 1909 by Caratheodoryl and later reviewed by Born.2 They showed that the heat absorbed in a reversible change, dq, can be expressed by a Pf& differential equation. As a result of their formulation of the second law of thermodynamics the absolute temperature T is an integrating factor of the equation and the entropy change, d q / T , becomes an exact differential. The Pfaff equation is then used to develop some of the fundamental thermodynamic functions. The method of Caratheodory has two disadvantages: (1) The mathematical treatment is too advanced for most undergraduate college students; (2) the function dq does not have the form of a Pfaff equation for an irreversible process, and consequently the method cannot he used for irreversible changes. Recently a formulation of thermodynamics in terms of intensive and extensive concepts of energy was presented by Leaf.s This treatment predicts that entropy increases during thermodynamic changes and results in the deduction of a generalized form of the second law of thermodynamics, a. generalization of the Carnot cycle, and in the derivation of Bronsted's system of energetics. Since this treatment is simple and straightforward and overcomes the ohiections of classical methods. i t seems to fulfill the neid for a more logical approach to the second law. The following treatment, which is essentially a simplification of the method of Leaf, has been found by the author to he quite successful in the teaching of chemical thermodynamics. (The author does not propose that this treatment entirely supersede the classical method of Planck, since there is fundamental material of value even to beginning students that may he gained by a consideration of the efficiency of reversible heat engines and its relationship to the second law of thermodynamics.) A system may be any part of the universe which we
may wish to consider. Actually, we shall consider a single phase for simplicity. The properties of a system depend on a few variables such as temperature, pressure, entropy, and volume. When these variables have definite values, the state of the system is defined. A thermodynamic process consists of the conversion of the system from one state to another. This definition leads to an important restriction on the treatment of thermodynamics-that it is impossible to treat a system while i t is in transition between one state and another. This renders impossible the treatment of fluid flow, diffusion, heat conduction, reaction rates, or any other condition of flux where the system is not in a definite state. Thus thermodynamics will be restricted to static systems and conditions of equilihrium. All forms of energy may he considered the product of an intensive factor and an extensive factor; therefore, mechanical work or energy is equal to a force (intensive factor) times the distance (extensive factor) through which it travels. If, on the other hand, we take the force per unit area, or pressure, as the intensive quantity, the extensive quantity becomes the change in volume of the system. Thus PdV is a measure of mechanical work. The same holds for other forms of energy. For example, the change of electricaI energy of a charged particle moving in an electrical field is equal to the amount of charge on the particle (extensive quantity) multiplied by the difference of potential (intensive quantity) through which it moves. Similarly the electrical energy converted to heat in the flow of current through a resistance is the voltage drop (intensive factor) times the quantity of electricity (extensive factor) that flows. The same is true for heat energy. Here the temperature T is the intensive factor, and represents the intensity of thermal energy, while the entropy S is the extensive quantity. Thus the increase of heat energy is equal to T d S , and the increase in entropy of a system is usually defined by the equation: dq/T = dS
Let us assume that our system is in a definite state. We shall also restrict our consideration to two kinds of energy, mechanical work and thermal energy. When energy enters the system against the prevailing intensity factors, the increase in energy of the system is given by the sum of the products of each intensity and the change in the corresponding extensive property. The increase in energy for an infinitesimal change is given by dE = TdS - PdV
(1)
TdS is the increase of heat energy. Since expansion of the system against a pressure involves doing work, the product Pd V indicates mechanical work done, or loss of energy; therefore, -Pd V gives the increase of mechanical energy of the system. Let us assume the system to be surrounded by an infinite reservoir of heat and mechanical energy. We are not concerned with interconversionsin the surroundings between the two forms of energy, and we shall use it merely as an energy reservoir. For the surroundings of the system the increase of energy, dE', is given by dE' = TUS'
- PUV'
(2)
According to the principle of conservation of energy, dE
+
dE'
= 0
The energy gained by one is lost by the other. stituting, TdS - PdV + T'dS' 1PUV' = 0
(3)
Sub(4)
The increase of entropy, dS, of the system is due to the entropy, 6S,gained from the surroundingspluswhatever entropy may be produced within the system during the change. Since we are not concerned with interconversions of energy within the surroundings, we may consider that the increase of entropy of the surroundings, ds', is due entirely to a transfer of thermal energy from the system; therefore, we shall not distinguish between dS' and 6s' for the surroundings. Since the entropy entering the system is the entropy leaving the surroundings, 6S = -dS'
Adding (9) and (lo),
Equation (11) indicates that in any exchange of heatand mechanical energy between a system and its surroundings, the entropy gained by the system is equal to or greater than the entropy lost by the surroundings. The entropy function i s always increasing. This is the fundamental statement of the second law of thermodynamics. At equilibrium between the system and its surroundings, T = T' and P = P'. Equation (7) becomes: TdS
Since P
=
- T'6S
P', TdS
=
- P')dV
(P
- T'6S
=0
T(dS - 65') = 0
(12)
Since the value of T is always greater than zero, dS = 6S (equilibrium)
(13)
According to equation (13) the entropy gained by the system is equal to the entropy lost by the surroundings; therefore, a t equilibrium there is no change in entropy. From (13) and (5)
(5)
The total entropy increase when the system is in equilibrium with surroundings is equal to zero. When the system is not a t equilibrium with its surroundings, T Z T'. In that case equation (12) does not hold and dS is not equal to 65; hence according to equation (11) dS > 6.9. The conventional forms for the first and second (6) laws of thermodynamics may be obtained by delining the function q, the heat absorbed, and w, the work done by the system. The heat absorbed is equal to the heat lost by the surroundings, and by equation (5)
It must be emphasized that all of the volume increase of the system must be gained by an equivalent loss of volume of the surroundings. This is an important diierence in the behavior of the entropy and volume functions: dV = -dV'
Substituting, TdS
- PdV - T'6S
+ P'd V = 0
TdS - T'6S = PdV - PUV
(7)
dq =
- TVS'
=
T'SS
(15)
A gradient of intensity function T determines the The work done by the system on the surroundings is force on an element of the extensive factor dV. If P is the increase of work energy of the surroundings, and by greater than P', the volume of the system increases and equation (6) d V is positive. If P is less than P', the volume of the dw = -P'dV' = P'dV (16) system decreases and dVis negative; therefore, ( P - P ~ 5:Vo (8) Rearranging equation (7), Combining (7) and
(a),
TdS
-
- PdV
=
(9)
Substituting in equation (I),
A gradient of the intensive function T determines the force on an element of extensive function 6sthat is, if T is less than T', heat flows into the system and 6S is positive. If T is greater than T', heat leaves the system and 6S is negative; therefore,
dE = T'SS
TdS
T'6S 9 0
a~
=
ap
T'6S
-
-
P'dV
-
aw
PUV
(17)
Equation (17) is the usual form for the first law of thermodynamics and states that the increase of energy of a system is the heat absorbed minus the work done. Substituting equation (15) into (9),
SUMMARY
Substituting equation (16) into (8),
Equations (18) and (19) are the usual expressions for the second law of thermodynamics, i. e., that the heat absorbed from the surroundings divided by the temperature of the system is equal to or less than the increase of entropy of the system, and the work done is equal to or less than the maximum work. At this point, if desired, the concepts of reversible (equilibrium) and irreversible (spontaneous ) processes may be developed more fully.
The energy of a system is the sum of the various forms of energy present, each form being defined in terms of the product of an intensive function and an extensive function. This method of treatment limits thermodynamics to consideration of systems in definite energy states. Using the law of conservation of energy and the fact that the gradient of each potential function regulates the direction of flow of energy, the property of the entropy function of always increasing is derived. The behavior of the entropy function during equilibrium and nonequilibrium changes is deduced, and the ordinary equations for the first and second laws of thermodynamics are derived.
