T H E SECOXD LAW OF THERMODYXAMICS I F CHEMISTRY BY R . C. CASTELO
The Second Law of Thermodynamics is one of the most precious of the laws of chemistry as well as of the other sciences. Its statement, however, is usually such that its fundamental nature in chemistry is concealed. Such expressions as: Heat of itself cannot flow from a lower to a higher temperature; the entropy of an isolated system tends always to increase; the mathematical expression derived from Carnot’s cycle; and even the more readily grasped idea that in equilibrium, the free energy is a minimum a t constant temperature and pressure; do not reveal themselves as of importance to chemical science. The fundamental idea of the Second Law is given by the statement: Heat cannot be converted into work without compensation; and from this expression of the Law, follow all the ideas as expressed above. There are two possible cases: I. Heat is converted into work at constant temperature, Le., an isothermal process by which heat is converted into work. U’hat is the compensation? This can best be made clear by examples. (a) A compressed perfect gas expands isothermally and does work. The initial state of the gas may be defined by the equation f(m,T,V1,) = 0, and the final state by f(m,T,V2,) = 0, the result being that heat has been converted into an equivalent amount of work, but this energy change has been accompanied by a change in state of the gas itself. (b) Isothermal evaporation of a liquid against a pressure less than its vapor pressure. Again we may define the initial and final states by the equations: f(m1,m2,T,V) = o f(ml
- dml,m2 + dm2,T,S’+ dv)
= o
Again a change in state is the compensation. (c) Finally, consider the isothermal production of electricity by means of a voltaic cell. We consider that the electricity is produced as a result of a chemical change, which is itself the summation of the two electrode reactions, the chemical change being accompanied by an evolution of heat or heat is absorbed from the surroundings. Here again we have a simultaneous production of energy from heat and a change in state. So that, from experience, we may say that an isothermal process can convert heat into work only if that isothermal process be accompanied by a change in state. This may be called conveniently the First Half of the Second Law of Thermodynamics. Since an isothermal process must be accompanied by a change of statc, it follows a t once that an isothermal cycle cannot convert heat into work.
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A cycle involves no change in state. If it were possible by such an isothermal cycle to convert heat into work, we could (to use the classical example) utilize the heat of the ocean and convert this immense reservoir of heat into useful work. The first half of the Second Law says that this is impossible. Such perpetual motion obtained from an isothermal cycle has been called “perpetual motion of the second kind.” Hence, the statement of the Second Law found sometimes: “Perpetual motion of the second kind is impossible.” From the First Half of the Second Law arises also the concept of a reversible process. For example, the expansion of a perfect gas from state A to state B may be carried out in such a way that a definite maximum amount of work can be obtained from it. Kow since work cannot be obtained from heat by means of an isothermal cycle, it will require as a minimum amount of work to restore the gas from State B to State A, a quantity of work equal to the maximum obtained by the expansion. Otherwise, if it were possible to return to the original state by expending a smaller amount of work, the difference would be gained as useful work by means of an isothermal cycle. This is contrary to experience. We define, therefore, a process as reversible if the maximum amount of work is obtained from it, or the minimum is expended on it. Otherwise the process is irreversible, and here again experience has shown that all naturally occuring spontaneous processes are irreversible. Consequently, the work-producing power of the universe (as we know it) is continually decreasing; for to restore things to their original state, we would have to expand a greater amount of work than can be obtained from the irreversible processes. This is the idea contained in the Principle of the Degradation of Energy, sometimes given as a statement of the Second Law. Now, since any definite change in state is capable of producing a definite maximum quantity of work, a quantity that is never exceeded no matter what the process be by which the change in state takes place, we may say that a system in a given state possesses a definite capacity for doing work. I t has a definite work content, A, and this changes by a definite amount in going from an initial to a final state, where its work content is AB. Then AI - AZ is the maximum work obtainable by the change in state from state I to state 2 . AI - A2 = -AA = W R, where WR represents the reversible work, and we may write where W represents the work obtained by any process:
W = -AA W< - AA W> AA
for a reversible process for an irreversible process NEVER
So that a mathematical expression for the First Half of the Second Law may be given as - A A = WR. The second case we have to consider is that in which heat is con2. verted into work and is accompanied by no change in state, Le., the case where
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C. CAXTELO
we have a cycle, and obviously from the First Half of the Law, this cycle cannot be an isothermal one. What is the compensation, therefore, when heat is converted into work by a cycle which is not isothermal throughout? Again from experience, we find in this case that always a certain amount of heat may be taken up at a given temperature; a part of this heat may be converted into work, but the remainder of i t must be given up a t a lower temperature. This rejection at a lower temperature of part of the heat taken up at the higher is the compensation. This case finds its mathematical expression in the familiar form derived by means of a Carnot or other cycle:
The rejection of the quantity of heat Q2 at the temperature TPis the compensation. The two halves of the law may be considered together best in the following form :
First Half An isothermal cycle cannot convert heat into work. Consider a process in which the working fluid is an ideal gas. Carnot’s theorem has shown that the conclusions drawn with an ideal gas as working fluid will hold independently of the material used. Then for a perfect gas, the change in internal energy is given by: and aU/dv =
0,dV/ap
= 0,and
:.
for d T =
0,dU
=
0.
Le., for an isothermal process. Now consider an isothermal process I in which a gas expands from A to B. The First Law says: and for an isothermal process
Consider the isothermal process I1 in which the gas is compressed B to A. Again by the First Law: o = Q2 - W Zor Q2 = WP Then Q1 - QZ = W, - W2 by the First Law. Xow the Second Law applies and says that Q1 - Q2 = W l- W s > 0, i.e., we cannot absorb a quantity of Heat Ql a t a temperature TI, and by reversing the process give up a smaller quantity of heat Q2a t the temperature Ti, thereby converting the quantity of heat Q1 - Q2 into the equivalent quantity of work R1- WP a t the constant temperature Ti, or an isothermal cycle cannot convert heat into work. Again &I - QZ = !,Vi - KZ o but Q1 - QZ = Wl - IV2 may = o
SECOND L A W OF THERMODYNAMICS IN CHEMISTRY
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and if W1 - Wz = 0,W1 = WZand the process is reversible and W1 = maximun work done by the system going from A to B; and WZis the maximum amount of work that must be done on the system to restore it from B to A. The Second Half of the Second Law is as follows:
If we allow the expansion A to B to take place isothermally a t a higher temperature T1 again for process I at TI, Q1 = W1.
If we consider the process B to A to take place at lower temperature Tz, for process 2 Qz = R z From the First Law as before Q1 - QZ = W1 - WZand the Second Law now says Q1 - QZ = W1 - W Zcan be > o i.e. Q1, can be greater than Q2] or we can absorb a quantity of heat Q1 a t the higher temperature TI, and then reversing the process can give up a smaller quantity of heat Q2 a t the lower temperature Tz, thereby converting the quantity of heat Q1 - Qz into the equivalent quantity of work Wl - Wz. So far we have been able to correlate several of the expressions for the Second Law with the fundamental statement: Heat cannot be converted into work without compensation. It remains to show now that those expressions involving the ideas of entropy, and of free energy also follow. The expression Q1 - QZ =
ZW = QI T1 ___ - TP has always been derived Ti
by the use of a reversible cycle, and i t is shown in elementary books upon z and that for a Thermodynamics that for a reversible cycle Q1/Tl Q ~ ~ =T 0, cycle in which there is an element of irreversibility Q1/Tl+Q2/T2 < o . Further it is shown that for any cyclic process involving only reversible elements Z Q/T = 0,orforaprocess consisting of infinitesimal elements f Q/T = 0.Then
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0
for a cycle consisting of the change in state A to B by the path A C B followed by change in state B to .A by the path B D A, f Q / T = 0. 0
This may be broken up into the two parts as follows:
i.e.
or
fQ j T depends upon the initial and final states alone. = SB- S.4 where S has been called the entropy of the system in the
;.e, the value B
:.
QjT i
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R. C. CANTELO
given state, and dS from the above is seen to be an exact differential. Again it has been stated that for a cycle which involves an element of irreversibility Z Q / T < 0,or f Q j T < 0,i.e. in such a cycle the entropy of the system has 0
decreased. Suppose now that we have a cyclic process made up of the paths A B C, C D A, where A B C is an irreversible change in state, and C D h is a reversible path. Then for the whole cycle f Q/T < o and for the path C D A, 0
A
For the path A B C. 0
A
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or Q j T < dS i.e. TdS > Q or TdS = Q 6 where 6 = some positive quantity. Now for an isolated system Q = o as well as W, i.e. an irreversible change in an isolated system is always accompanied by an increase in entropy. I n some way, the system itself generates entropy. This is immediately recognized as another statement of the Second Law. It may be shown now, quite readily, that the spontaneous transfer of heat from a hotter to a colder body is an irreversible process and as such is accompanied by an increase in entropy. From the fact that this is an irreversible process follows a t once the well-known statement of the Second Law; Heat cannot of itself pass from a colder to a hotter body. The most general expression for the Second Law now becomes 6U S T6S - p6V, where the equality sign applies t o reversible changes in an isolated system, and the inequality to irreversible processes; and in addition this equation applies to a system of invariable mass. Now in the science of chemical thermodynamics, we find ourselves at once at variance with classical thermodynamics, for chemical processes do not occur in isolated systems, but in systems surrounded by a medium (e.g. air, the containing vessel, etc.). It is necessary therefore, to extend the equation 6U S T6S - p6V before it can be used successfully in chemistry. Our extension is, that the system plus the surroundings constitute an isolated system. Then if S be the entropy of the system and So that of the surroundings, and if we also assume that all changes in the surroundings take place reversibly, the relation 65 SS, 5 o holds. Then if in an infinitesimal stage of a process the heat q is absorbed from the surroundings, from the first law, 6U = q - p6V.
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SECOND LAW OF THERMODYNAMICS IN CHEMISTRY
Again the increase in entropy of the surroundings SS, =
:.
6S - 3
2 0 or
T-
from the equation for the first law. 6S -
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- q/T 6U
+ p6V T
L
O
or 6U Z T6S - pbV, an expression of exactly the same form as we have obtained above for an isolated system. Now it is quite possible by such devices as the van’t Hoff Equilibrium Box and semi-permeable membranes, to bring about our desired changes of state reversibly, so that we may drop the inequality sign and write our first fundamental equation of chemical thermodynamics as 6U = T6S - pSV, for a system of invariable mass. Now the energy of a system may be expressed U = f(S1 VI mt, mz . . . mk) and for any reversible change in state for which the composition of the system remains invariable 6U = T6S - p6V. Now it is possible to change the entropy by the addition or subtraction of heat, the volume may be altered b y work done on or by the system, both types of changes producing corresponding changes in energy. It is possible, however, to increase or diminish the energy, entropy and volume of the system simultaneously by increasing or diminishing its mass while the internal physical state as determined by p and T remains the same Then for a change in mass dm for any one of the components we write dU = 6V U; dm d S = 6s Sodm dV = 6V V, dm and solving for 6U, 6S, 6V and substituting in our original equation 6U = T6S - p6V we obtain dU = TdS - pdV (V, - TS, pV,) dm. where U,, So,V, represent the energy, entropy and volume of unit mass under the specified conditions of temperature and pressure Now letting U, - TS, pVo = p, Gibbs chemical potential for the substance, we have finally pdm dU = T d S - pdV The general equation, then, extended to any number of independently variable components becomes, dU = TdS - pdV MI dmt p~ dm2 . . . pk dmk This equation has been of fundamental importance in chemistry because of the variety of forms into which i t may be transformed. For example, dU - TdS = - pdV pl dml pr dmr . . . If T and V are constant dU - T d S = Zpdm. or d(U - TS)T,V= Zpdm
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The function U - TS has been called the Free Energy Function, but is known in this country as the work function, or work content A.
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h’ow
C.
Ai Az
CANTELO = =
Ui - TSi Uz - TSz
Therefore if by a reversible process we go from state I to state 2 , Ai - Az = (Vi - Uz) T(Sz - Si) = - AU Q = WR a conclusion which we had reached from a fundamental consideration of the Second Law. Again our fundamental equation may be put into the form. dU - TdS pdV = p1 dml pz dmz . . . For constant temperature and pressure. d (U - TS ~ V ) T=! ~Z p dm and the function 77 - T S pV has been known as the thermodynamic potential, but in this country is known as the free energy function F. Since dA and d F are exact differentials, A and F are functions of the state of the system only. This means that - A A and - AF are independent of the path by which the process is carried out. Evidently F1 - Fz = -AF = hi - hz pV1 - PVZ = - A h - PAV = M’R - ~ A v . Since chemical reactions are regularly carried out a t constant temperature and pressure, this function has found ready application in chemistry as will be shown below , F1 = Vi pl‘i - TSi FZ = Uz pVz - TSz. .*.Fi - FB= (Vi - Vz) p(V1 - V,) T(S2 - Si) = - AU [ T A S - PAVI
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= o
since A U = T A S - p A V for a finite reversible change a t constant temperature and pressure, but for a finite irreversible change a t constant temperature and pressure, T A S - p A V > A U :. F1 - FZ= 6, some positive quantity. :. For an irreversible change ( - AF)T,P> 0. And in this we have a criterion for an irreversible or spontaneous process. This is of supreme importance to the chemist for in this quantity (- AF)T!P > o , he has a criterion as to which of his reactions will go, a criterion as to which reactions may possibly be catalyzed, and, more important still, a quantity which will enable him actually to calculate his equilibrium concentrations. Again he has in this quantity a criterion of physical or chemical equilibrium. For, imagine a system in equilibrium to undergo an infinitesimally small reversible change from equilibrium a t constant temperature and pressure, Then (dF)T,Pi = 0 Again we have seen that U = f ( S I Y 1 m l , m z . . . mk) pldml p ~ d m z$. . . . Then dU = TdS - pdY
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SECOND LAW OF THERMODYNAMICS IN CHEMISTRY
and
dU = a U d S
whence
Ts T = aU, -p
as
+ aU.dV aU dml + a U dmz + . . . - +as
aml
- aU, 111 = aU, .
=
av
amz
,
.
aml
Now U is a homogeneous function of the first degree in S, VI ml, m2, .mK. :. By Euler’s Theorem U = Sa U + V a U + m 1a U + m ~ aU+. . . as aTr am, amz and therefore from the relations given above U=TS-pV+plml+mmz+. . . Differentiating dU = TdS SdT - pdV - Vdp p1 dml ml d p l + PZdmz m2 dl*z . . . :. 0 = SdT - vdp ml dpl mz dpz . . . Again F = plml PZ mz 113 m3 . . . d F = p1 dml ml d p l + PZdm2 m2 dpz . . . :. d F = - SdT vdp p1 dml p~ dms. whence ml dpl m2 dpz . . . = Vdp - SdT. Collecting our equations involving F. we have, For equilibrium, (dF)TIp = 0. For a finite reversible change a t constant T and p,
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-AF=o For a finite irreversible change at constant T and P
- AF .> o
Again (dF)T,p = Bpdm. and ml dp1 mz dp2 . . . = Vdp - SdT. By an application to chemistry of these equations, obtained ultimately from the First and Second Laws of thermodynamics, such varied expressions as the following may be derived readily: The Mass Action Equation, the Phase Rule, the Equations for the Colligative Properties of solutions, the Clapeyron-Clausius Equation, the van’t Hoff Isotherm and Isochore; the electromotive forces of voltaic cells, the variation of the equilibrium constant with the temperature, the fugacity and activity relations, etc., etc., and of late years it has been possible to calculate tables of standard Free Energies of Formation (Standard Affinity Tables.) Thus from the fundamental statement of the Second Law: “Heat cannot be converted into work without compensation,” we have led ourselves finally to the familiar laws of physical chemistry and of chemical thermodynamics.
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Department of Chemistry, University of Cincinnati, Cincinnati, Ohio.
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