Reversible and Irreversible Changes Involving only Pressure-Volume Work J. KENNETH O'LOANE University of New Hampshire, Durham, New Hampshire
Tm
formulas for adiabatic changes seem to be stumhling blocks for students in elementary physical chemistry a t the junior level, and for seniors and even firstyear graduate students in elementary thermodynamics. This is a11 the more true because examination of several texts in physical chemistry and thermodynamics shows that these formulas are often left as, presumably, moderately difficult exercises for the beginner. It has been my experience that the beginner, struggling to keep his head above water as far as the first law of thermodynamics is concerned, will sink a t this point in nine out of ten cases. In this paper I should like to stress two simple devices which have proved of great value pedagogically in studying adiabatic changes. The first is to insist that derivations involving adiabatic changes begin with the first law of thermodynamics, preferably in the differential form, d E = Dp - Du, where the exact differential, d E , is the difference of the two inexact differentials, Dq and Dw. Unless this is done, the student with a good memory will probably begin with the formula, C,dT = - P d V
(')
The student with not so good a memory leafs through the book until he finds (2),not stopping to check the conditions under which the formula is valid. Perhaps the student is not entirely to blame for this, since most textbooks print such formulas so that they stand out plainly from the page, while the qualifying statements which restrict their use, which are as important as the equations themselves, are lost in the accompanying text. It would be a great help to beginners if important restrictions were placed close enough to any formula to form almost an integral Part of it. Beginning then with equation (1) we have, since, for an adiabatic change Dq = 0 d E = -&
In general, Dw = P .,. dV - 5 d L - 8 dZ - X d l - r d* ZY,dxi
(3) (4)
- mgdh -
(5)
where the terms on the right represent, respectively, the
work required to chauge the volume of a system; the work required to stretch a sy stem elastically; to change the electrical charge on a system; to vary the intensity of magnetization; to alter the surface area; to increase or decrease the gravitational potential; and to perform other kinds of work not included in the above. Usually only, one, or a t most two, of these I\-ork terms will be important in a given problem, but the student who has been compelled to consider these various work terms and to realize that they can he eliminated only when a t least one of the factors in them is zero, will be less likely to imagine that Dw is identical with P.,dV, and less flabbergasted when asked to handle a problem where the only ~vorkis, say, electrical. For adiabatic changes with both P-V and electrical work we have, d E = -Pa,., dV
+ & dZ
(6)
and for adiabatic changes with only P-V work, dE
=
-P.,dV
(7)
Use of the symbol P*,, for the pressure of the surroundings, the external pressure against which the system operates, is the second simple device which helps the beginner overcome the confusion which inevitably arises over the pressure of the system, P,y.,, occurring in the equation of state of the system, and the pressure of the surroundings, P.., occurring in the differential P d V . Without this simple difference in symbols I have found it well nigh impossible to prevent a semiautomatic substitution of p,., for p,,.. In problems involving irreversible changes, this error is fatal. A number of problems involving (7) can readily be devised. Although some of these are not of great practical value in themselves, they are of the greatest utility in helping the beginning student develop the ability to think physically about his calculus formulas, and to overcome the compartmentalization which bas usually arisen, since the student has all too often been required to juggle mathematical symbols with no physical referent. For example, if a block of aluminum undergoes an adiabatic change in which the internal energy has decreased five calories, while the external pressure is constant a t two atmospheres, doee the
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volume increase, decrease, or remain constant? Since may make the very important substitution for reverdE is negative here, dV must be positive. If desired, sible expansions: the actual volume change may be calculated from the P."" = Pant (15) data given, care being taken regarding dimensions. For an adiabatic reversible change of an ideal gas xith A troublesome problem for beginners is the calculation of the temperature change for an ideal gas in an only P-V worlc (9) becomes adiabatic, irreversible process involving only P-V work. Aside from exceptional cases, such as phase changes, it is usually convenient to take the internal energy, E, as a function of the temperature and volume, so that we have, Carrying along the n with the C , and with the equation of state helps the beginner, who needs this reSubstituting in (7) we obtain a general expression peated emphasis, to avoid ending a t this point with an from which we can calculate the temperature change in equation which would imply that the temperature an adiabatic change, involving only P-V work: change depends on the number of mols of gas present. This is particularly likely to happen if the double substitution P = Po,, = nRT/V is made in (2). Physically the beginner could easily check to see that the temperature change should he independent of the number of mols present by considering tu70 difFor only P-V work, ferent containers, one with twice as many mols of gas as the other. By putting the confined gases through the same relative volume change, it is easy to see physiwhere n is the number of moles and C* is the molar heat cally that the temperature change should he the same capacit,~. in both cases. For cases where other than P-V work is involved we Integration of (17) gives, for the adiabatic, reversible would have change of an ideal gas involving only P-V work (C, constant), Ideal gases have (bE/bV), = 0, so that (9)becomes, for an adiabatic change of an ideal gas with only P-1' work:
The diierence between (18) and (13) and (14) emphasizes the necessity of care in distinguishing irreversible, (13)and (14),and reversible, (IS), behavior. The importance of regarding equation (9), rather than (2) or (12) or (16), as the immediate point of departure for adiabatic changes involving only P-V work can be seen again if it is a question of calculating, say, the temperature effect for a van der Waals' gas. One who starts with (16), thinking that it is necessary only to use
This equation is useful in pointing out that the temperature drop depends upon the amount of work done by the gas during this change. I n a free ezpansion, where Pa.., is zero by definition, there will be no temperature change. In irreversible processes other than free expansion, where P.,., is either constant or variable, equation (12) is the starting point for the calculation of the temperature change. has already thrown away the (bE/bV), term of (9). The general integrated formula will be To apply (9) we first get (bE/bV),in somewhat more usable form from a "thermodynamic equation of state," In the special case of the adiabatic change of an ideal gas, with heat capacity independent of both temperature and colume, with only P-V work against a constant external pressure we obtain,
In reversible expansions the external pressure must always be infinitesimally less than the pressure of the system. Neglecting this infinitesimal difference, we
("1a v
T =
T
(st) aT v - pan,
(20)
From (19))the equation of state for a van der Waals' gas, we have
SO
that (20) becomes nRT V - nh
nRT + -an2 = - ana V - nh V* V'
(22)
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Substituting (22) int-o (9) we get for the irreversible adiabatic expansion of a oan der Waals' gas rith only P-V work: an
dT
=
-
nC.
(23)
Comparing (23) with (12) we see that while, for an ideal gas, the temperature change depends only on the external work done (if C , is constant), there will be a temperature change for a van der Waals' gas even with free expansion. In a reversible expansion we again have P,.. equal to P,.,, and substituting from (19), (23) becomes for the reversible, adiabatic expansion of a van der Woals' gas, with only P-V work, dT =
- nC.(VnRT- nb) -- CdVRT- nb)
(24)
Up to this point we have always considered C, a constant. To help drive home to the beginner that we have been dealing with a special case, it is useful now to consider C, as a t least a linear function of T. Substituting C.=a+BT
from (9) if (i)E/i)V)V), can be obtained from P-V-Tdata using equation (20). A final stumbling block occurs in connection with the integrated form,
(25)
into (24) and integrating we have finally, with the restrictions applying to (IQ),
For real gases other than "van der Waals' gases" adiabatic temperature changes may be determined
If the temperature change in terms of the pressure rather than the volume is desired, it seems natural for the beginner to substitute from P,V1 = P2V2. Unfor77
tunately, this short cut, while "eliminating" 5,does V, not give the correct result,
The reason that it does not is that one cannot use Boyle's law, which holds at constant temperature, to substitute into the righthand side of an equation of which the lefthand side involves a change in temperature. The correct substitution is
+
When this is made, (27) results, using Ce = C, R. In this paper I have mentioned the advisability of starting from the first law, using the general work term in beginning the derivation of formulas for adiabatic changes. The value of using different symbols for pressure, according to whether it is the pressure of the system or external pressure of the surroundings, has heen stressed.
