edited by
J. DUDLEY HERRON Purdue University West Lafayene. Indiana 47907
The Solution: "Derivation of the Ideal Gas Law" Compiled by the Column Editor from suggestions suhmitted by: W a r r e n L. Bosch, Florida Institute of Technology Crayton M. Crawford, Mississippi State University Walter J. Gensler, Boston Uniuersity Albert Haim, State University of New York a t Stony Brook Ira N. Levine, Brvoklyn College P e t e r F. Linde, S a n Francisco State Uniuersity J o h n C. Salzsieder, Phillips Unrversity Waldemar Silberszyc, Dartmouth College L a r r y A. Viehland, Parks College of St. Louis Uniuersity J u r g Waser, Calrfornia Institute of Technology, Retired
and we conclude that the product of the volume and Kelvin temperature of a gas is always the same! We know that this is not true. Other manipulations of the original equations lead to similar falsities. Solutions that Work Several contributors provided a solution similar to the following argument by Haim. A rigorous mathematical derivation of the equation P V = nRT f n m [the simple gas laws] makes use of calculus. T h e total differential of the equation of state
V = V(P, T , n )
(1)
is given by
(2)
There is many a slip 'twixt the cup and the lip. -Palladas. l i w ~ kA n l h d n ~ hk.X. ~ . epigram 82. A very nnrient proverb. 8"mefimer a t t r i h u i ~ dt o Homer.
This shows how much easier it is to he critical than to he correct. Renjsmin Ilisraeli.Speech,dsnua~P4,1860.
The following is presented in response to the misinterpretations that. appeared in the August Forum.' Most of those who contributed responses to Mr. Vaitkunas' question concerning a procedure for getting from Charles', Boyle's and other gas laws to the Ideal Gas Law, pointed out the basicerror in the suggestions that were added to his article. The Error in Logic Quoting from Ira Levine's contribution, [the error] is to combine equations that hold under different conditions. T h e equation P V = hl holds only when T and n are constant; the equation V = h2T holds only when P and n are constant; the equation P = k 3 n holds only when V and Tare constant. Since the conditions for each of these three equations are different, you cannot combine these equations. When P V = h l is valid, the equation V = kzT is not valid, since P is varying when P V = hl holds, while P must he constant for V = k2T to hold. Therefore, the manipulations.. . carried out in the approach laheled 1) are quite wrong, just as the multiplication that Mr. Vaitkunas carried out to get P3V = klkzh3nT is wrong.
To further emphasize the problems that one can encounter by forgetting that kl, k 2 and ks are not true constants but functions of other variables, Peter Linde suggested the following manipulations. From the equations labeled (4) in the Vaitkunas' article, we have P=h ~ l V and P = knT
so we can say h~lV = h?T
rearranging:
' Vaitkunas, John J., d. CHEM. EDUC.,56,530 (1979). ' Hermn. d. I)., S r i e n w A~.tiuitiw,3, 18, (March, 1970).
"For an early paper on the subject of this column see: Roseman. R. and Katzaif, S., J. CHEM. EDUC., 11,350 (1934).
Each of the partial derivatives in eqn. (2) is obtained from the laws of Boyle (eqn. (3))Gay-Lussac (eqn. (5)) and Avogadro (eqn. (7)) as indicated below K V=2 T and n constant D
. . V = K2 T
P and n constant
. . V = KBn
P and T constant
. .
Substitution of eqn. (41, (6), and (8) into eqn. (2) followed by rearrangement yields eqn. (9) ~~
-
Definite inteeration of eon. (9) readily yields the desired equation:
An alternative approach that can be presented without use of differential eauations was sueeested by several people. Consider a change in the stateof a gas from &arbitrary initial state V i , Pi, Ti, ni to a final state Vf, RE,Tt, nt. The overall change is broken down into three partial processes, A n Editorial Comment: Editors are aware that gremlins lurk in dark earners of editorial offices, waiting to work their mischief as harried editors rush to meet deadlines. Although we take considerable steps to guard against their attacks, we occasionally fall victim to their unwitting ways. This was, unfirrtunately, what happened to the John J. Vaitkunas article which appeared in the August column of Forum. The staffwishes to apologize to all concerned. To thnse who took the time to point out our errors and suggest proper solutions, we thank you. Arlyne M. (Mickey) S a r q u i s Hie11 Srhuol Editor.
Volume 57. Number 3.March 1980 / 201
each one corresponding to the conditions specified for each of the three laws. The changes in state are specified in the figure. It is seen that the first process is a pressure-volume change a t constant T and n. The resulting pressure and volume are P, and V', respectively. Fur this process PjV; = PtV'
or
PiVi V' = -
111)
Pt The second process is a volume-temperature change for which .
"I
V"
Final State
Vf
PI
TI
nt
Chart showing that each partial change is governed by one of the thee laws (eqns. (11).(12).and (13))
(12)
.
Finally, the third process is a volume-number of moles change for which V" -
Boyle's Law
V'
"I$
=
5 vr
(13)
The approaches suggested above may he difficult for some high school students. Teachers may find it useful to simply state that P V = nRT has been found to hold for most eases in those empirical relationships:
Equating the V' terms of eqn. (11) and (14) and rearranging our variables yields the desired result: . .
The following is another alternative. It was suggested by both Gensler and Viehland. By experiment, the volume V i s a function of the independent variables n, p, and T, where n is the number of moles, P is the pressure, and T is the Kelvin temperature. Thus, V = *(n,P,T) (16) also by experiment,
v = kn (P.n constant) Charles Law: T V nR ldeal Gas Law: - = T P v = k2 but when n and P are constant, T
(28)
Some Additional Comments
Crayton Crawford has pointed out that in the most common . . . terminology, a scientific law is a summary of exoerimental data exoressed bv a mathematical eauatian. In this
and V = T f2(n,P) (18) (Note that eqn. (17) is not a statement of Boyle's law unless n and T a r e held constant, and that eqn. (18) is not a statement of Charles' Law unless n and P are made constant.) From eqns. (17) and (18)
multiplying through by PInT
The second expression in eqn. (20) is explicitly not a function of the independent variable P, but only of n and T. The third expression in ean. (20) is ex~licitlvnot a function of the independent variable T , hut only of and P. There is only one way that two functions based on different sets of independent variables can he equal, and this is to have the two functions both equal to the same constant. Therefore, from eqn. (20):
n
Looking a t only the first and last terms of eqn. (Zl),we arrive at
which, when rearranged and the constant labeled as R, results in the familiar
202 / Journal of Chemical Education
Boyle's Law: PV = k l (n.T constant) ldeal Gas Law: P V = nRT but if n and Tare constant then P V = k,
m =
mass of gas.
This relationship is based on experimental evidence. Similarly, it may be useful to indicate that, empirically, it is the mass rather than number of moles of the gas that is held constant when Boyle's Law and Charles' Law are derived. In a comment concerning terminology, Peter Linde says, I wish to call attention to the growing abuse of the term 'dimensions' by chemists.. .Dimensions are descriptors of the t y p e uf measurement that is made (eg., length, mass, time); units are labels attached to the numerical results.. . (e.g., mile, gram, second). Dimensional analysis is a much broader and more abstract topic than unit analysis.
Finally a word from your editor. In all of this discussion, I trust that we do not lose sight of the fact that we must also teach what is meant when we say "directly proportional," "this product is a constant," or "this relationship holds when we keep T and m constant." The meaning of such statements is not self-evident and needs to he explained or demonstrated. I have, for several years, demonstrated the relationship hetween the volume of a cylinder and the height or diameter of the cylinder when other variables are hold c o n ~ t a n t My .~ students seem to find the exercise instructive. I hope that readers have learned from these suggestions on teaching the gas laws. If you have questions concerning other topics, send them to Forum and we will attempt to get you an answer.
