Warren Hirsch Edward R. Murrow H.S. Brooklyn, NY 11230 Polyfechnic Institute of New York Brooklyn, NY 11201 Vojtech Fried' Brooklyn College of the city University of New York Brooklyn. NY 11210
I I Gas ~ fusion-A f I
Relaxation methods such as temperature jump and pressure jump, developed by Eigen and his associates, are important tools in today's study of chemical kinetics (2,3).However, when teachers wish to explain the concept of relaxation to students, they frequently turn to a physical process as an analog-the exponential decay exhibited during capacitive discharge, for example. The mathematical techniques usually employed in the treatment of transport phenomena in terms of relaxation are complicated and cannot be dealt with in an undergraduate physical chemistry course (4). In this note we present a method based on simple kinetic molecular theory and rate equations used in relaxation kinetics to demonstrate that gas effusion is a relaxation nrocess. Suppose two bulbs of equal volume contain unequal numbers of gas molecules a t low pressures and constant temperature. Let the bulbs he connected by a valve with an orifice whose length and diameter are small compared to the mean free path. These conditions make collisions between molecules an unlikelv event while as sine through the orifice (5.8). .. . Let tbeie he a total o i Nt ~ o l e c u l ein~the whole system, with N molecules in the right-hand hulb and (Nt - N) molecules in the left-hand bulb. The Second Law of Thermodynamics predicts that the molecular concentrations will equalizekhen the valve is opened-in order to maximize the entropy of the system. The rate of the process is, however, not predicted by thermodynamics. The dependence of the rate of effusion on temperature and molecula~mass was shown by the pioneers Graham and Knudsen (67). These two factors determine the average speed of a molecule according to ij = ( 8 k T l ~ m ) l /where ~, k is Boltzmann's constant, T is the Kelvin temperature, and m is the molecular mass in crams. The number of mol&ules effusing out of an orifice per second is given by dNIdt = ijANI4V where A represents the area of the orifice and V represents the volume of the hulb (8). The factor of 1'4 arises from the fact that the component of the molecular speed normal to the orifice is one half the average speed and from the fact that molecules have an equal a priori ~
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To whom correspondence should be addressed.
706 1 Journal of Chemical Education
Relaxation Process
probability of traveling in either direction along the axis normal to the orifice. Therefore, the number of molecules leaving the left-hand bulb per second is (ijAl4V) (N, - N). Similarly, the rate of molecules leaving the right-hand bulb is iiNAI4V. Thus, the net rate of molecular flow is given by
AB 4v
dNldt = -( N , - 2N)
(1)
Since the net flow rate is zero a t equilibrium, the equilibrium value of N is NJ2. Equation (1) can he modified easily for cases of unequal bulb volumes, provided the volume ratio is known. Suppose avariable ANis defined indicating the perturhation of N from the equilibrium value. Then AN = N - NJ2 or AN = (2N - Nt)/2. Since Nt is a constant, dAh'ldt = dNldt. I t follows from eqn. (1)that dANldt = -AiiANl2V. Separating variables and integrating provides
or, l n ( A N l ~ i N ~=) -Aiitl2V. Expressing the preceding equation in exponential form yields (2) AN = AN0 e-(AW2Vl Equation (2) is analogous to the Eigen equation for relaxation of a chemical equilibrium. Relaxation time (T)is defined as the time needed for a perturbed system to travel lle of the path back to equilibrium. Hence, if ANIANo = Ile = e-(AE'lzv), then T = 2VIAiT. The relaxation time is therefore independent of the variables Nt, N, and AN. It depends only on hulb volumes, orifice area, and average molecular speed. Literature Cited (1) Eigen, M., Disc olfomdoy Soc.. 24.25
(1957).
(2) DeMayer,L..Z. Eiecfroehem.. 64.73 (19601. (3) Chapman, S.. and Cowling, T. G.,"ThcMathematieal Thwryof Non-Uniform Garn. Cambridge Univ. Press.New York. 1970. pp. 106108. (4) Mason, E. A,, and Evans, R. B., J. CHEM. EDUC., 46.358 (1969). (5) Gcsham,T., Phil. Trans.Roy. Sac., 136.573 (1646). (6) Knudnen, M., Ann. Physik., 28.99 (1909). (7) Fried, V., Blukis, U.,Hameka. F., '"Physical Chemistry..' MacMillan Co., New York, 1979. pp. 57-58.
