J. Chem. Sci., Vol. 121, No. 5, September 2009, pp. 607–615. © Indian Academy of Sciences. Graham’s law of diffusion: Quantum analogy and non-ideality† CHANDRACHUR DAS1, NABAKUMAR BERA2 and KAMAL BHATTACHARYYA3,* 1Department of Chemistry, B.C. College, Asansol 713 304 2Department of Theoretical Physics, Indian Association for the Cultivation of Science, Jadavpur, Kolkata 700 032 3Department of Chemistry, University of Calcutta, 92 APC Road, Kolkata 700 009 e-mail: [email protected] Abstract. We focus attention on two equivalent forms of Graham’s law of diffusion that is valid for an ideal gas mixture. This equivalence is shown to be lost by the empirical equations of state in presence of an attractive nonideality. The modified forms are noted. We then construct a simple quantum mechanical model to simulate these results and obtain a one-to-one correspondence. We see how these equations of interest may be extended to D-dimensions. By employing the quantum model, we next observe the equivalence of the results found above with those obtained via statistical mechanics. As an added advan- tage, we demonstrate that the virial theorem for confined quantum stationary states retains its validity in the statistical domain too, though here the averaging scheme is correspondingly different. Keywords. Graham’s law; nonideality; confined quantum system; statistical mechanics, quantum ther- modynamics. 1. Introduction tions among the particles and thus the notion of an ‘excluded volume’ appears. On the other hand, Graham’s law of diffusion is well-known. It states attraction is dealt with more carefully and elabo- that the rates of diffusion RG of different gases in a rately. It is also directly linked with a reduction of 1/2 mixture satisfy the proportionality RG ∝ (P/d) , pressure at the walls. An added advantage is that, where P is the gas pressure and d the density. An while the role of size is difficult to decipher in the equivalent expression in terms of temperature T quantum domain, the attractive force can be mod- and molar mass M is also available in the form elled in a very rational manner. So, we shall mainly 1/2 RG ∝ (T/M) . For gas mixtures in thermal equilib- focus attention on this part of the interaction. rium, therefore, the rates of diffusion of different Considerable recent attention has been paid to components are inversely related to the square roots quantum-mechanical analogs of several macroscopic of their molar masses. Isotopic separation turns out processes and concepts. The primary motivation has to be an important application of this tenet. been provided by Bender et al1 who defined quan- The law is simple and useful. There are, neverthe- tum isothermal and adiabatic processes, and next less, three important assumptions in the above proceeded to define2 entropy and temperature in the statement of the law. First, we consider the process quantum domain. They employed the particle-in-a- to be too slow to affect the equilibrium distribution. box (PB) model to arrive at the quantum analogs. Secondly, we disregard the fact that the system is We later extended3,4 their analysis to include some open. Thirdly, we neglect any interparticle interac- effect of nonideality. Work along this direction of tion. It is this last point that we like to concentrate quantum thermodynamics is continuing.5 An inter- on. esting point in this context is that, one has to enclose In phenomenological equations of state for real the system in finite boundaries to define the pres- gases, we usually take care of repulsion among the sure.1,6 This naturally takes us to studies on confined particles through a hard-sphere model. This is basi- systems, another important area of current concern. cally a mechanical size effect. One avoids penetra- An exhaustive review of early works on such sys- tems may be found from Fröman et al.7 Interesting † works of more chemical relevance include studies Dedicated to the memory of the late Professor S K Rangarajan 8 9 –1 *For correspondence on confined H atom, H2 molecule, H2 ion, He 607 608 Chandrachur Das et al 10 atom and quantum dot, etc. Physical effects at the δδδVtea/ = Ac/4, (1) 11 boundaries of enclosed quantum systems have been investigated in a variety of ways. Mostly, simple where ca is the average speed. A similar relation for potentials are chosen. Among others, the Woods– 12 diffusion holds, the only difference being that here Saxon potential has received some attention. How- we have a number of such holes and hence ever, the oscillator problem has turned out to be the most favourite choice.3,4,13–18 Apart from purely nu- 13 δδVtcda/ = (/4)∑ δ A j . (2) merical works on confined systems, a confined j 1-D oscillator has been found to be efficient in fur- 14 nishing energy eigenvalues of near-exact quality Thus, for a given container, we have the following when confinement is gradually withdrawn. Peda- 15 rate expression in either case: gogic works concentrate on force and view it as a two-mode system16 to introduce an ‘oblique’ basis RVtcGa=∝δδ/ . (3) for computations. In case of D-dimensional oscilla- 17 tors of this type, one looks for degeneracies and 18 Let us remember at this point that fractional dimensions. Studies on chemical poten- 19 tials of fermionic and bosonic confined systems cc∝〈21/2 〉 form yet another class of endeavour. We have seen a . (4) earlier4 how the virial theorem20 (VT) and Hell- mann–Feynman theorem21 (HFT) help in such situa- Since the gas is in thermal equilibrium at tempera- tions. Indeed, the modification of the VT under ture T, the equipartition principle applies to the confinement is of very general interest.22 However, situation, yielding to the best of our knowledge, the implications of 13 nonideality on Graham’s law have not been explored mc〈〉=2 κ T. (5) so far. Therefore, it will be worthwhile to investi- 22 gate it in some detail here, both from macroscopic and microscopic standpoints. This would enrich our From (5), one obtains understanding of the correspondence between quan- tum concepts and the empirical classical notions. 〈〉=cTmRTM2 3/κ = 3/, (6) The behaviour of confined systems will also be con- solidated. Additionally, one may appreciate the im- where M is the molar mass of the constituent of the portance of the VT and the HFT, as applicable to gas. Using (3), (4) and (6), we thus get confined systems, again in the present case. 1/ 2 In view of the aforesaid discussion, purpose of the RTMG ∝ (/ ) . (7) present endeavour is to highlight the effect of intro- ducing an attractive nonideality in Graham’s law of For an ideal gas, on the other hand, the standard diffusion. This will be explored via three routes, viz. equation the phenomenological equations, the quantum mod- els and statistical mechanics. As we shall see, there 1 PV=〈〉 mN c2 , exist a few nice correspondences among the out- 3 (8) comes of these approaches. holds, with N defining the total number of particles. 2. Phenomenological outcomes This leads to We consider first the case of Graham’s law of effu- 2 Pd/ =〈 c 〉/3. (9) sion. The volume of a gas effusing out in time δ t δ through a hole of cross sectional area A is equiva- Coupling (3), (4) and (9), it follows that lent to the volume swept out by the particles during 1/ 2 this time. Assuming that no collision occurs within RPdG ∝ (/) . (10) δt, the result is given by the concerned area times 23 the average velocity along a particular direction. Note that (7) and (10) constitute the traditional We thus have Graham’s law and they are entirely equivalent. Graham’s law and non-ideality 609 However, we shall see below that this equivalence is is obeyed in the quantum domain too. Let us, how- lost once we go over to real gases. ever, highlight here the point that (5) continues to be Consider, for the sake of simplicity, a real gas true even when the gas becomes nonideal. Thus, a with only attractive force. We take first the case of a study of Graham’s law leads us to an important van der Waals gas for which one finds result: (7) is obeyed for real gases, but (10) is not. This loss of equivalence has not been noted before. PV+= a/, V RT (11) We shall here concentrate on this point in the course of discussion on nonideality. where V is the molar volume. This leads to It will be useful for the future to look at the forms of a few equations with some greater degree of gen- 2 Pd///.=− RTM adM (12) erality. So, we specifically remark here on the form of (16) or (18) for a D-dimensional ideal gas. From 2 A similar result follows for the Dieterici equation of PV= RT and M 〈〉c/2/2, = DRT the latter being a state that starts with consequence of the equipartition principle that retains its validity even when nonideality is present, PeaRTV V= RT, (13) we obtain Pd/ =〈 c2 〉/. D (19) yielding It is clear that the other relation, −ad MRT Pd/(/)= RTM e . (14) Pd/ = RTM/, (20) For ad << MRT, one obtains the same result as (12). does not depend on dimensionality. But, more important is to note that, in both the cases, we observe the inequality 3. Quantum models Pd//,〈 RTM (15) The PB model has long been used to describe ideal gases. The philosophy has been analysed in consid- 24 as (12) and (14) reveal. On the other hand, putting erable detail on several occasions.