Graham's Law of Diffusion

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 advantage, 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 thermodynamics.

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 systems 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

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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 quantum 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 models 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- Pd///.=− RTM adM2 (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. However, in a (6) in (9), the equality P/d = RT/M follows for the quantum-mechanical context, the temperature T does ideal gas model. The two observations may be cou- not enter the discussion in a natural way. So, on the pled by writing way to extend the PB model to real gases, we concentrate on (18), rather than (16). This is because, 2 P//. d≤ RT M (16) 〈c 〉 is directly linked with the average kinetic energy 〈T〉 and this is well-defined in the quantum In (16), the equality applies to the ideal gas case domain. only. An alternative is to proceed via the speed. To this end, we notice that, by virtue of (6), the inequal- 3.1 The ideal case ity (15) leads to We first consider a particle in a 1-D box of length L. Pd//3.〈〈 c2 〉 (17) The energy eigenvalues are given by 22 2 EnhmLn = /(8 ). (21) But, we already noted that (9) is obeyed for an ideal gas. Now, if we concentrate on the observations (9) The pressure is defined by the relation PEL=−∂n/ ∂ and (17), respectively applicable to ideal and real to yield gases, the net result can be cast in the form of the PnhmL= 22/(4 3 ), (22) following inequality: and this gives Pd//3.≤〈 c2 〉 (18) 1 PL==22 E m 〈〉 u2 , (23) The equality in (18) holds again only for an ideal n 2 gas. Indeed, we shall see below that such a relation with u as the velocity along x direction, whereby

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Pd/,=〈 u2 〉 (24) HddxVx=−22/ + (). (31) follows. Let us note that it agrees well with (19) Next, we confine the system in the specified region when we put D = 1. (–L/2, L/2). This amounts to the modification For a particle confined in a 3-D box of lengths Lx,

Ly and Lz, respectively along x, y and z directions, d422⎛⎞x M with υ = Lx Ly Lz, we would likewise have PE=−∂n/ ∂υ HVxT =− +() + lim⎜⎟ . (32) 22M →∞ =−(/)(/) ∂ELnx ∂ ∂ L x ∂υ =−(/)(/) ∂ELny ∂ ∂ L y ∂υ , etc. dxL⎝⎠ Thus, one obtains 2222 22 Thus, the effect of confinement is naturally taken nh111nhy nh P ===xz. (25) 444mL333LL mL LL mL LL care of via the last term in (32). In other words, we xyzyz xz xy may pretend that our system Hamiltonian now From (25), we get contains an auxiliary potential that is denoted by, say, V1(x), defined by the last term. This makes the 1 2 subsequent manipulations easy. For example, by 322PEυ ==n mc 〈〉. (26) 2 employing the VT to the total Hamiltonian HT, we It gives us directly obtain

Pd//3,=〈 c2 〉 (27) dV dV 2.〈〉=Tx + x1 n dx dx (33) that agrees with (9). Indeed, by following the same nn procedure in D dimensions with υ as the hyper- volume, one is easily led to the quantum analog of On the other hand, employing the HFT on (32), we (19). find A simpler way to obtain the above result proceeds υ L3 dEdV1 dV by taking the volume of the container as . Then, n ==−11x dLdLLdx. (34) we define the pressure by nn 22 5 PE=−∂∂=−∂∂∂∂=nn/(/)(/)/(12).υυ ELLnhmL (28) Combining (33) and (34), we obtain the following final expression for the VT in case of a confined This leads to system: Ppmυ =〈2 〉/3 , (29) dVdE 2 Tx=− Ln where p is the momentum of the particle. This again n dxdL. (35) n shows

2 Pd//3,=〈 c 〉 (30) The auxiliary potential V1(x) does not show up here. It is an advantage. However, the last term in (35) in- similar to (9) and (27). volves pressure. This term enters the expression be- In arriving at results (27) or (30), we deliberately cause of confinement. This is why, (35) becomes employed the symbol υ for the volume. The reason crucial in discussions on confined systems. The ex- is that, here a single particle is involved and it occu- tension to higher dimensions is pretty straightfor- pies the whole space. Conventionally, N particles ward. are kept in the volume V.

3.2 The real case 3.3 Attractive non-ideality

To account for the loss of ideality, we need to intro- In modelling cases of an attractive non-ideality, we duce a potential inside the container that the particle assume the presence of a potential that would some- will feel. This means, we do not have a free particle how reduce the probability of finding the particle in (–L/2, L/2) now. Instead, some potential field, near the walls of the container. In one dimension, symbolized by V(x), is present. So, we define our this is satisfied by simple potentials of the form x2J, system in this new situation by the Hamiltonian J = 1, 2, etc. This ensures an inward attraction, much

Graham’s law and non-ideality 611 like the ‘a’ constant in the empirical equations of and hence state like (11) and (13). In the quantum case, however, the virial theorem requires that (35) should be Pd/ <〈 c2 〉/. D (44) true, as shown above. Rearranging (35), we obtain The quantum analog of (19) is easy to get (see the dV PL=〈〉−2. T x (36) discussion below (27)) for the ideal case. However, dx here (44) shows an inequality owing to non-ideality.

2J Clearly, with V(x) ~ x , the second term at the right 4. Statistical mechanics side of (36) is positive definite. Noting that in 1–D, 2 we have 〈〉=〈Tmu 〉/2, it follows from (36) that In this section, we first take the simplest potential of the variety V(x) ~ x2J and proceed to estimate the PL<〈〉 m u2 . (37) pressure as well as the average energy. Then, we can check our results against those found via the quan- Thus, we obtain tum model and the phenomenological equations. The effect of other potentials of the same attractive Pd/.<〈 u2 〉 (38) nature will be transparent at the subsequent stage. Thus, we hope to extract some general features of This inequality is the 1–D equivalent of (17). This the whole endeavour. may be contrasted with the equality in (24) for the ideal case. 4.1 The harmonic potential in one-dimension Confining attention to 3–D, we rewrite (35) as Let us consider the PB system in (–L/2, L/2) and 2 ⎛⎞∂∂∂EEE study the effect of the potential λx on the energy −++LLLnnn ⎜⎟xyz levels and hence the partition function. Here λ acts ⎝⎠∂∂∂LLLxyz as a perturbation parameter. The overall Hamilto- =〈〉+〈〉+〈〉−〈⋅∇〉2(TTT ) rV . xn yn zn n (39) nian is given by

Following the definitions and usage that has been M ⎛⎞4x2 used previously in arriving at (25), we now see that Hx=−∇22 +λ + lim . ⎜⎟2 (45) the left side of (39) becomes M →∞ ⎝⎠L

⎛⎞∂∂∂EEEnnn −++=LLL3. Pυ The energy levels En of this system can be expanded ⎜⎟xyz∂∂∂LLL (40) ⎝⎠xyz as

Therefore, keeping it in mind that the second factor (0) (1) 2 (2) E = E + λE + λ E + ... (46) at the right side of (39) is positive definite and that n n n n

2 〈〉=〈〉+〈〉+〈〉=〈TTxyz T Tmc 〉/2, (41) where

2 2 we arrive at the inequality (0) n h E = , (47) n 8mL2 Pd//3.<〈 c2 〉 (42)

LL22 This is the quantum-mechanical analog of (17). (1) En =− , (48) Now, coupled with (27), (42) yields the inequality 12 2π 22n that is the microscopic equivalent of (18). D 6 For a -dimensional system, the left side of (39) (2)mL ⎛⎞7101 4 EOnn =−⎜⎟ + + (1/ ) , (49) becomes DPυ. The second term at the right side is 4hn22⎝⎠45 3 π 22 n again positive definite. Naturally, we then have

etc. Let us note that series (46) is convergent. In- DPυ <〈〉 m c2 , (43) deed, it has to converge at large L because then the

612 Chandrachur Das et al left side would basically correspond to the energy of We now define the harmonic oscillator. This is indeed the basis of h the work of ref. 14. With this assurance, we now λ = , (54) d 2π mkT proceed to write the partition function Z as

where λd is the thermal de Broglie wavelength. ∞ Since we consider here a set of N non-interacting ZEkT=−∑exp[n / ] (50) particles confined in a one-dimensional box of n=1 L length , there are two dimensionless parameters ∞ which define the thermodynamic and the classical (0) (1) 2 (2) =−+++∑exp[ (EEnnλλ E n ...)/ kT ] . limits. These are respectively given by L/λd and nλd, –1 n=1 where n = N/L is the number density, so that n sig- nifies the mean separation between particles. The thermodynamic limit is obtained under the condition ∞ L/λd >> 1. If this is satisfied, the energies of the par- ≈−exp[EkT(0) / ]exp[ −λλ EkT (1) / ]exp[ − 2 EkT (2) / ]. ∑ nn n ticles can be approximated as continuously distrib- n=1 uted. The classical limit is obtained by requiring that

nλd << 1. In this limit, quantum correlation is insig- For small λ, one can expand the last two exponen- nificant and one can safely use the Maxwell– tials to obtain Boltzmann statistics. Anyway, we now go back to

∞ (54) and employ the thermodynamic limit, i.e. (0) (1) λ L → ZEkTEkT≈−∑exp[nn / ][1 −λ / + ...] d/ 0. Under this limit, the first sum can be inte- n=1 grated (see, e.g. ref. 25 for a discussion on this point) to yield L/λd, while the second one can be ap- [1−+λ 2(2)EkT / ...] 2 n proximated by π /6. Thus, we arrive at

Now, for simplicity, we retain terms up to first order ⎛⎞⎛⎞LL222 Lπ Z ≈−1 λλ + in λ only. This yields ⎜⎟⎜⎟2 ⎝⎠⎝⎠12kT λd 2π kT 6

∞ 22 ZEkTEkT≈−exp[/][1(0) −λ (1) /]. ⎛⎞λλLL L ∑ nn (51) =−⎜⎟1. + n=1 ⎝⎠12kTkTλd 12

To explicitly work out, we now put the energy ex- However, the second term at the right side of the pressions from (47) and (48). Thus, above equation is negligible compared to the first one. So, we may go one step further to retain the ∞ ⎡⎤nh22 Z ≈−exp most significant contribution to the partition func- ∑ ⎢⎥8mL2 kT n=1 ⎣⎦ tion: ⎛⎞λLL2 ⎛⎞LL22 Z ≈−⎜⎟1. (55) ⎜⎟1−+λλ . (52) ⎝⎠12kT λd ⎝⎠12kT 2π 22nkT In (55), the first term refers to the pure PB model One has to separate here two different types of and the λ-dependent part accounts for the primary terms: non-ideality correction factor. This is the simplest possible form for the partition function that one can ∞ ⎛⎞⎡⎤Lnh222 use profitably. Z ≈−1expλ − ⎜⎟12kT ∑ ⎢⎥8mL2 kT ⎝⎠⎣⎦n=1 From (55), one can quickly estimate the quantities of interest. We find

⎛⎞L2 ∞ 1 ⎡ nh22 ⎤ + λ exp − . (53) ⎛⎞λLL2 ⎜⎟2π 2kT ∑ nmLkT22⎢ 8 ⎥ ⎝⎠n=1 ⎣ ⎦ lnZ =− ln⎜⎟ 1 + ln , ⎝⎠12kT λd

Graham’s law and non-ideality 613 that can be further approximated to statistical equivalent of (16) as well. Also, using kTmu=〈〉2 , we obtain 2 LL 2 lnZO=− lnλλ + ( ), (56) 2 λd 12kT Pd/ <〈 u 〉, (64) so that we keep consistently terms up to first order which is the statistical analog of (38). Further, since 2 in λ only. We now define the pressure by the stan- V(x) = λx , we can rewrite (62) in the form dard equation dV PL=〈〉−2. T x (65) ⎛⎞∂∂A dx PNkTZ=− =−(ln), − ⎜⎟∂∂LL (57) ⎝⎠T This equation, we happily note, resembles exactly that gives, from (56), the relation the quantum-mechanical VT for confined systems, as given by (35), but here with statistical averaging. NkT⎛⎞λ L2 P =−⎜⎟1. (58) LkT⎝⎠6 4.2 Other potentials and higher dimensions

On the other hand, the average energy is obtained The two key points mentioned above, viz. obtaining from the relation the statistical equivalents of (15) and (17), which follow from the empirical equations of state, and ∂ ln Z demonstrating that the quantal VT is valid in the sta- 〈〉=ENkT2 . ∂T (59) tistical domain too, retain their validity in other potentials and in higher dimensions as well.

Consider first the 3-D extension of the results pre- Employing (56) here, we obtain sented in subsection 4.1. We have in this case, by

virtue of the approximation (55), ⎛⎞1 λL2 〈〉=ENkT2 + . (60) ⎜⎟2T 12kT 2 ⎝⎠ ⎛⎞λL2 ⎛⎞λL2 ZZZZ=≈−−11x y 3Dxyz⎜⎟⎜⎟ ⎝⎠12kTkT⎝⎠ 12 In (60), the first term arises from the kinetic energy part while the second one accounts for the potential ⎛⎞λL2 LLL 1− z xyz energy contribution. Thus, (60) may be rewritten as ⎜⎟3 . (66) ⎝⎠12kT λd

NkT Nλ L2 〈〉=ETV + =〈〉+〈〉. 212 (61) Approximating further [see (56)], one finds

V λ 222 2 Note that the averages in (59)–(61) now refer to sta- lnZLLLO3Dxyz=− ln ( +++ ) (λ ). (67) λ3 12kT tistical averages. Rearranging (58), we use (61) to d find The work function A takes then the form NLλ 2 PL=− NkT =〈〉−〈〉22. T V (62) VNλ 6 A=− NkTln Z =− NkT ln + 3D 3 12 λd 222 2 The result (62) is very relevant in the present con- ()().LLLxyz++ + Oλ (68) text. With d = Nm/L in 1-dimension, we note that it yields directly the inequality From (68), one gets the pressure as Pd//.< kTm (63) ⎛⎞∂∂ANkTNλ This provides the statistical support to (15) in 1-D. PLLL=− = −().222 + + ⎜⎟∂∂VV12 Vx yz (69) In case of ideality, we put λ = 0 in (62) to obtain the ⎝⎠T

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This finally leads us to The inequality states that the sum of all higher order corrections would be positive, so that the right hand NkTλ N side of (76) becomes an upper bound to the Helm- PLLL=−().222 ++ (70) VV6 xyz holtz potential. The length L of the box has not yet been chosen. Clearly, the best approximation is ob- The average energy likewise starts from tained by making the upper bound of A as small as possible. Thus, we choose L so as to minimize A, 2 ∂ln Z3D which then becomes the best available approxima- 〈E〉 = NkT , (71) ∂T tion to the Helmholtz free energy of the system. This leads to to yield finally ⎛⎞20kT 1/ 4 L = ⎜⎟. (77) 3 λN ⎝⎠λ 〈〉=ENkTLLL +()222 + + 212xyz This result determines the optimum size of a square- =()(). 〈TTT 〉+〈 〉+〈 〉 + 〈 VVV 〉+〈 〉+〈 〉 (72) xyz xyz well potential with which to approximate the thermal properties of the system, and it determines the Rearranging (70) and making use of (72), we then corresponding approximate Helmholtz potential. For obtain a system of N identical but distinguishable particles, (75) can be used to calculate the pressure, correct up 32(PV=〈〉+〈〉+〈〉−〈∇〉 T T T ) r. V . xyz (73) to first order, and it turns out to be

Coupling (39) and (40), we notice that (73) is the NkT⎡ λ L4 ⎤ D P =−⎢120⎥ . (78) 3– equivalent of (35) that is valid in the statistical LkT⎣ ⎦ domain. Additionally, we see clearly from (70) that

(63) is true. It is also notable from (73) that (44) is Here the interesting point is that, for the optimal size obeyed here. 4 of the square well (77), the resultant pressure disap- If we employ, for example, the perturbation λx 2 pears. This refers to the thermodynamic equilibrium. instead of λx in (45), the corrections (48) and (49) would change. However, one may notice that only the n-independent term finally survives in (55). To 5. Conclusions be explicit, (55) now takes the form Our purpose has been to investigate how far the pro- ⎛⎞λLL4 portionality of P/d and T/M retains its validity for Z ≈−⎜⎟1. (74) ⎝⎠80kT λd gaseous systems in presence of nonideality. We have chosen Graham’s law of diffusion as a physical From here, one can work out and check that all our process that employs it directly. A quantum model findings like (63) to (65) retain their validity. This to simulate the case of an attractive nonideality has reflects the generality of our endeavour. Further, the been put forward and analysed. The chief results are Helmholtz free energy, correct up to first order, can as follows: Pd/ ≤〈 c2 〉/ D and P//. d≤ RT M They are now be calculated from (74) as found to agree well with those obtained from phenomenological equations of state. To provide a more ⎛⎞LL4 AA(0)+=− (1) kTln⎜⎟ +λ . (75) concrete basis, we have next proceeded to check our ⎝⎠λd 80 conclusions via statistical mechanics, using the same quantum model in the course of constructing the 26 This agrees with the result obtained by Callen . partition function. A weakly perturbed system has Additionally, we may check that the Bogoliubov shown the correctness of all our previous findings. inequality [see ref 26] here becomes The proper reduction to the ideal case has always been shown to be maintained. Additionally, our ⎛⎞LL4 analysis has revealed that the VT for confined quan- AkT≤−ln⎜⎟ +λ . (76) ⎝⎠λd 80 tum systems, which is known to be valid only for

Graham’s law and non-ideality 615

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