Derivation of the Second Law of Thermodynamics from Boltzmann's Distribution Law P. G. Nelson The University, Hull HU6 7RX, England In a previous article ( I ) , I showed how chemical equilibria law ran be treated by starting from Bol~r~nann'sdistribution (2). Thisapproach has the merit of simplicity and provides a deeper insight into the nature tlf chemical equilihrium than does a theknodynamic treatment. At the same time, by taking Boltzmann's law as a law of nature (21, it avoids the foundational problems associated with a statistical mechanical treatment (3). In the previous article, I showed how the thermodynamic condition for equilibrium in an isolated system-that the entronv . . of the svstem be at a maximum-can be derived bv the applicatim of Boltzmnnn's law to the racemi7arion of U D I ~ C U i3omers. ~ In the oresent article. Iwould like to show hbw the same ~ o n d i t i o ~ c be a n derived by the application of Boltzmann's law to a simple physical system. This derivation is in some ways more illuminating than the one given earlier and anticipates some of the key results of a statistical mechanical treatment. It could therefore be usefully included in an introductory course on chemical equilibrium based on Boltzmann's law (I),where it would help to prepare the way for a statistical mechanical treatment given at a later stage (4).
where the differential is at constant volume because the energy levels change when the volume of the solid is changed. With suitable values of u, eq 5 or 6 gives values of Cv = (JU/aT)vfor metals that are in good agreement with experiment, except at very low temperatures, where they fall off too rapidly (5). Consider now the same block of metal removed from the heat bath and placed in thermal isolation a t the same volume. The distribution in the block can no longer be given exactly by eq 2 because for Q > U Pi must now bezero. However, the values of Pi given by eq 2 for ti > U are exceedingly small, and the change in distribution on isolation of the block does not lead to any observable effect at a thermodynamic level, so that eqs 2-6 can also be used to characterize the equilibrium state of the isolated system (6). Approach to Equilibrium If the oscillators in a block of metal that is in a state of thermal equilibrium are distributed according to eq 2, it follows that, when the block is not in thermal equilibrium, the oscillators have a different distribution from this, and
The Equilibrlum State of an Isolated Metal Block Consider first a block of metal that is in thermal equilibrium with an infinitely large heat bath at temperature T . Let the number of atoms in the block be Nat,and let the block be represented by N = 3Natharmonic oscillators of energy where t o is the zero-point energy of each oscillator, u is its fundamental frequency, and h is Planck's constant (5).Let the oscillators exchange energy with each other and with the heat bath by processes that are of negligibly short duration compared with the intervals between them. Then, according to Boltzmann's law, the fraction of time each oscillator spends with energy ri a t equilibrium ( F 3 ,and the proportion of oscillators having energy 6 ; a t equilibrium ( P Fq), are given by where k is the Boltzmann constant and q is the partition function
The average energy of each oscillator a t equilibrium is thus given by
and the internal energy of the block of metal by -r?,/kT
U = N ; = (NIq)
rie i
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Journal of Chemical Education
(5)
Numbers of osciilators in different states (a) in a black of metal at T = $0 (8 = hulk). , .lb\ , ~in, an identical block of metal at T = 28. . Ic\ . . in thetwo blocksol metal when first brought into contact (open bars) and after equilibration (solid bars). The numbers of oscillators are represented by the lengths of the bars. The equilibrium temperature is 1.2858 ~
~~
that the movement of the system toward equilihrium is characterized by a change of distribution towards a Boltzmann one. As an example of such a change, consider two blocks of the same metal (1 and 2), initially isolated at different temperatures, and then hrought into contact (see figure). Before contact, the internal energies of the blocks are given by eq 5 as
I
U,(TJ = N d T J UATJ = Nz&Tz)
This equation may be compared with that obtained by multiplying eq 15 by dPi (In PJdP; + (Inq)dP, + (~Jkr)dP,= 0
(19)
This immediately identifies a and 0, and makes f to be such that JfIJP; = In Pi
(20)
From eq 13 the required function is thus (7)
where ;(T)is the average energy of an oscillator a t temperature T. At the point of contact, the combined system does not have a Boltzmann distribution (Fig. lc, open bars), but moves toward one a t some new temperature T3 (Fig. lc, solid bars). The internal energy becomes
(plug an arbitrary constant, conveniently set equal to zero). That f is a minimum fur a Holt~manndistribution can be shown by considering a change in distrihution involving three states, j , k, and 1 (8).For such a change, eqs 21,16, and 17 give
But conservation of energy requires that U3(T3)= U,(T,) + UdTz)
(9)
which is positive. The approach to equilibrium is thus characterized by
Thus T3 must be such that
df 5 0 Since ;(T) increases smoothly with T (eq 4), this means that T3 must fall between TI and T2. Further, the change in internal energy of each block is given by
(U, V constant)
(23)
or more explicitly JfIJt 5 0
(U, V constant)
(24)
where t is time. This result corresponds to Boltzmann's celebrated H theorem (9). which means that, if Tz > TI, AUI > 0 and AU2 < 0, that is, energy flows from the hotter to the colder body. Characleriratlon of the Approach to Equillbrlum
T o quantify the way in which the distrihution of energy in the above system approaches a Boltzmann distrihution at equilibrium, we need a function of the distrihution that reaches a limiting value with the Boltzmann distribution. Let this function he f. Let Pi he the fraction of oscillators having energy ri at time t, averaged over the period t - 6t to t + 6 t , with 6t chosen so as to make Pi a smoothly varying function of time (i.e., neither so short that P; fluctuates nor so long that it does not appear to change at all). Then the function that we are looking for must have
when
Thermodynamic Condition for Equillbrlum The function f can he related to a macroscopic property of
the svstem as follows. Eauations 2 and 5 are suhstituted into eq 2 i to give f " = -UINkT- I n q
(25)
Equation6 is integrated by parts between T = 0 and T = T t o give In [q/q(O)]= -UlNkT
+ [S - S(O)]INk
(26)
where S is the thermodynamic entropy, defined by dS = 6Q..,/T
(27)
where 6 6 , is the heat absorbed by a system in an infinitesimal reversible change, here equal to (aU)v. Equation 26 is reduced by equating the temperature-independent terms: S(0) = Nk lnq(0) ( = 0)
(28)
+ SINk
(29)
This leaves In q = -UINkT
Substitution of this equation into eq 25 then gives f '9 (eq 2). Now changes in f are subject to the maintenance of &Pi = 1,that is, to
= -SINk
(30)
Equation 30 can be generalized to nonequilibrium states by defining entropy as S = -Nkf
In an isolated system, they are also subject to the maintenance of N2Jk = U, that is, to
By the method of undetermined multipliers (77, we therefore require (Jf/JP;)JP,+ udP; + 8qdP; = 0
(18)
(31)
In terms of 8, the approach to equilibrium in the system is characterized by (eq 23) JS r 0
(U, V constant)
(32)
In the special case that the change a is from a state that can itself exist as an equilibrium state to another such state, this equation can be written JS 2 0
( U , V constant)
(33)
the change either maintaining equilibrium and being reversVolume 65
Number 5
May 1988
391
ible (8.9 = O), or perturbing the equilibrium and being irreversible (aS > 0). Equation 33 cannot be applied to the system considered here as there is only one state that can exist as an equilibrium state-the final state. I t can, however, he applied to the racemization of optical isomers considered in ref 1 . In this case, if the reaction requires a catalyst, mixtures of the twoisomers having the compositions that are passed through as the reaction proceeds can each be ohtained in an equilibrium state in the absence of a catalyst. Entropy The physical significance of S can he seen by writing eq 29 as where
reflects the extent to which the oscillators are able to disperse themselves over a wider range of states than a single state for each oscillator. In the hypothetical situation that only states of energy ? are available to each oscillator, the number of such states is given by. exp(SINk). For copper at . 25 O C this equals 3.77 (18. The more general quantity 8 measures the degree of dispersal that the oscillators in a hlock of metal have achieved. Comparison with the value of S shows how much more dispersal has to take place for the hlock to reach equilibrium. ~
Llterature Cited 1. N e h n , P.G. J. Chem.Educ. 1986.63.852. 2. G~ggenheim.E. A. Bolfrmonnh Distribution Lam. 3rd ed.; North-Holland: Amsterdam, 1963. 3. See, for example, Penroae, 0. Rep. Rog.Phvs. 1979.44 1937. 4. Ref 2, preface fo 1st sd. 5. far example, ~ y r i n gH.: , ende ern on, D.:stover, B.J.; ~ y r i n g E. , M. Stotirricol MechoniraandDynomies. 2nd ed.: Wiley: New York, 1982; Section 10.2. 6. Andreas, F. C. Equilibrium Statistical Meehonies, 1st ed.: Wiley: New York, 1953:
see.
Chrpl"o
This is the partition function for an oscillator when the energy of each state is referred to the average energy of the oscillator as zero. If there was but one state, of energy 2, q* would he equal to one a n d S tozero. Themagnitude of S thus
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Journal of Chemical Education
7 Sr Ix rzample, t k m P W Ph:vrol Chrmwrr). 3rd c d . 0 x f . r . i I l u v r r ~ wNcu Y ~ c k l'>&, . pp :22 52?. 3 Cumware I*nbkgh K Pnnr~plerol Chem,col t.*utl.br.~n.. lli, rd Csmhndw Unl. $?tar\ Ucu Y ~ r k , 1 9 9 1 . ~ ~ 7 I G W 9 R d i . S e ~ t 8" 8 5 19-521 10 S u l l . 1, H I ~ ~ p h s14t .JAN:IFTnemr.orhrm,co. ?hbtv 2nd MI. M R D S K B i :.i Katmnsl U m s d o f irandarda W a r h n w t . D :, 1471.
