KENJIISHIDA
3806
Stochastic Approach to Nonequilibrium Thermodynamics of First-Order Chemical Reactions
by Kenji Ishida Laboratory of Radiochemistry, Japan Atomic Energy Research Institute, Tokai-mura, Naka-gun, Ibaraki-ken, Japan (Received April 22, 1966)
Since chemical reaction is in general a random process, the probability for a reaction state in t’he reaction system, in which the probability distribution for reaction states is determined by the stochastic theory of reaction process, is connected with the entropy change due to chemical reaction. It is then possible to formulate stochastically the nonequilibrium thermodynamics of chemical reaction. It is also shown that the relation between entropy and fluctuation is obtainable from such stochastic considerations.
Introduction Throughout this paper, attention is confined to homogeneous gas reactions of first order in a closed system maintained at uniform temperature T. It is assumed that the reactions proceed sufficiently slowly so as not to disturb seriously the equilibrium energy distribution of each component to any appreciable extent.’ Such a reaction system will be said to be in thermal equilibrium.2 This assumption usually has been made also in the study of nonequilibrium thennodynamics of chemical reactions,3 in which the notion of entropy production plays a central role. It is an interesting problem, however, to investigate how to include the condition of thermal equilibrium in the formula for entropy change due to chemical reactions. To obtain the answer to this question, we apply the theory of stochastic process to chemical kinetics. Then the chemical reaction in thermal equilibrium can be treated as a temporally homogeneous Markov p r o c e ~ s . ~ ~It~ is- ~possible, in some simple cases, to find the probability distribution for reaction states. I n addition, if it is physically justified that entropy is closely related to the probability of a reaction state, the nonequilibrium thermodynamics of chemical reaction may be stochastically constructed without starting from the thermodynamic Gibbs relation. Preliminary Approach According to the theory of irreversible thermodynamics,S~’Othe entropy production d,S resulting from a chemical reaction in closed system is given by The J O U Tof~Physical C h a i s t r y
d,S = Ad[/T 2 0
(1)
where A is the chemical affinity for the reaction and 5 is the degree of advancement. The last equality of (1) holds for the equilibrium state. Of course, as explained in the previous section, thermal equilibrium is tacitly assumed in eq 1. The integral of (1) may be written in the form
s=
$S 1
arbitrary reaction state
equilibrium state
Ad5 -I- Se
(2)
where S, denotes the entropy for the equilibrium state. We shall for the sake of brevity consider the nonequilibrium thermodynamics of the reaction A B. In this case A and d( are given by eq 3 and 4, respectively.
+
(1) R. H. Fowler, “Statistical Mechanics,” Cambridge University Press, Cambridge, 1936, p 700. (2) K. Ishida, Bull. Chenc. SOC.Japan, 33, 1030 (1960). (3) I. Prigogine, ”Introduction t o Thermodynamics of Irreversible Processes,” 2nd ed, Interscience Publishers, Inc., New York, N. Y., 1961, p 93. (4) G. F. Bartholomay, Bull. Math. Biophys., 20, 175 (1958). (5) A. T. Bharucha-Reid, “Elements of the Theory of Markov Processes and Their Applications,” McGraw-Hill Book Co., Inc., New York, N. Y., 1960, Chapter 8. (6) D. A. McQuarrie, J. Chem. Phys., 38, 433 (1963). (7) D. A. McQuarrie, C. J. Jachimowski, and M. E. Russell, ibid., 40, 2914 (1964). (8) K. Ishida, {bid., 41, 2472 (1964). (9) See ref 3, p 23. (10) $. R. de Groot, “Thermodynamics of Irreversible Processes,” North-Holland Publishing Co., Amsterdam, 1952, Chapter 9.
NONEQUILIBRIUM THERMODYNAMICS OF FIRST-ORDER REACTIONS
3807
(9)
and d4
(4)
-dnA = dnB
where n, (y = A, B) is the number of moles per unit volume. Substituting (3) and (4) into (2) and carrying out the integration, we obtain
=
%,e
where n =
-nRT In [I jY
fy,e
k1
-
PA'h - pBOkl' exp [ki hi'
+
(kl
+ h')t 1 (10)
exp(se/R) fY
1
TR(t)
(5)
Cn, is the total number of moles per unit Y
volume, f y is the mole fraction, and se = S,/n. If it is physically and stochastically justified that f, can be replaced by the probability p,(t) for a y molecule to be found in the reaction system at time t, then the entropy is closely related to the probability of a reaction state.
Stochastic Entropy Production To visualize the last statement, first of all, it is necessary to find the probability distribution for reaction states. A-ccording to the stochastic theory of chemic,al reactions in thermal equilibrium, the probability distribution P(NA,NB; t) for the reaction A B satisfies
+ ki(NA 4- ~ ) P ( N A + 1, NB - 1; t ) + k i ' ( N ~+ ~ ) P ( N A - 1, NB + 1; t) - ( ~ I N A+ ki'NB)P(NA,
NB;t)
N! NA' !NB0
= {pA(t)]NA(pB(t)]NB
(11)
represents the probability for a reaction state at time t. We are now in a position to introduce a mathematical expression for the condition that the reaction proceeds in thermal equilibrium. This condition implies that for the relaxation time T , which may be short compared with observed time but sufficiently long on the microscopic time scale,12one has
where qr (y = A, B) denotes the probability of a y molecule in thermal equilibrium and is independent of the initial condition. Therefore, the probability TT,E(~) for such a thermal equilibrium state in the course of reaction is given - by Let us define by the following relation the stochastic entropy S due to the chemical reaction in thermal equilibrium
(6)
where P(NA, NB; t ) denotes the probability of finding the numbers Na and NB of A and B molecules in the reaction system during the time interval from 0 to t, kl and ICl' are the transition probabilities for the forward and reverse reactions, respectively. Solving this system of differential-diff erence equations under the initial condit,ion P(NA0, ,YBo; 0) =
pB(t) =
In eq 8
S = - R x n , l n L +n S e Y
and
where k is Boltzmann's constant and the symbol ( ) stands for the mean with respect to the probability distribution for reaction states. Substituting (11) and (13) into (14), wehave
=
PYG) --kNCp,(t) In *r exp(se/k)
(7)
we obtain the binomial distribution" = T I
(16)
Y
ONA~~~ONBO
where (N,) = Np,(t) and Se = Se/N. The stochastic entropy (16) is in form identical with the deterministic entropy (5) and analogous to the so-called Gibbs
1Y !
~ ( N ANB; , t ) = NA!NB! - - - - ( ~ A ( O J ~ A { ~ (8) B ( ~ ) J ~ ~ (11) See the footnote for where N is the total number of molecules, pA(t) and p B ( t ) are given by, respectively
(19) and (20). (12) In other words, this means that the thermal equilibrium m a y be
instantaneously established owing to the rapid energy exchange between molecules.
Volume 70,Number 13 December 1966
KENJIISHIDA
3808
entropy postulate in nonequilibrium statistical mechanics.l3 It follows with the help of (13) that S approaches the value of entropy for the equilibrium state as t tends to infinity lim S = S,
t-
(17)
m
The differential of (15), Le., the stochastic entropy production, is given by dX
=
-kC
d ( N , ) In (p,(t)/q,)
Y
= -kC d
(NY)
In ((Ny)/’Wy)e)
(18)
Y
d S = A dt/T 2 0
(24)
which is on the average in agreement with the deterministic entropy production d,S. We need to emphasize a t this stage that in the present treatment all the variables are the means with respect to the probability distribution for reaction states, and that the stochastic entropy (16) enables us to formulate the nonequilibrium thermodynamics of chemical reaction without using the notion of chemical affinity.
Example In order to illustrate the validity of the stochastic
We now require the following differential equations with respect to the means { N A )and ( N B )
entropy production, we consider the reaction system of n components AI, Az, . . , , A,, between which all possible reactions of the type A, e A, occur. We then have ‘l2n(n- 1) of these possible reactions of which only n - 1are independent. The multidimensional Markov process for such a reaction system may be written in the formI6
and
d --P(N1, N2, . . . , N , , dt
where
(N,) d In p,(t) = 0, (N,) = Np,(t), and Y
= Nq, have been used.
. . ., N , ; t ) =
-CC’kij{NiP(NI, i
which can be easily derived by the method of momentgenerating function (mgf).14 Since, on the other hand, the degree of advancement 4 is defined by the relation d(N,) = where
VA
=
-1 and
YB
vY
d4
= -1, eq
(21)
18 becomes
1 d S = ---{kT In ((NA)/’(NA)J T
j
(Ni
N2,
.
* - 7
+ 1)P(Ni, N2, .
Nt, .
N , ; t)
+
Nt 1, . - 9 N , - 1, . . ., N , ; t)) .)
+ (25)
where N , is the number of A , molecules a t time t, ki, is the transition probability of A i 3 A , , and 2‘ denotes the sum of all j’s except j = i. As an initial condition, we assume P ( N , 0, . . . , 0; 0) = 1, where N is the total number of molecules. The solution of (25) is then given by the multinomial distribution (shown in eq 26)
kT In ((NB)/(NB)e)) d t (22) This can be also written as the function of time t
dS = kT(pAOk1
- p~Ok1’)exp[-(kl
+ k1’)tI X
Since in the theory of the Markov process time is no longer rever~ible,’~ it follows that dt > 0. Thus from (23) we have d S 2 0, which holds for all the values of t 2 0. On the right-hand side of (22) k T In ((NA)/(NA)~) - kT In ((NB)/(NB)e) corresponds to the chemical affinity A = C(A - c(B, because the chemical potential for a mixture of ideal k T In gases may be written in the form pY = ((N,)/(N,),). In consequence, we have
+
The Journal of Physical Chemiatrg
(13) S. R. de Groot and P. Mazur, “Non-Equilibrium Thermcdynamics,” North-Holland Publishing Co., Inc., Amsterdam, 1962, Chapter 7,p 126. (14) The mgf for P(WA, XB;t ) is defined by
where 81 and 6% are any real numbers and the summation is oyer all possible values of N A and N B . The differential-difference equation (6)is transformed into the partial differential equation, with the use of (a)
From the solution of this equation we can get the probability distribution (8). If (b) and the expansions of e-@1+02 and e@1--8? in these Taylor series are substituted into (e), eq 19 and 20 are obtained as the coefficients of 61 and 8 9 , respectively. (15) E. Parzen, “Stochastic Processes,” Holden-Day, Inc., San Francisco, Calif., 1962,p 187. (16) I. M. Krieger and P. J. Gans, J. Chen. Phys., 32, 247 (1960).
NONEQUILIBRIUM THERMODYNAMICS O F FIRST-ORDER REACTIONS
3809
i
where the probabilities p f ( t ) of finding an A , molecule in the reaction system a t time t satisfy x p f ( t ) = 1. i
Now, with the help of (28), we may write (30) in the form
The probability aR(t)for a reaction state is
(27)
aR(t) = n { p i ( t ) I N f i
n ._- I . n .. - 1-
6,= k N C xailAzexp(-Alt) X is1 1=1
Since, however, pi(t) is of the form n-1
+ E a f zexp(--XtO (28) where cqt = 1, eaiz = 0, and Xi > 0 for all Z’s, the P&)
=
qi
1=1
i
i
probability given by
P T , E ( ~ ) for
a state in thermal equilibrium is
n-1
=
(29)
i
Thus, applying the formula (14) to the present case, we get, as the entropy production 6, = dS/dt per unit time
where ( N , ) = Y p , ( l ) and ( N J e = Nql. From (25), on the other hand, we can derive the differential equations with respect t o ( N , ) d(N,) dt
=
-E’kij(Ni)
+ c‘k,,(N,)
j
j
(i
= 1, 2 ,
. . ., n)
(32)
Introducing the rates of reaction d(N,)/dt = -vl (i = 1, 2, . . . , n),we may write (31) as
n-1
Eail exp(-Act)
i = l 2=1
2=1
aT,E(t) = nqiNf
-e n-1
where e a n l exp(-A,t)
has
been used. It therefore follows from (34)-(37) that 6, 2 0, that is, the entropy production is always nonnegative.
Entropy and Fluctuation In the present section, we shall investigate whether the relation between entropy and fluctuation’’ may be obtained on the basis of the definition of stochastic entropy established in the previous sections. The rate equations 19, 20, and 32, which have been stochastically derived, are consistent in the mean with the corresponding deterministic rate equations. Since, however, chemical reaction is a random process, the chance fluctuation must be inherent in chemical reaction. It may be expected, therefore, that under certain circumstances the entropy change due to chemical reaction is related to such fluctuations. For this purpose, we rewrite the stochastic entropy (15) in t h e f m n a / k
=
- C ( N , ) In (W-r)/(Ny)e)
(39)
Y
n-1
where use has been made of
-cvf= on.
To prove
ill
the final equality and inequality of (33), we use the following observation. We first consider the behavior of (28). It is then shown that if p f ( t ) 2 qt, then
where AS = S - Se denotes the deviation of entropy from its equilibrium value. We return again to the reaction A B, for which A S / k is written with the use of ( N , ) - ( N , h = vY(E - l e ) as
n-1
Gait exp(-Azt) 2
1=1
while if p f ( t ) < lit, then
0
(34) (17) This is used in the sense of ( ( X - (X))m)or ( ( X - ( X ) ) m ( Y ( Y ) ) ” ) ,where X and Y are random variables and m and n positive integers. Refer also t o T . L. Hill, “Statistical Mechanics,” McGrawHill Book Co., Inc., New York, N. Y., 1956, Chapter 4.
Volume 70, Number 12 December 1966
KENJIISHIDA
3810
For small deviations from equilibrium this becomes1*
where u2,e2
=
= (NA'), - ( N A ) e 2 = N q ~ ( 1- Q A ) and ( N c ~) ~(Nc)e2= Nqc(1 - qc) are the vari-
Ul,e2
anCeS and -dqAqC/(l
where CY = t - te. The denominator ( N A ) e ( N B ) e / N on the right-hand side of (41) is connected with the variance in the following way. Since the variance ue2 = ( N A ' ) , - ( N A ) e 2 is given by ue2 = N q A q B = ( N A ) ~ ( N B ) ~from / N the binomial distribution for equilibrium state, eq 41 is brought in the form
AS/k =
(42)
-'/2(1/~e~)a~
This is in agreement with the deterministic formula1g
tlPel
=
Pe
-
((NANC)e
-qA)(1
+ ki')t]
exp[-2(kl
-1/2(~0/~e)2
- qc) is the correlation coefficient
On the other hand, differentiating the following chemical affinities with respect to 51 and t~
AI
=
kT In ((NA)/(NA)~) -
=
kT In
and
Az
((NB)/(NB)e)
- kT In ( ( N c ) / ( N c ) e )
where the degrees of advancement, 51 and t ~have , been introduced by the relations ( N A ) - ( N A ) e = -(ti [ 1 , e ) , ( N c ) - ( N c ) e = t z - &,e and ( N B ) = ( N B ) e - [2,e) (ll - &). From the trinomial distribution for equilibrium state, we can obtain the following formulas for fluctuations
+
(1
- Pe2)U1,e2
(1 -
-
(NA)e
Pe
Pe2)Ul,egZ,e
and
The Journal of Physical Chemistry
+-
1
(NB)e
-- 1
(50)
where the right-hand sides are obtained in the course of an entropy production such as (22), we find the relations 1
+- 1
(51)
and
These three relations may also be derived in conventional nonequilibrium thermodynamics, but it is possible from the stochastic point of view that the thermodynamic quantities ( b A l / b b ) e , ( i ) A l / b b ) e , ( d A z / d t i ) e , and (bAz/bt2),are connected with the fluctuations through relations 46, 47, and 48. Substituting (46)-(48) into (45), we obtain the expression in terms of the fluctuations c i , eand Pe as
where
--1
In ( ( N B ) / ( N B ) (49) ~
(44)
where ao = - 52 for the initial state has been introduced in the course of derivation. Equations 42 and 44, which tell us how entropy has a relation to fluctuation, result only from the stochastic considerations on the process of chemical reaction. For more understanding of this problem, we consider the consecutive reaction A $ B $ C, for which the probability distribution is given by a trinomial distribution. For this case, we have analogous to (41)
1
=
5 1).
with the chemical affinity A [cf. (51)l. Equation 42 is also expressed in terms of time t as AS/k =
(NA)e(NC)e)/gl,euZ,e
CY$
=
ti
- tie. This may reduce to AS =
(55)
-','2Xg*jataj %,3
which is in form completely identical with the general formula obtained on the basis of the Gibbs entropy postulate in nonequilibrium statistical mechanics,20 (18) Developing (40) in the Taylor series with respect t o we obtain
(46)
AS/k = -1/2(l/~ez)(€
- tee)'
X
(47)
(NB)e
Since ( ( N A
E - E*,
1 1 - ( N A ) ) ~ ) ~ / ( =c ~ s(p* ~)* -
of the second term in the bracket { with 1 for sufficiently large N . (19) Reference 3, p 47. (20) Reference 13, p 127.
i),
the absolute value
) is negligibly small compared
WETTABILITY OF POLYETHYLENE SINGLE CRYSTAL AGGREGATES
since we have begun with (16). We should note, however, that in the case of the stochastic nonequilibrium thermodynamics of chemical reaction, the coefficients
381 1
gu are explicitly expressed in terms of fluctuations with
respect to the numbers of molecules in equilibrium state.
Wettability of Polyethylene Single Crystal Aggregates
by Harold Schonhorn and Frank W. Ryan Bell Telephone Laboratories, Incorporated, Murray Hill,New Jersey
(Received M a y 4, 1966)
~~
The importance of describing fully the detailed physical properties (e.g., density, degree of crystallinity, and molecular weight distribution) of materials to be classified with respect to their critical surface tension of wetting (yo) is stressed. This is illustrated by determining the yofor a well-characterized preparation of polyethylene single crystal aggregates. The yc of the crystalline polyethylene is shown to be 53.6 dynes/cm compared to the generally accepted value of 31 dynes/cm. An analysis based on Fowked approach to wettability data is consistent with our results.
Investigations in surface chemistry as applied to the wettability of polymers' have failed generally to specify with any precision the detailed physical properties (e.g. density, degree of crystallinity, and molecular weight distribution) of the materials to be classified with respect to their critical surface tension of wetting (yo). In this communication we shall endeavor to demonstrate the importance of describing fully the preparation of samples employed in wettability studies. We shall demonstrate that, for example, a variation in the surface density ( p s ) of a polymer will change the critical surface tension of wetting. Recently, Roe,2 and Lee, Muir, and Lyman3 have called attention to the concept of the density of the surface layer of polymers as being important in determining their ultimate wettability. To obtain agreement between the accepted critical surface tensions of wetting ( y o ) at 20" and empirical calculations based on the parachor concept, the above authors2s3had to employ the amorphous densities of the polymers. However, there is no a priori reason for choosing the amorphous density since polymers may assume a range of densities depending upon their molecular weight and degree of
crystallinity, while retaining their chemical constitution. Polyethylene, for example, the subject of this investigation, has an amorphous bulk density ( p ~ of ) 0.855 g / ~ m and ~ , ~a crystalline bulk density of 1.OOO g / ~ m at ~ , 20". ~ Therefore, in principle, polyethylene should assume a spectrum of surface densities and yo values depending upon the ratio of amorphous to crystalline polymer present in the surface layer of the specimen. Invariably, the polymer specimens which are employed in wettability experiments are of the meltcrystallized variety. That is, they are molded in the melt against a smooth surface, then cooled. Polymer molecules which cannot be accommodated into the crystal lattice during crystallization are rejected to the (1) E. Wolfram, Kolloid-Z., 182, 75 (1962); K. L. Wolf, 2. Physik. Chem. (Leipzig), 2 2 5 , 1 (1964); V. R. Gray, Forest Prod. J., 1 2 , 452 (1962); A. V. Neumann and P. J. Sell, 2. Physik. Chem. (Frankfurt), 41, 183, 191 (1964). (2) R. J. Roe, J. Phys. Chem., 69, 2809 (1965). (3) I. J. Lee, W. M. Muir, and D. J. Lyman, ibid., 69, 3220 (1965). (4) G. Allen, G. Gee, and G. J. Wilson, Polymer, 1, 456 (1960). (5) P. H. Geil, "Polymer Single Crystals," Interscience Publishers, Inc., New York, N . Y., 1963.
Volume 70,Number 12 December 1966
