10 Chemical Equilibrium in Electrically Excited Gases
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M.
MANES
1
Mellon Institute, Pittsburgh, Pa.
A model chemical reaction system is assumed, in which energy is separately conserved within each degree of freedom of each molecular species, each such degree of freedom therefore having in effect its own "temperature" A statistical mechanical treatment of this model shows that the composition behavior at any fixed set of "temperatures" is the same as for an equilibrium system—i.e., the conventional expression for the equilibrium constant is retained—although the expressions for the magnitude and temperature dependence of the equilibrium constant become quite complex and no attempt is made to evaluate them. It is suggested that the model may simulate the behavior of chemical reactions in nonequilibrium steady-state electrical discharges, and therefore that such systems may resemble equilibrium systems in their responses to composition perturbations from the steady state.
'Tphe formation of methane, water, ethane, and acetylene from mixtures of carbon monoxide and hydrogen in microwave discharges have been observed by Blaustein and Fu (1). Such systems obviously are not in equilibrium with respect to energy exchange with their surroundings. Moreover, the acetylene yields that were obtained are not found in normal systems at thermal equilibrium except at considerably higher temperatures than was estimated. Nevertheless, the response of these systems to composition changes—e.g., removal of individual components by cold-finger condensation—is in keeping with Le Chatelier's principle applied to equilibrium systems. These systems therefore raise the ques1
Present address: Kent State University, Kent, Ohio. 133 Blaustein; Chemical Reactions in Electrical Discharges Advances in Chemistry; American Chemical Society: Washington, DC, 1969.
134
CHEMICAL REACTIONS IN ELECTRICAL DISCHARGES
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tion of whether or not chemical equilibrium can exist (at least in the sense that the activity products for the individual chemical reactions will be approximately constant over some reasonably wide range of compo sition) when the system itself is at some excited steady state. One approach is to consider excited steady-state systems in which equilibrium does not exist between the individual degrees of freedom. Since such systems are outside the domain of classical thermodynamics, we consider a model system and see whether or not a relation similar to the customary expression for the equilibrium constant can be derived from statistical mechanics. Our model system will be a multicomponent system in which: (1) a single chemical reaction can take place (generalization to multireaction systems is quite straightforward); (2) equilibrium exists within each degree of freedom (translational, rotational, vibrational, and elec tronic) ; and (3) equilibrium does not necessarily exist between different degrees of freedom. The assumptions state in effect that a "temperature" may be said to exist for each degree of freedom, but not necessarily for the system as a whole. It could be further assumed that the translational degrees of freedom are well equilibrated, in which case they would be a measure of the conventional temperature. The model could be a reason able one for an electrically excited system. One can imagine, for example, that a hierarchy of excitation "temperature" exists at the steady state between the individual degrees of freedom, each receiving and emitting energy at equal rates. It should be emphasized that we do not yet know the extent to which our model simulates real systems. However, since it is at least a step in this direction, let us examine its consequences. The derivation of the equilibrium constant expression for a multicomponent system will require the use of an appropriate modification of the single-molecule partition function, which we shall first derive. Con sider a one-component system in which the index / specifies the energy level of the molecule. If we assume the energies of the individual degrees of freedom to be additive, then the energy ej of the fth state is the sum of the energies €_,i, e , . . . e of the individual "substates" of the d degrees of freedom available to the molecule, where e is the contribution of the fc'th degree of freedom to the fth energy level. The index / may be considered as an abbreviation for the set /Ί, . . . U of energy substates which have the corresponding energies e . . . , e ; two such sets are identical only when they have every j (and e ) in common. The statisti cal weight, gj of the molecule is the product of the statistical weights g of the individual substates within the set that makes up J. Assuming corrected Boltzmann statistics (2), the number of states, t, available to a system of Ν identical particles, each with d degrees of freedom, is j2
jd
jk
jh
k
jd
jk
jk
Blaustein; Chemical Reactions in Electrical Discharges Advances in Chemistry; American Chemical Society: Washington, DC, 1969.
10.
MÂNES
135
Electrically Excited Gases = π %
ί
= π %
τ
(1)
Γ
We want the most probable distribution subject to the constraints: («)
%Nj = N = E ; k = 1,. . , ά(-β ,
XN
jejk
k
(2) (3)
. . ., -β )
ι
Λ
Downloaded by UNIV OF PITTSBURGH on May 12, 2016 | http://pubs.acs.org Publication Date: June 1, 1969 | doi: 10.1021/ba-1969-0080.ch010
3
where the corresponding Lagrangian multipliers are in parentheses. The constraints of Equation 3 express our assumption that equilibrium is attained in energy exchange within each degree of freedom but not between degrees of freedom. The usual application of Lagrange's method of undetermined multipliers leads to the equation N-jrf = e gjUe-^ = e Ug e~^ a
N
a
k
(4)
jk
k
and summing over Nj gives (5)
N = e XUg e-^ a
jk
3 k
Let us now define the single-particle partition function, q , for the k'th degree of freedom as 1* = %&ί*" "'* (β) k
β
3
The assumption of additivity of energies allows us to write, for the molecule, (7)
q*.=Uq k
k
which is analogous to the partition function of a normal molecule except that there is a separate β for each degree of freedom; the asterisk indicates lack of equilibrium between the individual degrees of freedom. (We can identify β with 1/kT only for those degrees of freedom that are equilibrated to the conventional temperature). Returning now to Equa tion 5 we see that XUgflfi'^ = UXg e~^ jk
J
k
=q*.
(8)
k3
The first equality follows from the fact that the summation over / indi cates that the products are to be taken over all possible combinations of the available substates, and the second equality follows from Equation 6. Before proceeding to the multicomponent case we digress to consider briefly the consequences of the nonequilibrium assumption on the distri-
Blaustein; Chemical Reactions in Electrical Discharges Advances in Chemistry; American Chemical Society: Washington, DC, 1969.
136
CHEMICAL REACTIONS IN ELECTRICAL DISCHARGES
bution of states. Returning to Equations 4 and 5 we find that e = and that a
= ±
N/q*
.
Ν
(9)
q*
K
'
The distribution of the sublevels of each degree of freedom may be found by noting that P(j) ( = Nj/N), the probability of finding the fth level, is the product of the probabilities of finding each of the d sublevels—i.e., PU) = P(h)P(i ) ·. · P(U) = π ~" gjke
2
Downloaded by UNIV OF PITTSBURGH on May 12, 2016 | http://pubs.acs.org Publication Date: June 1, 1969 | doi: 10.1021/ba-1969-0080.ch010
(10)
k€ik
k
qk
or (π)
P(h)= Ç^ Êli
Qk
The individual substates, therefore, follow the Boltzmann distribution, each with its own value of fi . Consider now the case of a chemical reaction at a nonequilibrium steady state. For simplicity, we consider the reaction k
A + xB = AB
X
i.e., the association of 1 atom of A with χ atoms of Β to form the com pound A B (which for ease of notation we shall call C ) . Then, following the derivation in Davidson's text (2) (with a slight change.in notation) we write X
(aA)Nj*
ΝΑ» J
j
iy
(a.B)Nj*
"
·
—ïyin 3
3
'
Ly
(g- ) J c
N
c
tf.ci
Π
3
3
iy
'
where the superscript letters refer to the individual molecular species and are not to be read as exponents. The index / again refers to individual (total) energy states; the use of a single index for all three molecular species is for simplicity of notation, it being understood that different molecular species in general have completely different sets of energetic states. We now maximize In t subject to the constraints ΧΝ^ + 1,Νρ = η
Α
(«A)
xZNf + iNf = n
B
(α ) Β
(13) ,
(14)
where n and n are the total number of atoms of A and Β in the system; this gives the Lagrangian multipliers a and « shown in parentheses. These are the same constraints as in the conventional derivation for an equilibrium systems. The energy constraints are A
B
A
%Nf
A
€flb
= * > E
A
* = 1 , · . · ,
B
