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Thermodynamic restrictions for chemical laser efficiency
Volume 209, number $6
16 July 1993
CHEMICAL PHYSICS LETTERS
Thermodynamic restrictions for chemical laser efficiency A.V. Eletskii and E.V. Stepanov institute ofApplied Chemical Physics, Russian Science Center “Kurchatov Institute”, Kurchatov Square, 123182 Moscow, Russian Federation
Received 3 1 March 1993; in final form 17 May 1993
A thermodynamic approach to the problem of the maximum effkiency for a chemical laser is developed. Assuming the laser radiation entropy to he negligibly small in comparison with the entropy of the chemically reacting compounds, the thermodynamic limitations for the laser efficiency are obtained. A number of kinds of chemical reactions, each one being a source of laser pumping, are explored. It is shown that the most significant restrictions appear for reactions of recombination type reducing the number of particles in the laser mixture. Photorecombination lasers are investigated in detail. The maximum efficiency for those lasers is computed over a wide range of proposed parameters of the initial mixture.
1. From the thermodynamic viewpoint, a chemical laser is a transformer of the energy released as a result of an exothermic chemical reaction into coherent radiation energy. Obviously, this transformation is accompanied by the loss of some part of the energy for heating, so that the efficiency of such a transformation is always less than unity. It is easy to show [l] that if the number of laser radiation quanta per one elementary act of the chemical reaction is about unity or less, the entropy of coherent radiation carrying away the beneficial part of the transformed energy is negligibly small in comparison with the entropy of the reagents and reaction products. This is caused by the small angular divergence and the small relative frequency width for laser radiation. Thus, a chemical laser should be considered as a device for converting a chemical energy to an energy whose carrier is characterized by negligibly small entropy. Under this consideration, the second law of thermodynamics should be taken into account, whereby the entropy change for a system as a result of any energy transformation must be positive. For a chemical laser, this means that removing laser radiation cannot be attended by a decrease in the entropy of the compounds. In turn, it is possible only if the chemical reaction itself causes the entropy to increase. Otherwise, a part of the chemical energy has to be expended for heating the medium to make
the net entropy change positive. The latter case, apparently, is accompanied by a limitation to the maximum energy of the issuing radiation, or the effrciency of the chemical laser. In the present work, this kind of restriction is explored theoretically for various methods of laser pumping. 2. Let us define the chemical efficiency of a chemical laser by the usual relation: ?I= EJQ, where E, is the energy of the laser radiation, Q is the energy released in the chemical reaction used as a base for the laser involved. According to the energy conservation law, the expression for the efficiency can be written in the form I/=1-
E, -&I +A
Q
(E,, is the energy of the system until the chemical reaction has begun, Et is the energy of the reaction products, A is the value of the work done upon the products by external forces). The expression ( 1) is to be calculated assuming that the values of entropy before and after reaction, S, and S,, are linked to one another by the following relation: so
(2)
where S, is the entropy of laser radiation, considered to be negligible in comparison with S, and S,.
0009-2614/93/S 06.00 0 1993 Elsevier Science Publishers B.V. AU rights reserved.
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Thus, the problem should be formulated as follows: what is the maximum amount of the energy released as a result of the chemical reaction, which can be removed from the system by radiation in such a way that the entropy of the issuing radiation is negligibly small, and the entropy of the reaction products is not less than the entropy of the reagents? The response to this question depends on the kind of reaction involved. 3. We discuss initially some simple examples which do not require cumbersome calculations. For the first example, we consider a reaction Az+B2+2AB+Q,
(3)
which is in wide use as a chemical laser pumping source [ 2-41. Assume that the initial quantities of reagents are equal to one another, and the reaction is accomplished entirely, exhausting all the reagents. In terms of the canonical distribution [5], the expressions for energy and entropy for a two-atom molecular gas have the forms
X,~& I /
xexp
T
)I
'
j(j+ 1)B I Awv - ~ T
T
I
’
(10) (5)
where N is the quantity of reagents (AZ and B,), B is the molecular rotational constant, fiw is the vibrational quantum, j, v are rotational and vibrational quantum numbers. (Here and below, the gas constant R is omitted in calculations, so that temperature will be expressed in energy units, and specific heat capacity and entropy will be dimensionless.) Carrying out straightforward manipulations of (4) and (5), and taking into account the condition usually met:
526
where To (T,) and V,, ( V,) are the temperature and volume of the system before (after) the reaction, Bi, B2 and B3 are the rotational constants for AZ,B2 and AB molecules, and m,, m, and m3 are their masses, respectively_ The final temperature of the system T, as well as the work value A depend on the current conditions for the reaction being analyzed. To be more specific, let us consider isochoric and isobaric processes. In the former case, V, = Vuand A =O, and in the latter one, V, = VoT, / To and A = 2N( T, - To). For these processes, the condition S, BS,, is reduced to the relation
(4)
C C (2j+l) c ‘30 v>o
E=sNT+NT2$ln xexp
T
we derive the following:
(k= 5 for an isochoric, and k= 7 for an isobaric process), whence it is easy to obtain the limitation for the efficiency magnitude,
(2jfl)
j(j+l)B+% - ~
16July I993
Obviously, this limitation is valid only if the expression in braces is positive. The opposite case refers to an increase of entropy in the reaction itself, and no limitations occur. Under the conditions suitable for a chemical laser, the relation T,JQ<<: 1. This means that the restrictions (10) are rather insignificant. Indeed, for the reaction: Hz+Fz-12HF, which is widely used as a laser pumping source [2,4], the relation To/ Qz 5 x 10m3, implying a maximum efficiency magnitude of 0.994. 4. We come to the same conclusion analyzing the excitation mechanism based on a simple exchange reaction
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16 July I993
CHEMICALPHYSICSLETTERS
A+BC-+AB+C+Q
(11)
for which the approach developed above leads to the expressions
(17)
(12)
m’m2
’ 2nh2(m, (13) the value of k=4 for an isochoric, and k=6 for an isobaric process. Here,. ml, m2, m3 are the masses of atoms A, B, C; B,, B2 are the rotational constants for molecules BC and.AB, respectively. As can be seen, the small value of the relation T,JQ implies a conclusion about the insignificant limitation of laser efficiency in this case too. So for the reaction:, 0 + CS- CO + S, which is successfully used for laser pumping [2,4], the relation To/Qz8x 10m3, and expression ( 13) gives ~,,,zO.995. Now let us consider just another laser excitation mechanism, where the energy is provided with a recombination reaction,
where
AtB+ABtQ.
(14)
The above mechanism is not used in practice yet, but is intensively discussed in the literature at the present time [ 6-8 1. For this kind of reaction, we have
E. = 3NT, .
(15)
To derive the entropy in the final state, it should be taken into account that the last part of inequality (6) can be violated in this situation, where, as a rule, T, B fiw. Considering, for simplification, the ultimate condition T, XPfiw, we obtain
-1
T,(m, tm2) “* T: 2diz > Bftw ’
St = 5N+2Nln
E, =;NT,
,
which results in the following restrictions:
(16)
r2$?+
(18)
tm2)
Here, the values of k= 7, K= 3 are for an isochoric, and k=9, K=5 for an isobaric process. In this situation, the thermodynamic restriction depends on the density of the reacting compounds, which is caused by the change of the number of particles due to the reaction. The role of the above restriction increases gradually as the density drops, and, according to the expression ( 18)) its value can reach several tens percent. It should be noted that the assessment of T, and fulfilled above is rather rough. That is conman rt nected not only with the approximate character of conditions (6), but also with the inaccurate assumption of an absence of dissociating product molecules in the final state. Those assumptions are made to obtain qualitative estimations and, in general, can be violated, especially for recombination lasers. Besides, the role of neutral compounds which do not contribute to the chemical reaction, but are usually present in gas mixture, should have been explored. The more strict and detailed approach taking the above circumstances into account is developed beneath. 5. To analyze the thermodynamic restrictions for chemical lasers in general, we consider a thermo-insulated system opened for laser radiation output only, and consisting of a mixture of exothermically reacting gas compounds and some neutral gases, which is enclosed in a reservoir. The initial quantities of reagents are regarded to be in stoichiometric relation. This assumption permits us to simplify the calculations significantly without loss of generality, because if some reagents are in excess over the stoichiometric quantity, then the remainder can be conditionally attributed to a neutral gas. Let the initial temperature of the gas and reservoir be To, and 527
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the final state reported to complete thermodynamic and chemical equilibrium at the temperature T,, the energy of the laser radiation being removed from the system. The correlation between the temperatures T, and T,, is derived from the energy balance equation,
_- $ (1 i
16 July 1993
Vj
In V,(-
1
Vi
In V,)
i
_nOA!]n!!!
’
(22)
AS:(T) = 1 VjJp(T) - 1 v~s:( T) i j
(23)
I
(19) where the specific heat of reaction is
Q(T)= 1 v;h~(T)I
z v/i;(T) i
hy are the specific enthalpies of compound formation, E is the laser radiation energy per one molecule of reagent, ny is the initial molar concentration of one of the reagents, in terms of which the concentrations np and nj of other reagents and products are expressed, N, is the net number of moles in the initial mixture, < is the reagent conversion fraction in the final state (0 <<< I), c,, are the specific heat capacities of components, CR is the heat capacity of the reservoir, v are stoichiometric factors; here are below indices i, j, k refer to reagents, products and neutral compounds, respectively. The process is assumed to be isobaric. The difference between the net entropy of the system in the final and initial states, which has to be positive, is described by the following expression:
+zln?, o
(20)
To
where b,(T,)=hf(Ti)-Avlnp +$F(nt,:)+~(l-0, I
528
v,
v,
Here, A$‘( T) is the specific entropy of reaction, s”( T) are the specific entropies of compounds at the standard pressure (1 atm), p is the relation of the mixture pressure to its standard value, Au is the change of the number of moles in reaction. The first term in expression (20) describes the change of entropy stipulated by the qualitative change of the mixture composition due to chemical transformation, and the two last terms display the entropy increase caused by heating the mixture and reservoir. In turn, the expression (2 1) for A.sr,beside the specific entropy of reaction (23 ), is composed of the term connected with the dependency of translational entropy upon specific volume of a component (which contributes only in the case of reactions changing the number of particles), the function F(ny, r) describing the difference between entropies of mixing in the initial and final mixtures, and the term being responsible for incomplete combustion of reagents. The latter, as a rule, is negligible for highly exothermic reactions for which <- 1. The net contribution of the terms pointed out is of the order of unity (i.e. of order of the gas constant R, as is the convention here), while the value of As: can significantly exceed unity by its modulus. Therefore the value of As, is determined mainly by the kind of reaction and, to a lesser degree, by the conditions of its proceeding. 6. The relevance of the thermodynamic restrictions under one or another method of chemical laser excitation is connected with the sign of the value of As,. Indeed, as the energy of laser radiation increases, the amount of chemical energy available for heating the mixture vanishes, and the net change of entropy of the system is mainly determined by the first term in expression (20). If Ass,>0, then no restrictions related to the second law of thermody-
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CHEMICAL PHYSICSLETTERS
namics exist. Otherwise, the sum of the two last terms in expression (20) can decrease only down to a certain magnitude in order to keep the net entropy change positive. Just the latter case implies a limitation to the radiation energy. The maximum efflciency value can be found from the ultimate condition A.S=O, considering expression (20) and equation ( 19) combined with atomic balance equations and the e’quation linking the conversion fraction < with equilibrium concentrations of components in the final state. In common use, determining qIllaxby this procedure requires numerical computing. Nevertheless, the manner in which the maximum efficiency magnitude depends on the kind of reaction and initial conditions, can be generally demonstrated by example of an isothermal situation. Here the equations pointed out are split and can be solved analytically if the heat capacity of the reservoir is assumed to approach infinity. Eliminating the terms of first order in the vanishing virtual increment of temperature 6T in expressions (19) and (20), and neglecting those of higher order, we come to the following relation:
(24) Apparently, because of the small value of the relation T,/Qc 1, the thermodynamic restrictions can play a considerable role only for reactions with 1A$1 x- 1. Foremost, those are reactions reducing the number of particles, the simple example ( 14) for which has been considered above. Indeed, states with energy in continuous spectrum have the most statistical weight, which corresponds to the greatest contribution of translational degrees of freedom to entropy in comparison with internal ones. Thus, if a reaction causes the number of translational degrees of freedom to be reduced (and, obviously, the number of internal ones to be increased), then the specific entropy for such a reaction will be negative. Just the same situation takes place in reactions with reduction of the number of particles. The appropriate value of As: can roughly be estimated, using a perfect gas approximation, and also ignoring the contribution of internal degrees of freedom as welI as the
16July 1993
weak dependency on temperature, to be of order AvsEz+ 1, wheresOv z 20 is the translational entropy of a perfect gas. By contrast, specific entropies for reactions that do not change the number of particles can have any sign, and, as a rule, their moduli are significantly less than the value pointed out. For simple examples (3) and ( 1 1), this has been demonstrated above. The results obtained can be readily interpreted using a simple model of a chemical laser. Since a twolevel approach fails to describe the cw generation [ 4 1, a three-level consideration should be assumed as the simplest model conserving all features of the effect involved. This model consists of the upper and Iower laser levels (marked as 2 and 1 below), and the ground level. Assume that a chemical pumping reaction of rate R causes the formation of excited molecules. It results in an inverted population and cw amplification on the 2-l transition provided the following condition is met: &,>Noexp(-A&/T),
(25)
where 7sp is the spontaneous radiation time, N,-,is the ground level population, and A.!?,,,is the difference of the lower laser and ground level energies (the statistical weights of the levels are assumed to be unity). In accordance with the analysis fulfilled above, the value A&, is confined from below in such a way that the entropy of the final state does not prove to be less than the entropy of the initial state of the system. Otherwise, for instance, if A&,, 5 T, the population of the lower laser level N,-%exp(-A&IT)-& and, under the laser working conditions (25), the population of the upper laser level N2 is of the same order of magnitude. In its turn, that intensifies a reverse reaction, causing a shift of equilibrium for the pumping reaction, and thus preventing the formation of an inverted population, As a trial application of the theory developed, we have computed the maximum chemical efficiency for a photo-recombination chlorine chemical laser [ 681. The method of numerical computation is based on the simultaneous solution of eqs. (I9), (20) combined with the equilibrium balance equation by the manner described above. Values of efficiency obtained under various pressures and degrees of dis529
CHEMICAL PHYSICS LETTERS
Volume 209, number 5,6
solution of the initial mixture by Ar are exhibited in fig. 1 (the initial temperature is assumed to be room temperature), The decrease of chemical efficiency in the beginning of every curve is connected with adi-
loI 1
0.8
16 July 1993
abatic heating of the mixture; the decline at high dissolution degrees, when the situation is close to isothermal, is explained by the behaviour of the function F(np, i) describing the change of the entropy of mixing. As can be seen, the thermodynamic restrictions of efficiency for similar lasers prove to be quite essential. Perhaps this is one of the reasons for the lack of sufficiently reliable photo-recombination lasers at the present time.
References
c OO
10
Li -2
10
-’
1
10
dissolution
10
2 lo3
10' 10i 10B 10;
[Ar]:[CI]
Fig. 1. The efficiency of a Cl, photo-recombination laser in dependence on the degree of dissolution of the initial mixture with argonatvariouspressures (atm): (A) 10e4; (B) IO-‘; (C) IO-*; (D) 10-‘;(E) 1.
530
[ 1 ] M.A. Leontovich, Soviet Phys. Uspekhi 114 ( 1974) 555. [ 21 A.V. Eletskii, Soviet Phys. Uspekhi 134 (I 981) 237. [3] R.W.F. Gross and J.F. Bott, eds., Handbook of Chemical Lasers (Wiley, New York, 1976). [4] A.S. Bashkin, V.I. Igoshin, A.N. Oraevskyand V.A. Scheglov, chemical lasers, ed. N.G. Basov (Nauka, Moscow, 1982) (in Russian). [ 51 L.D. Landau and E.M. Lifshitz, Statistical Physics (Nauka, Moscow, 1976) (in Russian). 161 V.A. Kochelap, LA. Izmailov, L.A. Kemashitskii and V.V. Naumov, Study of the recombination gasdynamic lasers on electronic transitions (Visible GDL). Current Status, J. Phys. IV, Colloq. C7, Suppl. au J. Phys. III (199 I ) 637. [7] L.A. Kernashitskii, V.E. Nosenko, V.V. Naumov, V.A. Kochelap and LA. Izmailov, Chem. Phys. Letters I 16 ( 1985) 197. [S] P. Prigent and H. Brunet, SPIE 1031 ( 1988) 408.
16 July 1993
CHEMICAL PHYSICS LETTERS
Thermodynamic restrictions for chemical laser efficiency A.V. Eletskii and E.V. Stepanov institute ofApplied Chemical Physics, Russian Science Center “Kurchatov Institute”, Kurchatov Square, 123182 Moscow, Russian Federation
Received 3 1 March 1993; in final form 17 May 1993
A thermodynamic approach to the problem of the maximum effkiency for a chemical laser is developed. Assuming the laser radiation entropy to he negligibly small in comparison with the entropy of the chemically reacting compounds, the thermodynamic limitations for the laser efficiency are obtained. A number of kinds of chemical reactions, each one being a source of laser pumping, are explored. It is shown that the most significant restrictions appear for reactions of recombination type reducing the number of particles in the laser mixture. Photorecombination lasers are investigated in detail. The maximum efficiency for those lasers is computed over a wide range of proposed parameters of the initial mixture.
1. From the thermodynamic viewpoint, a chemical laser is a transformer of the energy released as a result of an exothermic chemical reaction into coherent radiation energy. Obviously, this transformation is accompanied by the loss of some part of the energy for heating, so that the efficiency of such a transformation is always less than unity. It is easy to show [l] that if the number of laser radiation quanta per one elementary act of the chemical reaction is about unity or less, the entropy of coherent radiation carrying away the beneficial part of the transformed energy is negligibly small in comparison with the entropy of the reagents and reaction products. This is caused by the small angular divergence and the small relative frequency width for laser radiation. Thus, a chemical laser should be considered as a device for converting a chemical energy to an energy whose carrier is characterized by negligibly small entropy. Under this consideration, the second law of thermodynamics should be taken into account, whereby the entropy change for a system as a result of any energy transformation must be positive. For a chemical laser, this means that removing laser radiation cannot be attended by a decrease in the entropy of the compounds. In turn, it is possible only if the chemical reaction itself causes the entropy to increase. Otherwise, a part of the chemical energy has to be expended for heating the medium to make
the net entropy change positive. The latter case, apparently, is accompanied by a limitation to the maximum energy of the issuing radiation, or the effrciency of the chemical laser. In the present work, this kind of restriction is explored theoretically for various methods of laser pumping. 2. Let us define the chemical efficiency of a chemical laser by the usual relation: ?I= EJQ, where E, is the energy of the laser radiation, Q is the energy released in the chemical reaction used as a base for the laser involved. According to the energy conservation law, the expression for the efficiency can be written in the form I/=1-
E, -&I +A
Q
(E,, is the energy of the system until the chemical reaction has begun, Et is the energy of the reaction products, A is the value of the work done upon the products by external forces). The expression ( 1) is to be calculated assuming that the values of entropy before and after reaction, S, and S,, are linked to one another by the following relation: so
(2)
where S, is the entropy of laser radiation, considered to be negligible in comparison with S, and S,.
0009-2614/93/S 06.00 0 1993 Elsevier Science Publishers B.V. AU rights reserved.
525
CHEMICAL PHYSICSLETTERS
Volume209,number5,6
Thus, the problem should be formulated as follows: what is the maximum amount of the energy released as a result of the chemical reaction, which can be removed from the system by radiation in such a way that the entropy of the issuing radiation is negligibly small, and the entropy of the reaction products is not less than the entropy of the reagents? The response to this question depends on the kind of reaction involved. 3. We discuss initially some simple examples which do not require cumbersome calculations. For the first example, we consider a reaction Az+B2+2AB+Q,
(3)
which is in wide use as a chemical laser pumping source [ 2-41. Assume that the initial quantities of reagents are equal to one another, and the reaction is accomplished entirely, exhausting all the reagents. In terms of the canonical distribution [5], the expressions for energy and entropy for a two-atom molecular gas have the forms
X,~& I /
xexp
T
)I
'
j(j+ 1)B I Awv - ~ T
T
I
’
(10) (5)
where N is the quantity of reagents (AZ and B,), B is the molecular rotational constant, fiw is the vibrational quantum, j, v are rotational and vibrational quantum numbers. (Here and below, the gas constant R is omitted in calculations, so that temperature will be expressed in energy units, and specific heat capacity and entropy will be dimensionless.) Carrying out straightforward manipulations of (4) and (5), and taking into account the condition usually met:
526
where To (T,) and V,, ( V,) are the temperature and volume of the system before (after) the reaction, Bi, B2 and B3 are the rotational constants for AZ,B2 and AB molecules, and m,, m, and m3 are their masses, respectively_ The final temperature of the system T, as well as the work value A depend on the current conditions for the reaction being analyzed. To be more specific, let us consider isochoric and isobaric processes. In the former case, V, = Vuand A =O, and in the latter one, V, = VoT, / To and A = 2N( T, - To). For these processes, the condition S, BS,, is reduced to the relation
(4)
C C (2j+l) c ‘30 v>o
E=sNT+NT2$ln xexp
T
we derive the following:
(k= 5 for an isochoric, and k= 7 for an isobaric process), whence it is easy to obtain the limitation for the efficiency magnitude,
(2jfl)
j(j+l)B+% - ~
16July I993
Obviously, this limitation is valid only if the expression in braces is positive. The opposite case refers to an increase of entropy in the reaction itself, and no limitations occur. Under the conditions suitable for a chemical laser, the relation T,JQ<<: 1. This means that the restrictions (10) are rather insignificant. Indeed, for the reaction: Hz+Fz-12HF, which is widely used as a laser pumping source [2,4], the relation To/ Qz 5 x 10m3, implying a maximum efficiency magnitude of 0.994. 4. We come to the same conclusion analyzing the excitation mechanism based on a simple exchange reaction
Volume 209, number 5,6
16 July I993
CHEMICALPHYSICSLETTERS
A+BC-+AB+C+Q
(11)
for which the approach developed above leads to the expressions
(17)
(12)
m’m2
’ 2nh2(m, (13) the value of k=4 for an isochoric, and k=6 for an isobaric process. Here,. ml, m2, m3 are the masses of atoms A, B, C; B,, B2 are the rotational constants for molecules BC and.AB, respectively. As can be seen, the small value of the relation T,JQ implies a conclusion about the insignificant limitation of laser efficiency in this case too. So for the reaction:, 0 + CS- CO + S, which is successfully used for laser pumping [2,4], the relation To/Qz8x 10m3, and expression ( 13) gives ~,,,zO.995. Now let us consider just another laser excitation mechanism, where the energy is provided with a recombination reaction,
where
AtB+ABtQ.
(14)
The above mechanism is not used in practice yet, but is intensively discussed in the literature at the present time [ 6-8 1. For this kind of reaction, we have
E. = 3NT, .
(15)
To derive the entropy in the final state, it should be taken into account that the last part of inequality (6) can be violated in this situation, where, as a rule, T, B fiw. Considering, for simplification, the ultimate condition T, XPfiw, we obtain
-1
T,(m, tm2) “* T: 2diz > Bftw ’
St = 5N+2Nln
E, =;NT,
,
which results in the following restrictions:
(16)
r2$?+
(18)
tm2)
Here, the values of k= 7, K= 3 are for an isochoric, and k=9, K=5 for an isobaric process. In this situation, the thermodynamic restriction depends on the density of the reacting compounds, which is caused by the change of the number of particles due to the reaction. The role of the above restriction increases gradually as the density drops, and, according to the expression ( 18)) its value can reach several tens percent. It should be noted that the assessment of T, and fulfilled above is rather rough. That is conman rt nected not only with the approximate character of conditions (6), but also with the inaccurate assumption of an absence of dissociating product molecules in the final state. Those assumptions are made to obtain qualitative estimations and, in general, can be violated, especially for recombination lasers. Besides, the role of neutral compounds which do not contribute to the chemical reaction, but are usually present in gas mixture, should have been explored. The more strict and detailed approach taking the above circumstances into account is developed beneath. 5. To analyze the thermodynamic restrictions for chemical lasers in general, we consider a thermo-insulated system opened for laser radiation output only, and consisting of a mixture of exothermically reacting gas compounds and some neutral gases, which is enclosed in a reservoir. The initial quantities of reagents are regarded to be in stoichiometric relation. This assumption permits us to simplify the calculations significantly without loss of generality, because if some reagents are in excess over the stoichiometric quantity, then the remainder can be conditionally attributed to a neutral gas. Let the initial temperature of the gas and reservoir be To, and 527
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the final state reported to complete thermodynamic and chemical equilibrium at the temperature T,, the energy of the laser radiation being removed from the system. The correlation between the temperatures T, and T,, is derived from the energy balance equation,
_- $ (1 i
16 July 1993
Vj
In V,(-
1
Vi
In V,)
i
_nOA!]n!!!
’
(22)
AS:(T) = 1 VjJp(T) - 1 v~s:( T) i j
(23)
I
(19) where the specific heat of reaction is
Q(T)= 1 v;h~(T)I
z v/i;(T) i
hy are the specific enthalpies of compound formation, E is the laser radiation energy per one molecule of reagent, ny is the initial molar concentration of one of the reagents, in terms of which the concentrations np and nj of other reagents and products are expressed, N, is the net number of moles in the initial mixture, < is the reagent conversion fraction in the final state (0 <<< I), c,, are the specific heat capacities of components, CR is the heat capacity of the reservoir, v are stoichiometric factors; here are below indices i, j, k refer to reagents, products and neutral compounds, respectively. The process is assumed to be isobaric. The difference between the net entropy of the system in the final and initial states, which has to be positive, is described by the following expression:
+zln?, o
(20)
To
where b,(T,)=hf(Ti)-Avlnp +$F(nt,:)+~(l-0, I
528
v,
v,
Here, A$‘( T) is the specific entropy of reaction, s”( T) are the specific entropies of compounds at the standard pressure (1 atm), p is the relation of the mixture pressure to its standard value, Au is the change of the number of moles in reaction. The first term in expression (20) describes the change of entropy stipulated by the qualitative change of the mixture composition due to chemical transformation, and the two last terms display the entropy increase caused by heating the mixture and reservoir. In turn, the expression (2 1) for A.sr,beside the specific entropy of reaction (23 ), is composed of the term connected with the dependency of translational entropy upon specific volume of a component (which contributes only in the case of reactions changing the number of particles), the function F(ny, r) describing the difference between entropies of mixing in the initial and final mixtures, and the term being responsible for incomplete combustion of reagents. The latter, as a rule, is negligible for highly exothermic reactions for which <- 1. The net contribution of the terms pointed out is of the order of unity (i.e. of order of the gas constant R, as is the convention here), while the value of As: can significantly exceed unity by its modulus. Therefore the value of As, is determined mainly by the kind of reaction and, to a lesser degree, by the conditions of its proceeding. 6. The relevance of the thermodynamic restrictions under one or another method of chemical laser excitation is connected with the sign of the value of As,. Indeed, as the energy of laser radiation increases, the amount of chemical energy available for heating the mixture vanishes, and the net change of entropy of the system is mainly determined by the first term in expression (20). If Ass,>0, then no restrictions related to the second law of thermody-
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CHEMICAL PHYSICSLETTERS
namics exist. Otherwise, the sum of the two last terms in expression (20) can decrease only down to a certain magnitude in order to keep the net entropy change positive. Just the latter case implies a limitation to the radiation energy. The maximum efflciency value can be found from the ultimate condition A.S=O, considering expression (20) and equation ( 19) combined with atomic balance equations and the e’quation linking the conversion fraction < with equilibrium concentrations of components in the final state. In common use, determining qIllaxby this procedure requires numerical computing. Nevertheless, the manner in which the maximum efficiency magnitude depends on the kind of reaction and initial conditions, can be generally demonstrated by example of an isothermal situation. Here the equations pointed out are split and can be solved analytically if the heat capacity of the reservoir is assumed to approach infinity. Eliminating the terms of first order in the vanishing virtual increment of temperature 6T in expressions (19) and (20), and neglecting those of higher order, we come to the following relation:
(24) Apparently, because of the small value of the relation T,/Qc 1, the thermodynamic restrictions can play a considerable role only for reactions with 1A$1 x- 1. Foremost, those are reactions reducing the number of particles, the simple example ( 14) for which has been considered above. Indeed, states with energy in continuous spectrum have the most statistical weight, which corresponds to the greatest contribution of translational degrees of freedom to entropy in comparison with internal ones. Thus, if a reaction causes the number of translational degrees of freedom to be reduced (and, obviously, the number of internal ones to be increased), then the specific entropy for such a reaction will be negative. Just the same situation takes place in reactions with reduction of the number of particles. The appropriate value of As: can roughly be estimated, using a perfect gas approximation, and also ignoring the contribution of internal degrees of freedom as welI as the
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weak dependency on temperature, to be of order AvsEz+ 1, wheresOv z 20 is the translational entropy of a perfect gas. By contrast, specific entropies for reactions that do not change the number of particles can have any sign, and, as a rule, their moduli are significantly less than the value pointed out. For simple examples (3) and ( 1 1), this has been demonstrated above. The results obtained can be readily interpreted using a simple model of a chemical laser. Since a twolevel approach fails to describe the cw generation [ 4 1, a three-level consideration should be assumed as the simplest model conserving all features of the effect involved. This model consists of the upper and Iower laser levels (marked as 2 and 1 below), and the ground level. Assume that a chemical pumping reaction of rate R causes the formation of excited molecules. It results in an inverted population and cw amplification on the 2-l transition provided the following condition is met: &,>Noexp(-A&/T),
(25)
where 7sp is the spontaneous radiation time, N,-,is the ground level population, and A.!?,,,is the difference of the lower laser and ground level energies (the statistical weights of the levels are assumed to be unity). In accordance with the analysis fulfilled above, the value A&, is confined from below in such a way that the entropy of the final state does not prove to be less than the entropy of the initial state of the system. Otherwise, for instance, if A&,, 5 T, the population of the lower laser level N,-%exp(-A&IT)-& and, under the laser working conditions (25), the population of the upper laser level N2 is of the same order of magnitude. In its turn, that intensifies a reverse reaction, causing a shift of equilibrium for the pumping reaction, and thus preventing the formation of an inverted population, As a trial application of the theory developed, we have computed the maximum chemical efficiency for a photo-recombination chlorine chemical laser [ 681. The method of numerical computation is based on the simultaneous solution of eqs. (I9), (20) combined with the equilibrium balance equation by the manner described above. Values of efficiency obtained under various pressures and degrees of dis529
CHEMICAL PHYSICS LETTERS
Volume 209, number 5,6
solution of the initial mixture by Ar are exhibited in fig. 1 (the initial temperature is assumed to be room temperature), The decrease of chemical efficiency in the beginning of every curve is connected with adi-
loI 1
0.8
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abatic heating of the mixture; the decline at high dissolution degrees, when the situation is close to isothermal, is explained by the behaviour of the function F(np, i) describing the change of the entropy of mixing. As can be seen, the thermodynamic restrictions of efficiency for similar lasers prove to be quite essential. Perhaps this is one of the reasons for the lack of sufficiently reliable photo-recombination lasers at the present time.
References
c OO
10
Li -2
10
-’
1
10
dissolution
10
2 lo3
10' 10i 10B 10;
[Ar]:[CI]
Fig. 1. The efficiency of a Cl, photo-recombination laser in dependence on the degree of dissolution of the initial mixture with argonatvariouspressures (atm): (A) 10e4; (B) IO-‘; (C) IO-*; (D) 10-‘;(E) 1.
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[ 1 ] M.A. Leontovich, Soviet Phys. Uspekhi 114 ( 1974) 555. [ 21 A.V. Eletskii, Soviet Phys. Uspekhi 134 (I 981) 237. [3] R.W.F. Gross and J.F. Bott, eds., Handbook of Chemical Lasers (Wiley, New York, 1976). [4] A.S. Bashkin, V.I. Igoshin, A.N. Oraevskyand V.A. Scheglov, chemical lasers, ed. N.G. Basov (Nauka, Moscow, 1982) (in Russian). [ 51 L.D. Landau and E.M. Lifshitz, Statistical Physics (Nauka, Moscow, 1976) (in Russian). 161 V.A. Kochelap, LA. Izmailov, L.A. Kemashitskii and V.V. Naumov, Study of the recombination gasdynamic lasers on electronic transitions (Visible GDL). Current Status, J. Phys. IV, Colloq. C7, Suppl. au J. Phys. III (199 I ) 637. [7] L.A. Kernashitskii, V.E. Nosenko, V.V. Naumov, V.A. Kochelap and LA. Izmailov, Chem. Phys. Letters I 16 ( 1985) 197. [S] P. Prigent and H. Brunet, SPIE 1031 ( 1988) 408.