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Temperature dependence of the probability of vibrational energy transfer between HF and F
Volume
14, number
1 May 1972
CHEMICAL PHYSICS LETTERS
1
TEMPERATURE OF VLBRATIONAL
DEPENDENCE OF THE PROBABILITY ENERGY TRANSFER BETWEEN I-IF AND Fi H.K. SHIN
Department of Chemistry fj, ihiversity ofNevada. Rerto, Nevada 89507, Received 9 February
USA
1972
The temperature dependence of the efficiency of vibrational energy transfer in the cohinear FH + F and HF + F collisions has been investigated over the temperature range 300” -1500°K by assuming the (8-4) inverse-power potential. The probability Pro(T) of vibrational de-excitation (1 + 0) is found to be always very large for FH + F compared to HF + F. At low temperatures?rc(T) for FH + F is large; as temperature increases, however, it decreases to a minimum value of =8 :: 10e3 at 900°K. and then slowly increases at higher temperatures. No such unusual temperature dependence is found for HF + F.
In recent years particular attention has been given to the problem of vibrational ener,v transfer in coiIision systems involving hydrogen fluoride molecules [ 141. The fluorides are important gases involved in CO2 chemical lasers for which continuous-wave laser action is achieved by energy released from chemical reaction [S, 61 . A detailed understanding of the operation of HF + CO2 (or DF f CO2) lasers requires knowledge of the rates at which the excited HF molecule exchanges its vibrational energy with atoms and molecules present in the system. One of the important species present in the system is the fluorine atom. In the present letter we report the temperature dependence of the probability of vi‘orational de-excitation (1 * 0) of the hydrogen fluoride molecules in the FH + F and HF + F collinear collisions over the tem-
perature range of 300” to 1500°K. To formulate the probability of vibrational transitions we assume the potential function to be an (8-4)
inverse-power form rather than the conventional (! 2-6) Lennard-Jones type whose repulsive part may be too steep for the present system. For the FH t F and HF + F collinear collisions, we thus express the 7 This work was supported by the Air Force Office of Scientific Research, Sf 64
Theoretical
Grant AFOSR-58- 1354 and 72-223 1.
Chemistry
Group Coptribr\tion
No.
1036.
potential energies by
(14 and
(lb) respectively, where D, re are the potential parameters, and rl, r2 denote the distances between the incident F atom and the H atom of the molecule and between the incident atom and the F atom of the molecule, respectively. ‘Ihe general expressions for the two distances
are
“2 , , , which, for the collinear alignments, reduce to r1 2 = f~I+x). Here r is the distance between the inrTYZJ\
‘I,2
= [r2 + 7; 1(&.V)2 - 2T2 l(d+x)‘cos~l
cident atom and the center of mass of the molecule, x is the displacement of the vibrational amplitude from the equilibrium bond distance d, 0 is the angle between the incident atom and the molecular axis, and ~1,2 = ~H,F/(~H+~F)We have chosen the r L4 dependence rather than rr6 for the attractive part to emphasize the importance of the attractive energy for rl > rle. For the present (5-4) form we estimate the equilibrium
distance between the incident atom and the center of mass of the molecule
as 1.5 8, and the energy param-
Volume 14, number 1
1 May 1972
CHE!+lICALPHYSICSLE’ITERS
Fig. 1. Contour plots of the potential energy u = UHF + UFF as a function of the orientation angle o and relative separation r between the center of mass of the HF molecule and the incident F atom. The large and small circles represent F and H, respectively. The dotted line connects the potential minima for given 0 and r.
eter D L as 6 kcal/mole from the equilibrium bond distance of HF, 0.917 A, and information on the hydrogen bond [7]. Note that the method used below for the calculation of vibrational transition probabilities is equally applicable to the (8-6) or any other type of interaction potentials. For HF + F, the two potential parameters are estimated as 2.5 a and 2 kcal/rnole. The totai energy u = U[rP + UP, at any orientation angle 19and relative separation rr can then be calculated. Fig. 1 shows the contour plots of the energy U for the oscillator which is in the ground state. The center of mass is taken as the origin. In the figure, for example, the potential energies at 30’ represent the interaction of the incident atom with the mo!ecule which is oriented at 30”. At 0 = O”, the collision partners are aligned collinearly as FH + F, and at 180” as L-IF-I-F_ Although we shall be concerned only with
these two types of collisions, the contour potential plots are useful in understanding the general behavior of the interaction energies. The figure shows a sharp increase in the overall potential energy in the neighborhood of the FH t F collinear orientation for r < 2.4 A. The strongest interaction comes from the orientation angle range of t30” to -30”. The depth of the potential well of U for FH f F is 7.95 kcal/mole which occurs at 2.4 A. These two parameters are significantly different from D, and rle given above due to the important contribution of the F-F interaction at the equitibrium separation. The distance r1 is a function of r and x, and for FH f F collisions, we shall express the potential energy given by eq. (la) in the form UI-IF(r 1) =.uL-&)
- F,,(r)x
,
(2)
i.e., the oscillator is driven by the perturbing force 65
Volume
14,
FHF(r). Since ‘r = I- - ~~(dix), can be exprcsscd simply by
the perturbing
force
In calculating the potential energy U shown in fig. I, it is found that UFF makes only a small contribution in the repulsion region. For example, at f = 2 A, UHF = 20.79 kcal/mole while UFI: is only i .017 kcal/mole. At r = 1.5 ~3, we find UHF = 5868 kcal/mole and UFF = 64.08 kcal/mole. In the formulation of vibrational transition probabilities for FH + F presented below, we shall ignore the effect of the F-F interaction. Similarly, the transition probability for HF + F will be derived solely with UF,(r7)_ Note that at F-= 1.5 A. kcal/mole U FF = 1 17.9 kcal/mole and UHF = -1.769 for HF + F. In a recent letter [S] we have shown the derivation of the 0 + n vibrational transition probability from *be solution of the time-dependent wave equation for the perturbed oscillator. The straightforward application of the method to the present model gives the transition probability P On = ClIrt!)+
: May 1972
CHEMICAL PHYSICS LETTERS
number 1
exp(-Q
,
(4)
of vibrational energy transfer. The expression e. for the HF t F collinear collision takes the same form except that D r , -y2, and rle a;e replaced by D2, yl, and r2e, respectively. To investigate the temperature dependence of the probability of vibrational de-excitation (1 + 0), we shall average PO1 obtained above over a Boltzmann distribution of the initial energies:
PJT)
= @I-)_’
J 0
eO exp(-co-E/W)&!?
.
(6)
Because of a complicated dependence of eO on E, the integration may have to be carried out numericahy. The de-excitation probability Plo(T) can be calculated as PO1(T) exp (h/W). Numerical integrations often have the disadvantage of giving only particular solutions for given initial conditions. Even if the solutions are approximate, on the other hand, analytical soiutions can yield detailed information on the temperature dependence of PI”(T). In the temperature range of 300’ to 1500°K, L\E,, is significantly smaller than tlw, so that such an approximate solution can be obtained from eq. (6) by expanding exp(-eo) in a power series and taking the first several terms. We shall show such a solution here. Utilizing the expansion ~~ exp(-go)
= e. - 6: + ;E;
- i.5: f .__
5with leads to
Pal(T) = (k~)-l X exp
=A
D9/”
*56’le
,
1
lT(rz--:) 4 -f-(n) ET’2
c
[
a17l8
1’
n
-&?rr~)“*
az1t8
,-6L lo2dle
013/B 1
. .-.
and M, w are the reduced.m& and frequency cf the o&zillator, respectively. !n eq. (S), LtEHF is the amount
s
(e,-ei+iei-$i+...)
0 (7)
In each integral, we notice that as the energy increases the amount of energy transfer increases as factor rapidly exp(-const Em51a) while the Boltzmann decreases, so that there is an energy (say E*) at which the integrand takes a maximum value. For the first integral, namely J eO exp(-eo) exp(-E/kT)dE, or Pal, the maximum occurs at
66. I
._
., 1.
.‘.,
exp(-E/kT)dE
‘. ,.
Volume
E*
14, number
CHEMICAL PHYSICS LEmERS
;‘;[+$$qx(o))1’2
(0) = $0’ _ _
where x(o)
1
=
[
1
8
wa cm-oL’2D;‘80rlekT
R/13
re,>
I
.
The second-, third-, fourth-, . . . . integrals take the maximum values at energies which are of the same form as eq. (8) except that x(O) is replaced by 281 lJx(0) 9 38’1 3X(O) 4S/13x(O) 7 ---, respectively. The integrals can therefoie be evaluated by de?ermining the contribution from the neighborhood of E* with the Laplace method [9]. The result is
Fig. 2. Calculated values of the vibrational transition lity for FH f F as a function of temperature.
Xv where K is the pre-exponential
part of eo, namely
The factor exp(-fiw/2kT) in eq. (9) is due to the symmetrization of the relative energies (velocities). To obtain the probability appropriate to a de-excitation of the oscillator, Plo(T), rather than to an excitation, we need to replace this exponential factor by exp (+fiw/2kT). fn fig. 2 we plot the calculated values of Plo(T) for FH + F as a function of temperature from 300” to !500”K. We found that there is no significant difference between the values obtained from eq. (9) including the first four term of the i-sum and those obtained numerically up to 1300°K. We shall discuss
probabi-
the importance of the higher-order terms of the sum in a later paragraph. Although the numerical method is desirable at higher temperatures, we shall use eq. (9) throughout this work. [Note that for HF f F eq. (9) is entirely satisfactory because mFF
exponent changes only slightly over the temperature range considered, because the leading term is negative while the others are positive. At 400”, 600”, SOO”, 67
Volume 14, number 1
1 May 1972
CHEMICAL PHYSICS LE?TERS
lOOO”, 1200”, and 1 SOO”K, the exponential of P$) __ takes the values exp(-11.54), exp(-13.34), exp(--13.81), exp(-13.85), exp(--13.72), and exp(-13.42), respectively. Beyond SOO”K, the ex-
ponential part changes very little with temperature; this variation reflects a weak temperature dependence of Plo(T) above 500% as shown in fig. 2. A sharp rise in P,,(T) below 500°K is due to the contribution of attractive energies. We note that the pre-exponential part of Pi:) depends but little on temperature. The importance of the second- and higher-order ?erms in eq. (9) are included in the values plotted in fig. 2. At low temperatures, they are very small and can be neglected. For example, at 400°K P$j is 5.76 X 10-2, while the second term Ply is only 4.68 X IOm5. At lOOO”K, the two terms are, respectively, 8.67 X 10m2 and 3.70 X 10m4. Therefore, it is app;irent that at stilI higher temperatures, we must include more terms in eq. (9); e.g., at 1500”K, Pi? = 1.44 X 10-2,P$) = 2.72 X 10W3, andP# = 1.18 X 10S3. At temperatures above 2000”K, it appears that we must carry out the integration of eq. (9) numerically without the use of the expansion. It should be mentioned that when the effect of the F-F interaction is introduced into UHF, the factor [I - (o2/~l)(~llT2X’lel’2e)] 2 = 0.979 appears in the pm-exponential part, while the exponential part remains unchanged. Because of the strong attraction, the probability of energy transfer is so large that the above formulation may not be used at low temperatures (< 300’K). Such a strong attraction may lead to a “sticky” collision at low temperatures which would require new approaches to calculate energy transfer probabilities. As seen in fig. 2 P,,-,(T) calculated from eq. (9) tends
to exceed unity below room temperature, the situation which is physically unacceptable; the method developed by Sharma and Brau [lo] may be used for such low-temperature energy processes. The result for the HF + F collinear collision is shown in fig. 3. The transition probability is very small compared to FH + F over the entire temperature range considered. Furthermore, it.,always increases with temperature. At 400”K, PIG(T j =4.86 X IO-rl, and it increases to 3.87 X IO-8 at 1500°K. Since DEFF G&o, the second- and higher-order terms in eq. (9) are completely negligible compared to the fist. Therefore, the calculation for HI; c F is carried out
68
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”
”
m
”
”
”
”
’
T(‘K)lWO
Fig. 3. CaIcu!ated values of the vibrational transition probability for HF + F as a function of temperature. with P#. The pre-exponential part of P# now contains the factor (D2 rl /r2$ which is very small compared to that of the FH + F case. In addition, in the exponent’the leading term 2.6 x(O)/kT takes a significantly large value. For example, at SGO”K,2.6 x(O)/kT for HF + F is 42.25; it is 33.58 for FH + F. Although the second term 1.397(0, X@))“‘/ kT contributes to reduce the difference, ‘rhe net difference is still very lar e; e.g., at SOO%, we find the exponential part of P,,(8 to be exp(--27.67) for HF + F, while it is as large as exp(-12.73) for FH + F. At lSOO”K, the two exponential factors are exp(-21.77) and exp(-13.42), respectively. A large difference in the probabilities of energy transfer of FH f F and HF + F follows the general arguments presented in a previous paper on the problem of preferential orientations for vibrational transitions [l 1] . Because of strong attractive energies between the F and H atoms, it is likely that the colliding partners keep the preferential orientation of FH i F during collision. The above results show that such preferred orientation is very efficient for vibrational energy
transfer and that the temperature
dependence of the
Volume
14, number
1
CHEMICAL
transition probability for the collinear collision over the temperature range 300°-1500% is anomalous. I wish to thank Professor S.H. Bauer for suggesting the form of the interaction potential.
References [l] J.R.Airey ar.d S.F.Fried, Chem. Phys. Letters 8 (1971) 23. [2] W.C.SoIomon, J.A.Blauer, F.C.Jayc and J.G.Hnat, Intern. J. Chem. Kinetics 3 (1971) 215. [3] J.F.Bott and N.Cohen, J. Chem. Phys 55 (1971) 3698.
PHYSICS
1 iMay 1972
LETTERS
Chem. Phys. Letters 10 (1971) 81. ’ and T.J.Falk, Appt Phys Letters 15 (1969) 318; T.A.Cool and R.R.Stephens, J. Chem. Phys. 51(1969) 5175. [6] J.R.Airey and S.F.McKay, AppL Phys. Letters 1.5 (1969) 401; 17 j G.C.Fimentel and A.L.hlcClellan, The hydrogen bond (Freeman, San Francisco, 1960) ch. 9; W.C.Hamilton and J.A.Ibers, Hydrogen bonding in solids (Benjamin, New York, 1968). [ 81 H.K.Shin, Chem. Phys. Letters 5 (1970) 13 7. [4j H.K.Shin,
[S] T.A.Cool, R.R.Stephens
191 A.Erdhlyi,
Asymptotic
expansions
(Dover Publications.
New York, 1956) p. 36. [lo] R.D.Sharma and C. A.Brau, I. Chem. Phys 50 (1969) 924. [ 1 l] H.K.Shin, J. Chem. Phys 49 (1968) 3964.
69
14, number
1 May 1972
CHEMICAL PHYSICS LETTERS
1
TEMPERATURE OF VLBRATIONAL
DEPENDENCE OF THE PROBABILITY ENERGY TRANSFER BETWEEN I-IF AND Fi H.K. SHIN
Department of Chemistry fj, ihiversity ofNevada. Rerto, Nevada 89507, Received 9 February
USA
1972
The temperature dependence of the efficiency of vibrational energy transfer in the cohinear FH + F and HF + F collisions has been investigated over the temperature range 300” -1500°K by assuming the (8-4) inverse-power potential. The probability Pro(T) of vibrational de-excitation (1 + 0) is found to be always very large for FH + F compared to HF + F. At low temperatures?rc(T) for FH + F is large; as temperature increases, however, it decreases to a minimum value of =8 :: 10e3 at 900°K. and then slowly increases at higher temperatures. No such unusual temperature dependence is found for HF + F.
In recent years particular attention has been given to the problem of vibrational ener,v transfer in coiIision systems involving hydrogen fluoride molecules [ 141. The fluorides are important gases involved in CO2 chemical lasers for which continuous-wave laser action is achieved by energy released from chemical reaction [S, 61 . A detailed understanding of the operation of HF + CO2 (or DF f CO2) lasers requires knowledge of the rates at which the excited HF molecule exchanges its vibrational energy with atoms and molecules present in the system. One of the important species present in the system is the fluorine atom. In the present letter we report the temperature dependence of the probability of vi‘orational de-excitation (1 * 0) of the hydrogen fluoride molecules in the FH + F and HF + F collinear collisions over the tem-
perature range of 300” to 1500°K. To formulate the probability of vibrational transitions we assume the potential function to be an (8-4)
inverse-power form rather than the conventional (! 2-6) Lennard-Jones type whose repulsive part may be too steep for the present system. For the FH t F and HF + F collinear collisions, we thus express the 7 This work was supported by the Air Force Office of Scientific Research, Sf 64
Theoretical
Grant AFOSR-58- 1354 and 72-223 1.
Chemistry
Group Coptribr\tion
No.
1036.
potential energies by
(14 and
(lb) respectively, where D, re are the potential parameters, and rl, r2 denote the distances between the incident F atom and the H atom of the molecule and between the incident atom and the F atom of the molecule, respectively. ‘Ihe general expressions for the two distances
are
“2 , , , which, for the collinear alignments, reduce to r1 2 = f~I+x). Here r is the distance between the inrTYZJ\
‘I,2
= [r2 + 7; 1(&.V)2 - 2T2 l(d+x)‘cos~l
cident atom and the center of mass of the molecule, x is the displacement of the vibrational amplitude from the equilibrium bond distance d, 0 is the angle between the incident atom and the molecular axis, and ~1,2 = ~H,F/(~H+~F)We have chosen the r L4 dependence rather than rr6 for the attractive part to emphasize the importance of the attractive energy for rl > rle. For the present (5-4) form we estimate the equilibrium
distance between the incident atom and the center of mass of the molecule
as 1.5 8, and the energy param-
Volume 14, number 1
1 May 1972
CHE!+lICALPHYSICSLE’ITERS
Fig. 1. Contour plots of the potential energy u = UHF + UFF as a function of the orientation angle o and relative separation r between the center of mass of the HF molecule and the incident F atom. The large and small circles represent F and H, respectively. The dotted line connects the potential minima for given 0 and r.
eter D L as 6 kcal/mole from the equilibrium bond distance of HF, 0.917 A, and information on the hydrogen bond [7]. Note that the method used below for the calculation of vibrational transition probabilities is equally applicable to the (8-6) or any other type of interaction potentials. For HF + F, the two potential parameters are estimated as 2.5 a and 2 kcal/rnole. The totai energy u = U[rP + UP, at any orientation angle 19and relative separation rr can then be calculated. Fig. 1 shows the contour plots of the energy U for the oscillator which is in the ground state. The center of mass is taken as the origin. In the figure, for example, the potential energies at 30’ represent the interaction of the incident atom with the mo!ecule which is oriented at 30”. At 0 = O”, the collision partners are aligned collinearly as FH + F, and at 180” as L-IF-I-F_ Although we shall be concerned only with
these two types of collisions, the contour potential plots are useful in understanding the general behavior of the interaction energies. The figure shows a sharp increase in the overall potential energy in the neighborhood of the FH t F collinear orientation for r < 2.4 A. The strongest interaction comes from the orientation angle range of t30” to -30”. The depth of the potential well of U for FH f F is 7.95 kcal/mole which occurs at 2.4 A. These two parameters are significantly different from D, and rle given above due to the important contribution of the F-F interaction at the equitibrium separation. The distance r1 is a function of r and x, and for FH f F collisions, we shall express the potential energy given by eq. (la) in the form UI-IF(r 1) =.uL-&)
- F,,(r)x
,
(2)
i.e., the oscillator is driven by the perturbing force 65
Volume
14,
FHF(r). Since ‘r = I- - ~~(dix), can be exprcsscd simply by
the perturbing
force
In calculating the potential energy U shown in fig. I, it is found that UFF makes only a small contribution in the repulsion region. For example, at f = 2 A, UHF = 20.79 kcal/mole while UFI: is only i .017 kcal/mole. At r = 1.5 ~3, we find UHF = 5868 kcal/mole and UFF = 64.08 kcal/mole. In the formulation of vibrational transition probabilities for FH + F presented below, we shall ignore the effect of the F-F interaction. Similarly, the transition probability for HF + F will be derived solely with UF,(r7)_ Note that at F-= 1.5 A. kcal/mole U FF = 1 17.9 kcal/mole and UHF = -1.769 for HF + F. In a recent letter [S] we have shown the derivation of the 0 + n vibrational transition probability from *be solution of the time-dependent wave equation for the perturbed oscillator. The straightforward application of the method to the present model gives the transition probability P On = ClIrt!)+
: May 1972
CHEMICAL PHYSICS LETTERS
number 1
exp(-Q
,
(4)
of vibrational energy transfer. The expression e. for the HF t F collinear collision takes the same form except that D r , -y2, and rle a;e replaced by D2, yl, and r2e, respectively. To investigate the temperature dependence of the probability of vibrational de-excitation (1 + 0), we shall average PO1 obtained above over a Boltzmann distribution of the initial energies:
PJT)
= @I-)_’
J 0
eO exp(-co-E/W)&!?
.
(6)
Because of a complicated dependence of eO on E, the integration may have to be carried out numericahy. The de-excitation probability Plo(T) can be calculated as PO1(T) exp (h/W). Numerical integrations often have the disadvantage of giving only particular solutions for given initial conditions. Even if the solutions are approximate, on the other hand, analytical soiutions can yield detailed information on the temperature dependence of PI”(T). In the temperature range of 300’ to 1500°K, L\E,, is significantly smaller than tlw, so that such an approximate solution can be obtained from eq. (6) by expanding exp(-eo) in a power series and taking the first several terms. We shall show such a solution here. Utilizing the expansion ~~ exp(-go)
= e. - 6: + ;E;
- i.5: f .__
5with leads to
Pal(T) = (k~)-l X exp
=A
D9/”
*56’le
,
1
lT(rz--:) 4 -f-(n) ET’2
c
[
a17l8
1’
n
-&?rr~)“*
az1t8
,-6L lo2dle
013/B 1
. .-.
and M, w are the reduced.m& and frequency cf the o&zillator, respectively. !n eq. (S), LtEHF is the amount
s
(e,-ei+iei-$i+...)
0 (7)
In each integral, we notice that as the energy increases the amount of energy transfer increases as factor rapidly exp(-const Em51a) while the Boltzmann decreases, so that there is an energy (say E*) at which the integrand takes a maximum value. For the first integral, namely J eO exp(-eo) exp(-E/kT)dE, or Pal, the maximum occurs at
66. I
._
., 1.
.‘.,
exp(-E/kT)dE
‘. ,.
Volume
E*
14, number
CHEMICAL PHYSICS LEmERS
;‘;[+$$qx(o))1’2
(0) = $0’ _ _
where x(o)
1
=
[
1
8
wa cm-oL’2D;‘80rlekT
R/13
re,>
I
.
The second-, third-, fourth-, . . . . integrals take the maximum values at energies which are of the same form as eq. (8) except that x(O) is replaced by 281 lJx(0) 9 38’1 3X(O) 4S/13x(O) 7 ---, respectively. The integrals can therefoie be evaluated by de?ermining the contribution from the neighborhood of E* with the Laplace method [9]. The result is
Fig. 2. Calculated values of the vibrational transition lity for FH f F as a function of temperature.
Xv where K is the pre-exponential
part of eo, namely
The factor exp(-fiw/2kT) in eq. (9) is due to the symmetrization of the relative energies (velocities). To obtain the probability appropriate to a de-excitation of the oscillator, Plo(T), rather than to an excitation, we need to replace this exponential factor by exp (+fiw/2kT). fn fig. 2 we plot the calculated values of Plo(T) for FH + F as a function of temperature from 300” to !500”K. We found that there is no significant difference between the values obtained from eq. (9) including the first four term of the i-sum and those obtained numerically up to 1300°K. We shall discuss
probabi-
the importance of the higher-order terms of the sum in a later paragraph. Although the numerical method is desirable at higher temperatures, we shall use eq. (9) throughout this work. [Note that for HF f F eq. (9) is entirely satisfactory because mFF
exponent changes only slightly over the temperature range considered, because the leading term is negative while the others are positive. At 400”, 600”, SOO”, 67
Volume 14, number 1
1 May 1972
CHEMICAL PHYSICS LE?TERS
lOOO”, 1200”, and 1 SOO”K, the exponential of P$) __ takes the values exp(-11.54), exp(-13.34), exp(--13.81), exp(-13.85), exp(--13.72), and exp(-13.42), respectively. Beyond SOO”K, the ex-
ponential part changes very little with temperature; this variation reflects a weak temperature dependence of Plo(T) above 500% as shown in fig. 2. A sharp rise in P,,(T) below 500°K is due to the contribution of attractive energies. We note that the pre-exponential part of Pi:) depends but little on temperature. The importance of the second- and higher-order ?erms in eq. (9) are included in the values plotted in fig. 2. At low temperatures, they are very small and can be neglected. For example, at 400°K P$j is 5.76 X 10-2, while the second term Ply is only 4.68 X IOm5. At lOOO”K, the two terms are, respectively, 8.67 X 10m2 and 3.70 X 10m4. Therefore, it is app;irent that at stilI higher temperatures, we must include more terms in eq. (9); e.g., at 1500”K, Pi? = 1.44 X 10-2,P$) = 2.72 X 10W3, andP# = 1.18 X 10S3. At temperatures above 2000”K, it appears that we must carry out the integration of eq. (9) numerically without the use of the expansion. It should be mentioned that when the effect of the F-F interaction is introduced into UHF, the factor [I - (o2/~l)(~llT2X’lel’2e)] 2 = 0.979 appears in the pm-exponential part, while the exponential part remains unchanged. Because of the strong attraction, the probability of energy transfer is so large that the above formulation may not be used at low temperatures (< 300’K). Such a strong attraction may lead to a “sticky” collision at low temperatures which would require new approaches to calculate energy transfer probabilities. As seen in fig. 2 P,,-,(T) calculated from eq. (9) tends
to exceed unity below room temperature, the situation which is physically unacceptable; the method developed by Sharma and Brau [lo] may be used for such low-temperature energy processes. The result for the HF + F collinear collision is shown in fig. 3. The transition probability is very small compared to FH + F over the entire temperature range considered. Furthermore, it.,always increases with temperature. At 400”K, PIG(T j =4.86 X IO-rl, and it increases to 3.87 X IO-8 at 1500°K. Since DEFF G&o, the second- and higher-order terms in eq. (9) are completely negligible compared to the fist. Therefore, the calculation for HI; c F is carried out
68
G’
”
”
”
m
”
”
”
”
’
T(‘K)lWO
Fig. 3. CaIcu!ated values of the vibrational transition probability for HF + F as a function of temperature. with P#. The pre-exponential part of P# now contains the factor (D2 rl /r2$ which is very small compared to that of the FH + F case. In addition, in the exponent’the leading term 2.6 x(O)/kT takes a significantly large value. For example, at SGO”K,2.6 x(O)/kT for HF + F is 42.25; it is 33.58 for FH + F. Although the second term 1.397(0, X@))“‘/ kT contributes to reduce the difference, ‘rhe net difference is still very lar e; e.g., at SOO%, we find the exponential part of P,,(8 to be exp(--27.67) for HF + F, while it is as large as exp(-12.73) for FH + F. At lSOO”K, the two exponential factors are exp(-21.77) and exp(-13.42), respectively. A large difference in the probabilities of energy transfer of FH f F and HF + F follows the general arguments presented in a previous paper on the problem of preferential orientations for vibrational transitions [l 1] . Because of strong attractive energies between the F and H atoms, it is likely that the colliding partners keep the preferential orientation of FH i F during collision. The above results show that such preferred orientation is very efficient for vibrational energy
transfer and that the temperature
dependence of the
Volume
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CHEMICAL
transition probability for the collinear collision over the temperature range 300°-1500% is anomalous. I wish to thank Professor S.H. Bauer for suggesting the form of the interaction potential.
References [l] J.R.Airey ar.d S.F.Fried, Chem. Phys. Letters 8 (1971) 23. [2] W.C.SoIomon, J.A.Blauer, F.C.Jayc and J.G.Hnat, Intern. J. Chem. Kinetics 3 (1971) 215. [3] J.F.Bott and N.Cohen, J. Chem. Phys 55 (1971) 3698.
PHYSICS
1 iMay 1972
LETTERS
Chem. Phys. Letters 10 (1971) 81. ’ and T.J.Falk, Appt Phys Letters 15 (1969) 318; T.A.Cool and R.R.Stephens, J. Chem. Phys. 51(1969) 5175. [6] J.R.Airey and S.F.McKay, AppL Phys. Letters 1.5 (1969) 401; 17 j G.C.Fimentel and A.L.hlcClellan, The hydrogen bond (Freeman, San Francisco, 1960) ch. 9; W.C.Hamilton and J.A.Ibers, Hydrogen bonding in solids (Benjamin, New York, 1968). [ 81 H.K.Shin, Chem. Phys. Letters 5 (1970) 13 7. [4j H.K.Shin,
[S] T.A.Cool, R.R.Stephens
191 A.Erdhlyi,
Asymptotic
expansions
(Dover Publications.
New York, 1956) p. 36. [lo] R.D.Sharma and C. A.Brau, I. Chem. Phys 50 (1969) 924. [ 1 l] H.K.Shin, J. Chem. Phys 49 (1968) 3964.
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