VIBRATIONAL RELAXATION TIMES

OF GASEOUS MIXTURlS OF DIATOMIC

MOLtCULIS AND THEIR EFFECT

ON ROCKET PERFORMANCE

Harmon C. Penny

iii'

i

1

VIBRATIONAL RELAXATION TIMES

OF GASEOUS MIXTURES OF DIATOMIC MOLECULES

AND THEIR SFFECT ON ROCKET PERFORMANCE

S854on spine:

PENNY

1954

THESISP33

Letter on front cover:

VIBRATIONAL RELAXATICN TIMES OF

GASEOUS MIXTURES OF DIATOMIC

MOLECULES AND THEIR EFFECT C^N

RCC'rCST PERFORI'IANCE

HARi40N C. PENNY

vy

squiremente

California Institute of Technology

Pasadena, California

1954

ffhesis

(I

ACKNOWLEDGEMENT

The author wishes to thank Dr. S. S. Penner for helpful

discuasion and Dr. Henry Aroeste for active 'assistance, instruc-

tion, and helpful criticism in the preparation of this tliesis.

25056

-li-

SUMMARY

The present methods available for tlie determination of the

magnitudes of the vibrational relaxation tinies of molecules are

discussed and the recently developed theory of Schwartz, Slawsky,

and Herzfeld is used to compute the variation with temperature of

the collisional vibrational excitation probabilities of mixtures of

N2 and O,* and of H2 and HF. The resulto are used to estimate

the extent of vibrational temperature lag in the hydrogen-fluorine

rocket motor.

i

-ili-

TABLE OF CONTENTS

PART TITLE PAGE

1. INTRODUCTION 1

n. EXPERIMENTAL METHODS 3

A. THE SOUND VELOCITY METHOD 3

B. IMPACT TUBE METHOD 5

C. EXPERIMENTAL RESULTS 9

m. THEORETICAL CALCULATIONS 10

IV. VIBRATIONAL LAG IN THE HYDROGEN-FLUORINE ROCKET MOTOR 16

-1-

I. INTRODUCTION

The internal energy, E, of a gas inay be written, approxi-

mately, as the sunr] of energies of translation, rotation, vibration,

and electronic motion. That is to say, we write that

where T,T^ ,Ty , and Te represent, respectively, the absolute temp-

eratures associated witii the trans lational, rotational, vibrational,

and electronic energy states. Under some conditions these energy

states will be in equilibrium such that T'T^'T^-T^ . If , in the flow

of a gas, teiiiperature changes occur at rates sufficiently fast to ex-

ceed the time required for these internal energy states to readjust

themselves, the gas v/ill depart from its equilibrium partition of

energy.

The terzn 'Vibrational relaxation time" has come into use as

a measure of the time required to establish equilibrium witli respect

to the internal vibrations of the molecules. The other internal states

adjust themselves so rapidly that we can usually neglect tlieir relaxa-

tion times. The vibrational relaxation time, T » is defined by the

relation

dfv.bt'T/dt = T-*[^,;^(T)-E.aTv)]. (2)

Working fluids such as steam, air, and exhaust gases Iiave

considerable vibrational heat capacity at higli temperatures. If these

gases have relaxation times comparable with or shorter than the in-

tervals during which temperature changes occur in the gas, losses

<

<

i

must be expected. For turbine working fluids* Kantrowitz^ ' esti-

xnatee that in aorae cases the losses at high teir.peratures due to tiiis

adjustment la(;j can be comparabie with the losaes due to slcin fric-

tion. Unfortunately, there exists a scarcity of data on the magni-

tudes of the relaxation tinges of most gases and, without these values

being Icnown at various temperature levels, their effects cannot be

analyzed.

Various eaqserimental methods have been devised to measure

relaxation times of gases. Of these, only the two discussed here

have been used with any degree of success. A tliird method involv-

ing a shock tube has been proposed, but has as yet not actually been

used.

-3-

II. EXPERIMENTAL METHODS

A. THE SOUND VEIX)CITY METHOD^ ^^

The sound velocity method can be adapted for the determina-

tion of the relaxation tin:ie8 of gaaes up to temipcratures of the order

of 2300^K. The two essentiale are (a) the production of sound v/aves

of a definite frequency and (b) the accurate measurement of temper-

ature.

The apparatus, originated by Pierce* '» consists essentially

of a carbon tube which can be heated to a high temperature and in

which a stationary train of sound waves can be set up from a quartz

crystal vibrating picsso-electrically, A piston traverses the tube.

The position of the piston is read on an accurately graduated steel

rtile. This piston fxinctions as a reflector to set up a standing wave

systent in the tube. The amplitude of vibration of the crystal changes

as the reflector is moved along the tube. Resonance is detected by

observing the magnitude of the plate current of the oscillator which

maintains the quartz crystal in vibration. The half v/ave -length is

determined by recording the different positions of the reflector at

which resonance occurs. The temperature of the gas in the tube is

measured by sighting an optical pyrometer of the "disappearing fila-

ment'* type on to the face of the movable reflector. An electric fur-

nace surrounds tl:ie carbon tube so that the temperature may be ad-

justed to higher levels. Figure 1 shows the general layout of the ap-

paratus.

The velocity of sound, V, in the gas is computed from the

measured half-wave length in the tube amd from the Imown frequency

i

-4-

o£ vibration of the crystal. The ratio of specific heats is given by

the formula,

Y* v^h/rt. (3)

where M is the molecular weight, R is the gas constant, and T is

the absolute tenrperature. The correction necessary to the alK>ve

formula for departure fronn the perfect gas laws' is negligible, since

at the temperaturcB involved most gases are esnentially perfect

gases. From the observed data, an apparent ratio of specific heats,

which we may call 7', is coi"nputed from Eq. (3). Also from the

relationship,

Cv/R - 1/(^-1)>

(4)

an apparent specific heat at constant volunne, C , can be obtained.

The crystal used for the production of soiind waves, if cut

properly, can be made to vibrate in either the flexural mode or the

longitudinal mode by suitable electronic circuitry. The two fre-

quencies produced may vary by as much as 20, 000 ops or more.

The apparent specific heat obtained at each of these frequencies

(4)may differ considerably. I^neser^ ' has put forth the following ex-

planation for this phenomenon. As a sound wave of a given frequency

passes through a gas, a part of the vibrational energy fails to follow

the acoustic cycle, and the gas departs from its equilibriuni condi-

tion. In fact, a part of the vibrational energy vanishes from the

escpreasion for the adiabatic elasticity* thus giving an increased

i

-5-

sound velocity. Kneser's theory led to the following formula for

the apparent specific heat, derived from tlie velocity of sound data,

r - Cy -t- 4Trn T C4 , i5i

Cv t- 4tt n T Cdi

where Cv =* true specific heat

n s frequency of the sound wave

C4 = specific heat of translation and rotation (= ^^ for

a diatomic molecule).

The values of the apparent specific heats may be substituted in this

equation to give two equations at each temperature for the determina^

tion of T and Cy *

B. IMPACT TUBE METHOD

In 1942 Kantrowitz' ' proposed the impact tube method for

finding relaxation times and made sufficient measurements to veri-

fy that the experimental results checked with the theoretical calcu-

(5)lations that were available at the time. Griffith^ ' later extended

this work and in 1950 reported on the results of his investigations,

listing the relaxation times determined and expressing confidence

in the method.

In the impact tube method, gas at rest is allowed to expand

adiabatically through a nozzle, and the flow thus produced is brought

to rest again by compression at the nose of an impact tube (pitot

tube). The total head defect in the flow, as a measure of the en-

tropy increase, is directly associated with the heat-capacity lag of

the gas and can be used to determine the relaucation time. The pro-

cess is shown schematically in Figure 2.

I

(

-6-

The underlying principles involved may be summarized

briefly as follows. The equations of momentum, continuity* and

energy for a connpressible fluid with possible separate transla-

tional and vibrational temperatures, together with the standard

definition for entropy, may be combined to establish the rela-

tionships.

^4-CC,.bT,*^CpTvc,V2) 'O

(6)

(7)

The notation Cy;^ refers to the specific heat of vibrational

motion. We have then that C^\^- ^^Vib/'^*''»«b * *^^ ^ ^l» ^® ^®

specific heat at constant pressure excluding contributions from vi-

bration, we also have that C|» « C|p t C^ib •

(6)Using the theory of JLandau and Teller^ , the variation of

Cv\bTy\b with tizrie may be expressed as

on the basis of considerations involving molecular collisions, the

relaxation time is found to have the following functional dependence:

The quantities V , s, and m represent, respectively, vibrational

frequency, nnolectilar radius, and molecular mass.

The departure from equilibrium between vibrational and

I

.7.

translational temperatures was expressed by Kantrowitz as

e = C,-.^(T,;j,-T'). (10)

In this eaq^ertraent the changes in T are small compared to T itself,

and Eq. {6) is reduced to

**^ "l^T^ ' Z ^^ (H)

nr is taken as the average of TT^;^ over the whole process*

The total head defect Ah is measured as the pressure dif-

ference jp^ - p« , and since this difference is small compared to

b^ , the differential change in entropy, ds » ~ Rdp/p, naay be closely

approsdmated by

AS - AhR/po . (12)

Combining Eqs. (11) and (12) we obtain the relationship,

(13)

If the temperature is eliminated from EqB, (7), (3), and (10),

it is found that

+ - • flAIdt c;r Cp dt ^* '

For the fast stopping coimpression process at the nose of the impact

tube, the second terin of this expression becomes small in compari-

son with the other terms, and the chauige in c becomes

-8-

Since at the beginning of compression C is zero and the velocity

is q, , Eq. (15) becomes

Co ^(16)

With q = 0, Eq. (14) is reduced to

at

Cp6(H)

For instantaneous connpression the result is accordingly,

(13)

At tliis point it is convenient to introduce the following set

of dimensionless parameters:

t' ' <j.t/d , q' . <^/<,. ,

where d is the diameter of tlie impact tube.

If Eqs. (13), (13), and (19) are now combined, it is fotmd

that

(19)

Ah

Ah« — e dt .

(20)

In order to determine the value of C in this expression it is neces-

sary to solve Eq. (14) exactly, the integral being

€ = exp

J -^cj. J Ydtdt » (21)

which, stated in terms of the difnensionless parameters, leads to

the expression

±UUf.e. (22)

The integration of £q. (22) has been done numerically for

various values of T . The integration of Eq. (20) has been car-

ried out graphically, and froir these resulting values a cur'/e of

Ah/ Ah: v^« ^ ^^s been prepared. With this curve on hand an

e3q>eriznent may be conducted using any gas, and the relaxation

time may be determiined from the measured values of T\ , ^» »

atnd |»t . The values of Cy;^ , Cp , Cp , and R for the gas can

be found by the use of well-Icnovm statistical raethods.

The procedure then is as follows. ^y

a) Calculate T7 from the relationship, ^ " *^ \9«/Po) •

b) Calculate <^, from the relationship, c^i aI ^ Cp vJI©" *» )j

c) Calculate Ah,*^ from Liq. (18).

d) Calculate ^\\ from the relationship, AK = l^o ""P3 •

e) Use the ratio of aK/aKj^ to enter into the graph of

aU/aK*^ vs. T to find the corresponding value for T .

f ) Calculate the relaxation time, T , from the relationship,

T - r'Cpd/Cp<^, .

C. EXPERIMENTAL RESULTS

Table I lists tiie values of the relaxation times of twenty gases

determined experimentally by various Investigators. The informa-

tion in this table was obtained, for the most part, from Reference (5).

I

i

-10-

m. THEORETICAJ^ CAL-CULATIONS

The accumulation of a large body of experimental data led to

efforts on the part of investigators to determine vibrational relaxa*

tion times using theoretical calculations. The n:iethod of Landau and

Teller* ', developed in 1936, and refined by Bethe and Teller* ' in

1940, has received general acceptance. A more detailed method de>

vcloped by Schwartz, Slav/sky, and Herzfeld* * ' was published in

1952. This method of calculation is based on the fact that an equilib-

rixim between the various degrees of freedom can be reached only

by means of a transfer of energy taking place during a molecular col-

lision. Thus for a pure gas it can be said that

T '"^-.^P' (23)

where M^ ^ is the number of collisions a molecule experiences per

second and is proportional to the density, while P is the probability

that upon a collision a quantum of vibrational energy will be converted

to translational energy. For a gas composed of polyatomic molecrJes

or nnixtures of different types of molecules, there arises the possibility

of atn energy exchange between the different vibrational degrees of

freedom during a collision as well as exchanges v/ith the translational

degrees of freedonri, and a separate relaxation time must be ascribed

to each of these energy transfers.

The tlieory of Schwartz, Slawsky, and Herzfeld will be used in

this paper to compute the variation of vibrational excitation probabi-

lities with temperature of mixtures of O^ and N,* and of HF and H,*

For a mixture of molecules of type a and b, following the notation of

-11-

Schwartz et al, we may define the probabilities P \^Jl^ ,

In each of these cases, the molecule, the symbol of which appears

first in the parenthesis, was initially in the first excited vibrational

state and finally in the ground or zeroth vibrational state. The sub-

script designates the behavior of the vibrational quantum number o£

the molecule, the symbol of which appears second in the parenthesis.

Each of these probabilities has been tabulated and graphed as func-

tions of temperature for both the O2 - N^ and HF - H^ gaseous mix-

tures.

The probability P Cb,d^ may be obtained from the rela-

tion

The others arc deterxnined independently. These probabilities may

then be used to obtain the relaxation times from the relations

and

(26)^^^-Ma>P,.oC^b)(.-e'''*/^'^)

Here ^^a ^'^^ ^a b denote the number of collisions of tlie designated

type per second. T.e may define similarly T (b,b) and T (b,a).

For the complex excitation process we may write

-12-

with a similar definition for f^^C**i^ Here IC^andX^are respec-

tively the mole fractions of species a and species b in the mixture.

It should be noted that only under special circumstances caoi the re-

laxation times defined above be grouped together to represent an ef-

fective relaxation time which corresponds to a measured relaxation

time.

In general, we may write for the probabilities

P ^ CP' , (28)

where

(31)

(32)

(33)

and

^ ^ \1.5(34)

Here ia is the reduced mass of the two colliding molecules, AE.

represents the amount of energy exchanged between vibration and

translation^ Y^ and £ are the usual constants in the Lennard-Jones

interaction potential, and V(i) is the customary perturbation

I

-13-

integral over harmonic oscillator wave functions. For molecules

whose value of £ is not too great, it is sufficient to write

OC » l7.5^ro . In the case of HF, where sufficient virial

coefficient, or transport data are not available, the collision di-

ameter, iTe t v^as estimated by plotting V^ as a function of the

electronegativity of the halogen atoms. From Fig. 3, it can be

seen that the value of Ya for HF is approximately 1. 7A.

The condition

|AV*ye > /^E (35)

will not hold below a certain temperature for each of the probabili-

ties. We need this inequality to write the expansion

It is not necessary that

v*V^ ^^ ^EK'o

although the larger the inequality, the better the convergence.

Schwartz and Herzfeld, in the appendix of their paper, have given

a more complex method of computation which applies in the region

of or below the limit temperatures defined by Eq. (35).

The following derivation applies for the determination of the

temperature limit for which the basic equations used here will be

applicable. The condition required by inequality (35) may be re-

stated as ^ /^ a .r / V'**-

If we combine this expression witli Eq. (33) we obtain

I

-14-

(37)

If

47T^kAE -.- e-^r-«

.

L- - J and r^AE.i''"

inequality (37) is reduced to

(JT)'^* - By4(JTV''' > a ,

<38)

We may obtain from this inequality the lower temperature limit of

T such that

or

£q. (39) has been used to compute the values of the limit tempera-

tures shov«rn in Table II.

A compilation of the constants used in maldng the calcula-

tions indicated by the basic equations is shown in Table III. The

.(10)

constants in the Lennard-Jones interaction potential, -rr' and V^ ,

were obtained from data listed in the "Transactions of the ASME

with the exception of the value for ¥^ for HF which was estimated

in the nnanner already described. The vibrational constant \^ was

found in each case from "vSpectra of Diatomic Molecules" by Herz-

4

-15-

The results of the indicated computations are listed in Table

IV and shoviTi graphically in Figures 4 and 5. The values indicated

for the O^ - N^ rrixture agree within a few percent with those listed

in the paper by Slawsky and Hcrefeld. It is believed that where dif-

ferences do exist, they may be attributed to a variation in the selec-

tion of force constants and low-velocity collision diameters inasmuch

as the published values are not always consistent. The calctUated re-

laxation times also agree reasonably well with values determined

experimientally, as discussed in the paper of Schwartz et al. No

data are available for comparison with our results for the HF - H,

probabilities.

-16-

IV. VIBRATIONAL LAG IN THE HYDROGEN-FLUORINE

ROCKET MOTOR

It is now possible to present some brief considerations in

determining whether or not the adiabatic expansion through the

de Laval nozzle in the H, - F2 rocket xnotor niay be taken as vi-

brational near-equilibriuzn flow. The reactants are taken in such

proportion that H^ will be in excess, so that the products of com-

bustion passing through the nozzle will be mostly a mixture of

HF and H^.

(12)Following a procedure analogous to that of Penner^ ', in

his treatment of chemical reaction in nozzle flow, we may take as

an estimate for the vibrational temperature lag for near -equilibrium

flow that

AT ^ (DT/ot)T . ,4m

For representative nozzles in 1,000 psia thrust chambers, it has

been found^^^^ that the cooling rate (-DT/Dt) ^ <Tc-Tcj/t^* 3x10'^

K/sec. The use of very small nozzles will, of course, increase

the cooling rate above this value; however, for nozzles in larger

thrust engines, (-DT/Dt) will generally be less than 3 x 10~ ^K/sec,

thereby leading to better approximations of near-equilibriuzri flow

than we estin^te below.

From Figure 5 it can be seen that the probability ^^,(^i*)

is approximately as large as some of the other probabilities and

much larger than others. The smaller probabilities contribute

little to the effective relaxation time and may be neglected. The

i

-17

probabilities, which are approximately as large as ^^(O^.J^) *

all contribute to the effective relaxation time. For the purpose of

simplicity, however, it may be assumed that the maximum possible

value of the effective relaxation time can be approximated as

'^« = V * l/^«^bM«.bPo*.C«».b) , (41)

The actual value of T must, of necessity, be smaller than Tm as

the other probabilities, which are approximiately as large as P ^ C^t^)

have not been considered in computing Ym and would lower its

value. If we can show, however, that vibrational near -equilibrium

flow exists for the calculated upper limit, *Tp^ • the conclusion

will certainly hold for the real T which is smaller than ^Tgt^ .

Since f^^b ' '^^^ch denotes the number of collisions per

second between HF and H^ molecules, increases markedly with

pressure, it is apparent that if the conditions at the nozzle exit

are such that vibrational near -equilibrium flow exists there, then

it must also exist everywhere in the noszle. The value of i^'F

was tlicrefore computed at the nozzle exit, where it is a maximum. *

In the present computations, the nozzle exit pressure was

taken as one atmosphere. The product "X^ Xi will generally lie

between the comparatively narrow limits of 0. 25 and 0. 20, and

will usually be about 0. 23. The product of ^'|* u and f^^jC^.b)

•Jwill not vary appreciably from 4x10' between the temperatures,

1400 to 3000° K. Therefore, for exit temperatures within this

range, tL ^ i x lO sec and, for (-DT/Dt) ^ 3 x 10 **K/8ec,

'^For a more detailed description of iterative lag calculations dur-ing nozzle flow, see Reference (14).

-18-

The estimate that over the length of the nozzle the vibra-

tional temperature lag cannot exceed 3 K indicates that vibra-

tional near -equilibrium flow occurs during expansion in a de

Laval nozzle for representative hydrogen-fluorine rocket nno-

tora.

4

-19-

REFEK£NCZS ^

1. Kantrowits, . . : "Iieat Capacity L.ag in Gas Dynamics", Journalof Cher.dcal Physics , Vol. 14, p. 150, 1946.

2. Sheratt, G. G. , and Griffiths, E. : "The Determination of the Spe-cific Heat of Gases at High Ten-iperatures by the Soiind VelocityMetliod", Proceedings of the Foyal Society of L.ondon , Vol. 147A,p. 292, 193?:

'

3. Fierce, G. W. : "Piezoelectric Crystal Oscillators Applied to thePrecision MeasureiT^ent of the Velocity of Sound in Air andCO2 at High rrcquencies", Proceedings of the Arr:erican Acad -

emy of Arts and Sciences, Vol. 6(), p. 271, 1925.

4. Kneser, K. O. : "Zur Dispersionstheorie des dchaiies'", Annalender Physik , Vol. U, p. 761, 1931.

5. Griffith, \V . : "Vibrational Relaxation Times in Gases", Journalof Applied Physics , Vol. 21, p. 1319, 1950.

6. Landau, L.. , and Teller, S. : "Zur Theorie der Schalldisperdon '

Physikalische Zeitschrift der Sowjetunion , Vol. 10, p. 34, 1936.

7. Bethe, H. A., and Teller, E. : "Deviations from Thermal liquili-

brium in Shock V.aves ', Report X-1 17, Ballistics ResearchLtaboratory, Aberdeen Proving GroundJ

'

8. Schwartz, K. , Slawsky, Z. , and Herzfeld, K. : "Calculation of

Vibrational Relaxation Times in Gases", Journal of ChemicalPhysics , Vol. 20, p. 1591, 1952.

9. Schwartz, R, , and Hersield, K. : "Vibrational Relaxation Tinics inGases (Three Diiyienaional Treatment)", Journal of ChemicalPhysics , Vol. 22, p. 3763, 1954.

IG. Hirshfelder, J. O. , Bird, R. B., and Spotz, £. L. : "Viscosityand Other Physical Properties of Gases and Gas Mixtures",1949 Transaictions of the American Society of MechanicalEngineers , Vol. 71. p. 921, 1949.

11. Herzberg, G. : Spectra of Diatomic Molecules , Table 39, p. 501.

12. Penner, S. S. : Thermodynamics and Chemical Kinetics of One-dimensional Non-viscous Flow through a Laval Nozzle ', Journalof Cae-gjcal Physics , Vol. 19, p. 877, 1951.

13. Altman, D. , arid Penner, S. S. : "Chemical Reaction During Adia-batic Flow through a Rocket N.'zzle", Journal of CheniicaAPhysics , VoL 17, p. 56, 1949.

14. Penner, S. S. : 'Linearization Procedures of the Study of Cheixiical

Reactions Flow^ Systems", 1953 Iowa Thermodynamics Syrrioosium .

-20-

Gas T r K \cr^

"k sec 3 \

i

A. Sonic Velocity Methxodj

Hydrogen 298 0.018 (rot)

Oxygen 294 1000Carbon monoxide 1257 10Chlorine 294 18Carbon dioxide 294 10.8Sulfur dioxide 294 0.181Ammonia 294 0.04Ethylene 294 0.233Metiiane 383 0.84Carbon disulfide 294 0.70Nitrous oxide 292 0.92

B. Impact Tube Methodt

?

i

Hydrogen 287 0. 021 (rot)Nitrogen 289 0.002Nitrous oxide 291 1.12Carbon dioxid? (comm) 291 2.02Sulfur dioxide 293 0.37Amrrjonia 290 0.12Mythl chloride 295 Q. 202Ethane 237 0.003Etliylene 289 0.207Propylene 294 0.006Propane 294 0.007Ethylene oxide 303 1.23Butane 294 0.018Butadiene 294 0.013Metliane 289 0.48

1

TABLE I. iLxperimentai results reported in Reference 5

for the relaxation tinnes of gases

i

-21-

Temp *

^K

Probability

^2-^2 HF - H^

Po-oC«.a) 326^ 1390°

Po-^^C^b) 405° 2700°

Pi>-oCa.b) 330° 2380°

Po-*otb.d) 403° 2440°i

P— . U.b) 231° 592°i

Po-,.iCb.d^ 231° 592°1

TABLE II. Lower temperature limits for which

the probabilities can be calculated froiti the theory

of Schwartz, Slawsky, and Her zfeld.

•23-

jr>robabilityA

c/k

•K

9 » 10'*

O2-N2

1 f

3.50 114.0 4.75 3.14

Po-^oCto.*>) 3.69 93.7 7.10 4.70

Po-r«(«.b) 3.60 103.5 4.75 3.14

P. -• ( *».«^ 3.60 103.5 7.10 4.70

P— .C«>.^) 3.60 103.5 4.757.10

1.56

HF.K2 '

Po-»«C«*«) 1.70 378.0 12.4 8.22

P— ^b.b) 2.94 35.1 13.2 8.74

Po^eCd.b) 2.32 114.0 12.4 8.22

Pc^oCb.a) 2.32 114.0 13.2 3.74

. . .. t

2.32 114.0 12.413.?

.52

TABLE lU. Values of constants used in computations for

the vibrational excitation probabilities.

-22-

«M2:

rs3

^^V vO NO ir> Tjt T^ t '^

£v^

1

o1 1 1 1

o o o1

o1

o^4 ^4 r^ Fi4 «M »M ^H

m X M XXX X Xt

o o U^ fsj 03 00 r-

Q?t>H sO CO *-! rvj ^ r-

/r>D K) >o nO tn Nj< "* •^* 1 1 1 1 i 1 1V o o o o o o o

\^ *>4 PM —* m* mt Vi4 m4X a XXX X X

of

ir> o rsj o o f* wo f-4 CO tn »-i c^ m

i

1

^^ oV Pl« o v^ vO nO in in

Si1

o1

o1 i 1o <^ o 1

o1

os^ f>« »H ^^ J—• fM ^H «<-•

X « XXX X < •

tfM4 f- •^ <*4 00 r* if»

i

• * • • * • • !

Q?c^ to f^ ^4 ^ vH

1

1

f^0

CO f*. >0 If) ^ '?}« ^I 1 i I I « I

o o o o o o oXo X

O nO t«-

X X X Xr- ph o o Xo

m «-l Tj* O

o»-i O^ 00 t*- nO lO If)

I I I I i I Io o o o o o o^4 ^4 «>l l~4 fM iBi) (-I

X X X X Xfo nO m rsj c* fO

e^ M e* r-

X Xfo -^

fo »^ eo

^^1

^ CO r«- >0 If) <* -f <*)

s1

o1

ot 1

O O•

o1

o1

o»s •M ft p-< Wt ^4 ^HX X X M X X X

tCO 00 eo If) o <M o

Of 00 vO CO in fSJ IX)

•

704

9o o o o o o oo o O C3 o o o-^ ya O If) o l^ o

>-<i f>« N N f^

?

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FIGURE 5- VARIATION OF (a = HF,b^H^) PROBABILITIES WITHTEMPERATURE. LIMIT TEMPERATURES CORRESPOND TOTHE JUNCTION BETWEEN THE SOLID PORTION AND THEDOTTED PORTION OF THE CURVES.

^250561 enny

Vibrational relaxation

tines of gaseous mix-

tures of diatotdc mole-

cules and their effect

on rocket performance.

125056

P33 PennyVibraticnal relaxation bni.es

of gaseous mixtures of diatcmicl

molecules and their effect en

rocket performance.

thesP33

Vibrational relaxation times of gaseous

3 2768 001 97957 8

DUDLEY KNOX LIBRARY

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