Temperature and Unique Energy Level Populations C. S. McKee University of Bradford, Bradford BD7 IDP, England The Boltzmann distrihution law may he expressed in two basic forms, one of which demonstrates quite clearly that the relative population of any two of a set of energy levels is uniquely determined by the temperature: niln, = l ~ , l cexp ~ ) - (rj
- r,)lkT
(1)
where g is the degeneracy. It is possible, therefore, to determine T by measurement of (njlnj). If ti < e j andgi = g, = 1, it follows that nj > n,. Thus, for a system consisting of species such as the harmonic (ncillator we are accustomed to visualize individual level populations decreasing monotonically with increasing energy, a t constant temperature. As a corollary, one tends to assume that the absolute population of a given upper level will increase continuously as T increases, an impression which is reinforced by certain textbook statements; for example, ". . . the concentration of any quantum species will he nraoortional t o . . . exoi-EilkT) . . ." (1). It is interestine to find that the impressim is fallacious. Except for the level of zero enerev, -. the fraction of the total number of oarticles in level i, p, = (n,lN), passes through a maximum with increasing temperature. In general, therefore, the absolute populat~onn,, of a given level does not define the temperature uniquely. A qualitative explanation of the effect can he given most easily by considering the first excited level of any system. If the temperature is increased from 0 K, the population p l will increase at first from its initial zero value. At higher temperatures, however, transfer of particles out of level 1 into levels 2.3 etc. will occur and when the rate of this second process hecomes predominant, P I will decrease again. Without actually mentioning the effect, many texts show diagrams of translational. rotational. and vibrational oooulations a t dif.. ferent temperatures in which a maximum in certainp, is implicit (e.g. J = 6 or u = 3, Fig. 1). Nevertheless, a detailed discussion is useful as a teaching exercise, involving exploration of the Boltzmann distrihution.. .oartition functions. spectroscopy, and basic mathematics. Also, it is of some interest in relation to temoerature determination. The existence of a maximum in pi a t a given temperature follows from the alternative expression for the Boltzmann distrihution: p, = (",IN) = ( ~ , / qexp(-4kT) )
Figure 1. Variation of (a) vibrational and (b) rotational level populations p. as functions 01 the corresponding quantum number. Data for three different temoeratures are shown in each case.
Table 1. Population Maxima p.,,.. and Corresponding Temperatures T,,,.. for the Lower Energy Levels of a System of Harmonic Oscillators
"
,
2
3
d
6
(2)
where o is the nartition function and N the total svstem population. Since q = q(T), there are two temperature-dependent factors and the differential (ap;ldT);, will involve two additive terms. It can, therefore, take a zero value, giving a turnine ooint in .D.(T) and the oossihilitv of two real roots for . eqn.
(5:
The Harmonic Oscillator The first excited state of the harmonic oscillator provides a simple demonstration, since eqn. (2) has a straightforward analytical solution. In this case p,,.,
= I1 -
exp(-H,idTI expl-8,ihlT)
(3)
where Bvih = hulk, is the characteristic vibrational temperature and u is the vihration frequency. The solutions are (4) exp(-R,idT) = 11 ( 1 - 4~,,)'/~112 Real roots exist only for 4p..=1 5 1. The fractional population
Figwe 2.Variation of vihtional level p ~ u l a t i o np,, as a function ol tempratwe: O,,. is the characteristic vibrational temperature. Absolute temperatures are indicated for two panicular molecules.
Volume 58
Number 8 August 1981
605
of the first level will reach a maximum value p,.=l,,,, = 0.25, at a temperature T,.=I,,,, = H,ji,lln2. Populations less than 0.25 will occur a t two temperatures, given hy lTL,=),m8x/Tl = I - in11 i I1 - 4 ~ , , l ' ~ ~ I / h 2 ,?
I he general expressions for p,,,,,, tianal level are as fr,llows:
+
p,.mar = v ~ ' / I vI ) ' + '
and T,.,,,,
15)
for any vihra-
T,,,n,ax = l~,ih/lnll+ Ui.1
As T increases, therefore, a given level will accommodate a maximum fraction of the total vibrational energy in the system at a temperature lower than that a t which the absolute value of the energy in that level passes through a maximum. The Rigid Rotator
In the case of the energy levels of a heteronuclear diatomic treated as a rigid rotator where 0 , = h2/8?r21kis the characteristic rotational temper~ ature and I is the moment of inertia. The maximum i n p and the corresponding temperature are given by Values of these quantities for some specific species are shown in Tahle 2. In eqn. (8) the integral approximation has been used for the rotational partition function, which is therefore in error by -15% a t the temperature corresponding to maximum population in level J = 1, decreasing to -3% for J = 3 ( 3 ) .There will he associated errors in the calculated values of .DJ,,,, and TJ ,.,, for these levels. The general nature of the solutions of eqn. (8) may he
0
2
TIT,..
4
6
Journal of Chemical Education
(7.1IT)exp(1 - 7.1 ,max IT1
(10)
Data tor a plot of "reduced population" (p,~lp,~,,,,)as a function of "reduced temperature" (TIT,,,,,,,, are easily comouted. The curve is shown in Fieure 3 while ulots of o.riT) lation of a given level, [ p , J i ~ ) ] , Jis, quite distinct from the well-known maximum in the isothermal variation of the relative populations of rotational levels, [ p ~ ( J ) ] r(Fig. , Ih). The latter maximum arises hecause the degeneracy and exponential terms in eqn. (8) differ in their variations with J . It occurs in level J a t a temperature (TIU,) = %(2J + 1)'. which is alwaysgreater than the temperature (eqn. (9))a t which the ahsolute population of J attains its maximum value. At T > 0 K, the existence of a maximum in [pj(J)].ra t J > 0, is eviJ dent in Figure 4 from the fact that the curves of [ ~ J ( T ) ]cross one another. In the vibrational case (Fig. 2 ) the curves do not cross, and a t all temperatures the maximum in [p,.(rr)]~ occurs a t u = 0 (Fie. la). For a given level the most striking feature of Figures 3b and 4 is that. in the vicinitv of the maximum and aeain a t hieh ture sensitivity of the population ;s given quantitativeiy by
.
,
,.,
Around TJ a change in (TITJ ),,,, from 0.83to 1.23, produces a maximum change in p,, of -2% and an average change of only -1% (see Fig. 3h). Some examples of the temperature range involved in a population change of 270on either side of p~,,,, are given in Tahle 3 for particular species. Note that the average change in p ~ = l for o OH in the region of 3000 K, Table 2. Population Maxima p,,,., and Corresponding Temperalures TJ,,.. tor Energy Levels of Some Heteronuclear Diatomic Rigid Rotators
1
Figwe 3. Variation of rotational level population pJ, relativeto its maximum value as a function of temperature. The variation of the temperature-sensilivl~ p,,, S. 01 PJ is shown also.
606
I p ~ i p,n>sx , ~I =
lfi)
Some numerical values for these quantities are shown in Tahle 1. Curves of p,.(T)are shown in ~'Lgure2. As expected from the Boltzmann distrihution, the dependence of p , on T decreases at higher temperaures where allp,, convergetnward a limiting value of zero. A closely related distribution, that of energy f,, = ohsp,.N, among the levels of a system of harmonic oscillators appears diagramatically in one text (2). without discussion. The absolute values of hoth the energy stored in level u and the population p,,, must reach maxima at the same temperature, T,,,,,,. Relative to other levels, however, 6, will he maximized when ( & , . I d r ) = ~ 0,which occurs a t a temperature Tlll,ih = L,.Expansion of the logarithm in eqn. (6) gives
0.0
demonstrated graphically Suhstituting eqns. (9) into (8) and rearranging, gives
Fiue 4. Variation of rotational level population p ~as . a function of temperahlre: I), is the characteristic rotational temperature. Absolute temperatures are indicated for two particular molecules.
Table 3. Temperature Variation of pJ About Its Maxlmum Value, PJ.",~.. PJ~PJ,~~.
HCI ( H , = 15.24 4: J= 5 J = 10 = 24.97 K): OH (2X+,11, J = 10
TI K
0.98
100
0.98
377 1383
457 1676
562 2061
2266
2747
3378
is of the order of 0.001 % K - I . This radical is of interest because of its occurrence in flames and its particularly high characteristic temperature (0, = 24.97 K). The population is a t its most sensitive to temperature when
Spectroscopic Determination of Temperature
Introduction of this topic is useful for relating level populations to measurements of practical importance and also for providing an opportunity for direct reference to spectra. The intensity I,, of a line arising from a transition between levels n and m , is proportional to the fractional population p,, of the initial state (4): I,, = CAnmhimm~, (13) where C is a numerical constant, A,, is the transition probabilitv and G,, is the wave number of the radiation. Provided the system is in thermal equilibrium, the Roltzmann expression for p, can be introduced to relate I to temperature. ~emper'tures may be determined from intensitydata for electronic (51,vibrational (6) and rotational transitions (7), and the method has been applied in the study of flames (a), stellar atmospheres (4), and high speed gas flows ( 9 ) . Emission by radicals such as OH and C2, which occur in hydrogen-oxygen and hydrocarbon-oxygen flames, provides a classic example (7). Vibration-rotation bands are observed in the u.v. and visible regions due to ' 2 - "1 transitions in the OH species and in the visible due to TI,- q,, transitions in C2 (the so-called Swan hands) (10). For emission, A,, is proportional to Cnm3 and so for a given set of rotational lines I,, = C'G4,,g(W
2x
+ l)(H,IT,) exp(-c,lkT,)
(14) g(J) is the nuclear spin degeneracy, where C' = ChA,,l9,,, t, is the energy of the initial rotational level in the upper electronic state and T, is the rotational "temperature." (The prime used normally to denote the quantum number of the initial level involved in a transition is dropped for convenience.) In nrinciole. . . C' can be calculated or measured (11. . . 12) . and so T,could be determined from the intensity of a single line. Aoart from very low accuracv, this method suffers from a findamental objection. If the line used bad its population maximum in the region of the temperature to be measured, there would be ambiguity in identification of the appropriate temperature root of eqn. (8). A common procedure ( 6 )which avoids this amhiguity, is to compare a number of lines by plotting In(l,,IA,,G,,) against t, (eqns. (13) and (14); see Fig. 5b); the slope gives -llkT,. The temperature-sensitivity of a given line will be greatest if the corresponding E , value statisfies eqn. (12). For other lines, where the initial state population is in the region of its maximum value a t the flame temperature, thesensitivitg will be low and points on the In [(en) plot should be weighted accordingly. For measurement of individual line intensities, the variation of detector response with frequency must be known, but this requirement can he eliminated by considering the intensity ratio R , of two lines a and b of very similar frequency (12).
-
Figure 5. @)TheR2 branch of the 0.0 band inthe '11 eiectroniclransition of OH; lines of the R,. P,, and 0,branchesandsatellitelines have beenomined for clarity (AOapted f r m ref. 16). (b) Curva (i): iso4ntensity plot f a the Rz branch of the 0.0 band of OH in the outer cone of a 50-50 oxygen-natural gas flame: numbers in brackets indicate the values of J. and &corresponding to a given point. (Adaptedhom ref. 6). Curve (ii): abnorml rotatianal distibution in Vw inner cone of an oxygen-acetylene flame diluted wilh 90% argon. Data for lines in the P,.R,, and 0,branches of Vw 0.0 band of OH are plotled according to q n s . 113) and (14) (From ref. 14).
= Cib exp(-Acab/kTr)
(15b)
where C,b contains the transition probabilities and degeneracies. Equation ( H a ) applies to a rigid rotator. For greater accuracy Ae.b would be derived from experimentally measured term values. The temperature-sensitivity S, of the intensity ratio relative is given hy to its maximum value S,,,,
where Ts,,..
= Ar.b/Zk
(17)
and
decreases. T o a first approximation, assume J , in the rigid rotator case
>> Jb; then,
,,.
T.s = %J.(J.+ I)#, = %TJ,,,,, (19) The best choice of line-pair is that in which the population of the initial state for the line of higher frequency, reaches a maximum a t a temperature about twice as high as the temperature to be measured. Volume 58
Number 8
August 1981
607
Rotational temperatures in low-pressure acetylene-oxygen flames have heen determined (131 h v comoarison of two lines in the (0.0) Swan band nI Cn; J = 1 9 i n the K branch and J = 47 in the t'hranrh, which are separated h v about 0.1 nm. T h e corresponding value of Ac.~,/h is 4663 K and so T,ym. = 2331.5 K. T h e temueratures measured under various flame conditions lay i n i h e range 2500-5100 K, corresponding t o SIS,., values hetween 1 and O.fi (Fig. 3al. T h i s particular choice of lines. governed partly h y spectroscopic considerations, isseen t o have heen appropriate o n the grounds of temperature sensitivity also. Instead of considering just a single line-pair, a more satisfactory procedure is tn s&ct a number o f pairs of lines o f equal intensity. Intensity calibration of the detector is thus avoided. Moreover. i f a head-formine hranch. such as the R? branch in the (0.01 hand o f OH (Fig. is chdsen it is possihle t o select iso-intense lines which differ very l i t t l e in wave number. Hence, all detector parameters can he eliminated (6). For any isnintense pair of the same frequency it follows from eqn. (15a) that
;a),
A p l o t of this function for an oxygen-natural gas flame is shown in Fiaure 5h. F r o m the slope o f a "least-squares" line the rotational temperature was f o i n d to be 2600 K Usingeqn. (17) and Figure 3, it is found t h a t the line-pair J . = 14, J b = 3 has maximum temperature sensitivity near the flame temis lower, decreasing t o perature h u t for other pairs, SIS,,, - 4 . 4 for the pair 10,6. Appropriate weighting o f the points in Figure 51) would increase the slope slightly and lower the calculated temoerature h v -100 K. Thr fact Ihit;the i - w i n ~ e n s i tplot ~ 11) Figure 5h i s a straight h er e~l de c t r o n ~ o ~ l l y line indiratrs th;n i n the l l t l m ~ r l l n ~ i dt ~ excited O t i r;ldl(:uli do have a rotnli#mald i s t r i l ~ u t i o nof tht. t \ l ) r experled l o r a +?SfQm at t h t v n a l t.quilihrium. Similar pnwedure.; may he applied t o \,~l,ruti~,naland electrmiu spertra>sn,pic dntn. I t i s 1 4interest tc, we i f the vihrational and electrmic p o p u l i l t i ~ n salso reflect equilibrium situatimd and II the "remlwmtures" characteriring t hese distrihutinns are identical the n m r t i m a l "temprrnt~~rr"'I'heenergy released i n the tvpe d e x o t herniir reaction rrspons~hlefor f o r n ~ a t i o n o f elertroni~:allvt w i t e d radicals tnny gu preferentially i n t o sperit'it drgrtws d ' f r i ~ e d o m .Hy inelastir rullisions, thesepafrredgm then may hr brt,ught i n t u an equi. rate d r a r w s #>I IiI,num-type distril~utinn.I)ut e q u ~ p a r l ~ t i outnenergy among all the de~'rev>mat nut he e>tahl~ihed.S ~ e c i f i rdevintims from a situation o f complete equilibration can give important information on the chemical reactions occurrine in the flame 1.5. 6. 11). A n abnormal rutational distribution. in an oxveen-acetvlene - .. flame diluted w i t h 90% argon, is exemplified h y the second p l o t shown in Figure 5h (14). I n this case there are t w o segments, the first passing through points arising from low-lying rotational levels and indicatina a relativelv normal temperature o f -1400 K. T h e second segment passes through points arisina from hiaher levels and corresponds t o an abnormally high 'tcmpcrilturc" oiT,5(Hl K. I t indh1t.s that electrunically c x c ~ t e dO H is formed with high rotational energy i n the flame reartignn. 'l'hr t i ~ n c t i u no f the diluent gas may he t(, lower the. ovrrall 1rmper;lIllrQ i n the flame, thus r r d u c ~ n g the Cuncentrations o f particles, such as ground-state OH, H atoms, etc, which might he especially active in removing rotational energy from excited .. . ~. ~. O H . 0xyyen.hydrogcm flames present n nmtrastinp; example i n which rotatitmid t q u i l i h r i u m appcdri to be estahlished t151. A thermal r q u ~ l ~ h r ~distrihuti,m um is found also for vibrational and el&.ronic enereies in the outer cone o f the flame. ~~~~~b u t in the inner cone there is an excess population in the v i hrational levels u' = 2 and u' = 3. T h i s feature mav be due t o a reaction involving H and 0 atoms present in excess of their
I,,
~
~~
608
~
~~
~
Journal of Chemical Education
Figure 6. The 0.1 vibration-rotationlaser Raman spenrum of pure oxygen at 263 K. 1 atm. (From ref. 9).
thermal concentrations, followed b y a radiationless transition: 0H(2Xt) O(:JP)t HPSI OH(?-I where 2 2 + is the vibrationally excited ground state (8,15). A b u i l d up in the concentration o f atoms in the reaction zone of a flam;is n o t surprising in the absence of walls o n which recombination could occur easilv. T h e accuracy o f temperatuie measurement in flames is o f the order of 100 K and is limited by, among other factors. difficulties in obtaining long-term stability in the flame operatina conditions. W i t h non-interacting gases the problem does not ariiv nnd much higher accuracy is attainable. A t Iuu temperature. hou,ever, thermally r x r ~ t e electrtmic d lransitions are absent and so vibrational transitions cannot he observed in emission from homonuclear diatomics. Moreover, temperature determination from absorption data IS difficult (16). I n such cases Raman spectroscopy provides a convenient alternative. T h e 0-1 vibration-rotation Raman spectrum o f pure oxygen a t 1atmosphere is shown in Figure 6 ( 9 ) ;a p l o t of the line intensities accordine to eon. (14) vields a aas temuerature of 263 K, the estimatederror.being only f 2 ~ . In addition t o its practical importance, a discussion o f Raman scattering is useful for illustrating the significance o f nuclear spin degeneracy. Amonl: rentrosvmmetri~:linear molecule* the ground staw $11' '"02t"2';1 IS rather exceptional I n that i t is antisvmmetric w ~ t re3pQcI h toexchange ~ i n u r l e i 1171. As a amsequence. CIJIis 0 fir./ even and 1 for J odd and rotational lincs w i t h initial state5 of r v e n J are missing frum the spectrum (Fig. f i ~ For . m d r c u l r s such as COPthe ground state r ' 2 ' ; ~ is symmetric and o d d - J lines are missing.
-
-
Acknowledgment 1 should like to thank m y colleagues Dr. J. M. C. Turner and Dr. C. V. W r i e h t for their stimulatina comments a n d for detailed readingof the manuscript. Literature Cited I11 Rice. 0. K., "Stadstical Mwhenies. Thermodynamics and Kinetics."Freeman. 1967. p. 39. 121 Fast. J. D.. "Entropy:'2nd Fd.. Philipa Technical Library. N. V. P h i l i p G l a i l a m pdabrieken. Eindhnven. IMB.p.fi9. (31 K"". J. H., "Molecular Thcrm.dynsmim: A" lnlrductiun 10 Sfatisticel Mechanics L r ~hamisls.".Jahn Wilcy. 1971.p. 107. 141 H=rxbew. G.. "Meloeular S ~ t r and a Molecular Structure. I.Spectra of Diatomic Molwule~."2ndFd..Van Ncrtrand Reinhold. 1950, p. 20. 15) Rruida. H. P.snd Shuler. K. E..J Chem Phys.27.91111957). (fil Shuler, K. E.. J. Chrm. Phyr.. lX.1466 (19101. 171 Hroids. H. P.J. Chpm Phys.,21.340 09531: seealsoref. 161. (8) Gaydon, A. G.. "The Spectnaeupy alFlames." 2nd Ed.,Chapman and Hall,London. 1974. (91 H~kmsn,R.S. andLianp, L..Rw Sci Imlrum.45, l5W. (1974). 1101 Gsydtm. A.G.and Wnlfhard. H. G.. R o c . Roy. Soc A.ZOI.561 (19501. (11; Shuler. K. E.J. Chem I'h.w. 1%. 1221 (1910):J. Chom Ph.v#.. 19.888 119511. 1121 Antetell. J.and Learner. H.C. M.. Pwr. Roy. Soc A.301.3S5(19671. 1I:II Hnlida. H. P.and Heath. I). F.. J. C h m . Phya..26,223(195il. (141 Kene. W.R.and Broida. H. P.. J Them Phyr .21.347 (19i.l). llil Rndds. H. P.endShuler. K. E..J. Chem I'h.vs.. 20. 168 llYS21. (Ifil K
