REFRACTIVITY AND T H E MOLECULAR PHASE HYPOTHESIS. I. BY E. C. C. BALY AND R. A. MORTON
-_____ It is well known that the Sellmeyer dispersion formula in its simplest form expresses the refractivity of substances with considerable accuracy, provided that the characteristic absorption frequency lies in the extreme ultra-violet region of the spectrum. This has been clearly shown by Mr. and Mrs. Cuthbcrtson in the caSe of many gaseous elements and compounds.l The formula is usually written in the form
where n is the refractive index for light of lhe frequency v, N is a constant and vo is the characteristic frequency in the ultra-violet. In the case of gases where the indices are not much greater than unity, n-1 may be used in place of $-I. This formula does not apply at all well if the substance exhibits an absorption band in the near ultra-violet as well as in the estreme ultra-violet. This is particularly noticeable with ozone and the halogen elements, all of which exhibit absorption bands in the long-wave ultra-violet. I n Table I are given the observed refractivities of chlorine together with the values calculated from the formula 7 .3 I3 I x I OZ7 n-I= 9.6294X 1o30-v2 The constants were calculated by the method of least squares so as to obtain the best possible values.
TABLE I Wave-lengt h
(n-I) X Io8 Observed
(n-I) x Io* Calculated
Difference Ca1c.-Obs.
6707.85 6438.47 5790.66 5769.60 5460 74 5209.08 5085.83 4799.91
7 7 563 7 7 703 78121 78135 78400 7865 1 78791 79166
77556 77697 78123 78139 78402 78655
-7
'
78791
79156
-6 +2
+4 +2
+4
.
+I -IO
It will be seen from the last column that the agreement is very far from satisfactory, since the calculated dispersion curve is flatter than the observed curve. This divergence is commonly attributed to the omission of a second Phil. Trans., 213,
I
(1914).
6 60
E . C . C . BALY AND R. A . MORTON
term in the formula, but up to tho present no success, has been obtained with this dispersion formula with the addition of a second term. The molecular phase hypo thesis', however, leads to a simple modification of the Sellmeyer formula which expresses the refractivity of substances with remarkable accuracy and in the present paper its applicability to some gases may be dealt with. According to this hypothesis a pure substance a t a given temperature and pressure is an equilibrium mixture of.two or more phases of one molecule, each of these phases being characterised by a frequency which is an integral multiple of the fundamental frequency of the molecule. This fundamental frequency is situated in the infra-red region of the spectrum, whilst the phase frequencies lie in the visible or ultra-violet region. Since two or more phases are present the observed refractivity will be the sum of the effects due to these phases, the effect of each phase being proportional to its concentration. The following formula therefore is arrived at:V3N3 ( ~ - I ) ( V ~ + V ~ + V .~.+.). = ViNi - VzNz +
y12-v2
+
y22-y2
y s L y 2 +... .
where VI, VZ, Vd, etc. are the relative volumes of the phases present in the gas, V I , YZ, v3, etc. their characteristic frequencies, and N1, N P , N3, etc. are constants characteristic of the phases. Since the phase frequencies are integral multiples of the fundamental frequency charact,erist,ic of the molecule, the formula may be written in the form:ViNi VzN2 V3N3 (n-I)(V1+V:!+V3+. . . .) = (av,)z-v2 (bvx)2-v2 (CV,)~-V~ where v, is the fundamental frequency, and a., b, e, etc. are positive integers. If the whole of the molecules present in the gas exist in one phase the NX formula is simplified to :- n-1 = (x V,)Z-Y? so that the Sellmeyer formula in its simple form is applicable to this condition and this condition only. This formula for light of infinit,e wave-length be-
+-
+-
~
comes
NX -
(n-I,)=
(XUX)2
and we therefore have N, = (xvx)z(n-I), . Since the value of n-r for waves of infinite wave-length must be indcpendent of the phases in which the molecules exist, the values of N,, Nz, N3, etc. must' be equal to (av,)2(n-I), l(bv,)2(n-l), ,(cv,>z(n-r), respectively, and hence the complete dispersion formula becomes :(n-x)(VI+V,+VJ+.. , ) = o Vl(n-I)oc+ ' (-~z(n-x)~ + 0 2 ~ ~ ( n - - I ) .~ + . ,
(au+vf
( cux)z-v2
I n this form the physical significance of the formula is rendered more apparent. Now chlorine is known to exhibit only one absorption band in the ultraviolet at greater wave-lengths than 2 1 0 0 A,, which is the limit of a quartz spectrograph working in air. It must be remembered that t,his absorption Baly: Phil. Mag., 40, I; 15, (1920).
66 I
REFRACT1 VITY
band has no apparent connection with the fine line absorption in the visible region, which moreover seems to have no influence on the refractivity since the dispersion curve is perfectly smooth and regular over the region wherein the fine absorption lines can be observed. Since chlorine only shows one absorption band in the near ultra-violet it is probable that two terms in the above formula will be sufficient, that is to say there are only two phases present, one having its frequency in the extreme ultra-violet and not yet observed. For purposes of calculation the formula can therefore be simplified to NI + VN2 (n-I)(I+V) = (av,)2-v2 (bv,)2-v2 where V is the relative volume of the molecules in the phase with larger frequency, the volume of the molecules with frequencies in the near ultra-violet being considered equal to unity. Now the centre of the first absorption band of chlorine lies at the wave-length 3380 Angstroms' or the frequency 8 . 8 7 6 X 10~4, but it must be remembered that the accuracy of this measurement cannot be guaranteed to within one Angstrom. Assuming in the first instance that the frequency of this absorption band is correct we have VN2 N1 (n-I)(I+V) = ( 8 . 8 7 6 X 1 0 1 4 ) 2 - ~ 2 (bu,)2-u2 Where V, N1, N2, and bu, are unknown. With a single dispersion curve we have not found it possible directly to determine these unknown quantities but have employed t,be following method of solution. As may readily be seen, the value of bv, may be determined if it be assumed that N1=N2, since if the dispersion curve is expressed by the formula it will equally well be expressed by N VIN (n-I) ( I +VI) = (8-8)6 X I O ~ * ) ~ - Y ~ (bv,)2-u2 where V1 is a constant. From the refractivities of chlorine for four different frequencies it was found that all the observed refractivities are expressed with considerable accuracy by the formula 7.948XIO2' 4620.8 X 7 . 9 4 8 X 1oZ7 (n-1) (462 I .8) = 7 . 8 7 78 7 6 X I oZ9-v2 1 0 4 . 8 3 4 X 1029- v2 The values of the phase frequencies ul and u2 are therefore very near to 8 . 8 7 6 X 1014 and 3 ,23781 X IO]^ respectively, and these must therefore be integral multiples of the fundamental frequency of chlorine. Now the ratio of these two frequencies is 3 .6479 and the two smallest integers which have a ratio approximating t o this are 197 and 54, the ratio of these integers being 3.64815. Assuming this ratio to be exact it is easy t o calculate the constants anew in the formula ~
+
+-
+
G. Ribaud: Ann. Phys., 12,
107,
(1919).
662
E. C . C . RSLY AND R. A. MORTON
where v, is the fiindamental frequency of chlorine. The final values obtained were as follows :7.945035X1oZ7 + 4649.6X7.945035X1oz7 (n-1)(4650.6)= 7.873996X 1 0 ~ ~ - v ~ I04.79487xI029-v2 and the refract)ivities calculated from t.his formula are given in Table 11.
TABLE XI x
n calc.
Difference! ca1c.-obs.
1.00077563
1.000775645
+o.oooooooI~
1.00077703
1.000777020
-0.000000010
1.00078121 I .0007813j 1.00078400 1.00078651 1.00078791 L .00079166
1.000781204 1.000781365 1.000783984 r.000786511 1.000787906 1.000791643
-o.000000006 +o.oooooooI~ -o.000000016
n obs.
6707.87 6438.47 5790.66 5769.60 5460.74 5209.08
508.5'83 4799.91
+0,000000001
-o.000000004 -0.000000017
The value of (n-I), given by the above formula i s I 000760158,and since N1=(n-r), X7 873996X1oZq=5.98548XroZ6andN2=(n-r), X 104 79487 X 102"= 7 96606x 10*~, the value of V is at once found to be 349.6. The final dispersion formula for chlorine is therefore (n-1) (350 6) =
5.98548x 1026 349.6X7 96606 X 1oZ7 104 79487 X xoZ9-v2 7 .873996X10~~-~~
+'
The agreement between the calculated and observed values of the refractive indices i p remarkably good and decidedly better than that obtained by the use of the approximate formula first used. There is no comparison between the results obtained by the new formula and those given by the simple Sellmeyer formula and set forth in Table I. It may be noted that the first formula for chlorine was calculated on the assumption that the measurement of the central wave-length of the first absorption band was exactly 3380 Angstroms, but this is not necessarily correct to within one Angstrom. I n the second and final formula the integral relation between the frequencies of the two abeorption bands was taken as exact and the wave-length of the band was found to be 3380.8 Angstroms, the difference of 0 . 8 Angstrom being well within the limits of experimental error of measurement of this band, The agreement between the calculated and observed frequencies given by the formula establishes some confidence in its correctness, especially in view of the fact that thc formula involves no new assumption beyond that the simple law of mixtures holds good with diffwent molecular phases of the same molecule. Before discussing the deductions to be made from the dispersion formula of chlorine the validity of the new formula may be tested on other gases, but unfortunately t,he accuracy of the refractivity measurements of other gases
'
663
REFRACTIVITY
with absorption bands in the near ultra-violet is not as great as in the case of chlorine. The validity of the formula is however supported by gaseous bromine and by ozone. In the case of bromine the refractivities observed and those calciilated from a simple Sellmeyer formula are given in Table 111.'
TABLE I11 x 10'
x
(n-1)
(n-1)x107
6707.87 6438.47 6000 5 800
5750 5700 5600 5460' 74
Difference
calc.
obs.
11525
11518
-7
11570
11571
+I
+ I3
I 1662
11675
11735
11731
-4
11741 11762 11796 1 I849
11746 11762 1 I767 I 1842
+S 0
-29
-7
The absorption band of bromine vapour has its centre a t X=4210 AZ and hence the value of v12 is 5.077833x 1 0 2 ~ . Using the same method as in the case of chlorine the following formula is obtained:78.798X6.05302X10~~ j . 59538X (n-1)(79 ' 798)= 5.077833x 1 0 2 ~ - v ~ 54.9314x and the calculated refractivities are given in Table IV.
+
TABLE IV
x
(
(n-I) Xi07
Difference
calc.
6707.87 6438' 47
11520
-5 0
11570
6000
11672
+IO
5800
11730
-5 +S
5750 5 700 5600 5460 74 '
11746 11763 11799 11854
+I
+3 +S
The agreement is very distinctly better than in the case of the simple Sellmeyer formula. In the case of ozone the refractivities given by Mr. and Mrs. Cuthbertson are obviously somewhat untrustworthy as these authors themselves point out, since the observations do not lie on a smooth curve. Before any use can be 1
2
C. and M. Cuthbertson: loc. cit. Ribaud: loc. cit.
664
E. C . C. BALY AND R . A . MORTON
made of these it is necessary to obtain the best approximation of the true dispersion curve of ozone and after plotting the observed values on squared paper the best possible curve was drawn through them. Now ozone exhibits an absorption band with centre at A = 2550 Al whence v 2 = I .384083I x 103~. The dispersion curve is exactly expressed by the formula (n-I) (4.9006) =
x IOZ6
6.7573 I ,384083I
X 1030-v2
.47I59X1Oz8 + 3.9006X1 3.0142257x I03'-V2
The values of VI and v g are I ,176471 X 1015 and 5.49016X 1015 respectively, and the ratio of these two numbers is 4.667which is 14/,3.The formula therefore can be written 6.7573X 1oZ6 3.9006X1.47I59XIO2* (n-1)(4.9006) = (3 3 .92I57 I014)2-Y2 (I4x3 .92I57 1014)2-Y2 The values givcn by this formula, which lie exactly on the dispersion curve are given in Table V together with the Cuthbertson observed values. ,
x
x
+
x
TABLEV x 6707.87 6438.47 5790.66 5769.60 54.60.74 5209.08 5085.83 4799.91
(n-1) X 10s calc.
50764 50957 5 I567 51592 51993 52394 52657 53249
(n-1) X 10s
obs.
50764 so967
0
- IO
51514
+53
51624
-32
52000
52375 52621 53290
-7
+I 9 +36 -41
Let it be assumed that the fundamental inolecular frequency of oxygen is the same as that of ozone, and therefore the refractivity of oxygen should be represented by a formula analogous to that for ozone. On this assumption the ultra-violet absorption frequencies of oxygen will be integral multiples of the same fundamental frequency as that of ozone, namely 3 .92I 57 X 1014. It is well known that oxygen possesses an absorption band with centre near to 1900A. and that these rays photochemically convert it into ozone. Now 4X3.92157Xxo~~= I .568628X1015 corresponds to the wave-length 1912.5A. and this therefore may be taken as the centre of the band. The refractivity of oxygen can exactly be expressed by the formula 6.53793X 1 0 ~ ~ 9.41j7 X8.00896X1 0 ~ ~ (n-1)(10.4157)= ( I4x 3 .92I 5 7 x 1014)2-V2 (4x 3 .92I 5 7 x I 0 1 4 ) 2 - Y 2
+
as can be seen from Table VI, 1
Ribaud: loc. cit.
66 5
REFRACTIVITY
The more refrangible band of oxygen therefore is the same as in the case of ozone and the frequency of the less refrangible band is four times the fundamental freuqency instead of three times as in the case of ozone. The different values of Nzare due to the fact that (n-I)a is different for the two gases. Although perhaps too much may not be claimed from this result in view of the inaccuracy of the observed refractivities of ozone, yet it may be said that the general validity of thc new dispersion formula seems to he established.
TABLE VI x 6562.82 j790.66 5460.74 4861.39
(n-1) Xros
calc. 26974.9 27098.8 27170.2
27345.1
(n-r)XIos obs. 26975 27099 27'70 27345
As regards the constants N1, NP, etc. in the numerators of the formula it has already been shown that these constants are the products of the refractivity for infinite wave-length into the square of the frequency of the absorption band. It follows from this, since the refractivity of a gas is directly proportional to the pressure, that (nLI)a must be proportional to the density, the frequency of the absorption band being constant and independent of thc pressure. Again, (n-I)a must obviously depend on the nature of the gas and thu. we may write for any gas (n-I)m = D X C , where D is the total mass of the molecules in one linear centimetre and C is a fundamental constant characteristic of the gas. The value of (n-1); for oxygen is o.000265705 and taking the number of molecules in I cc of oxygen to be 2 .75X10l9 and the mass of one molecule to be I . 5 6 X 1 o - ~ ~ X 3 2the , value of C is found to be I ,7593 X 1012. The frequencies of the two absorption bands of oxygen, v1 and v 2 were ~ 1 4 X 3 . 9 2 1 5 7 X 1 0 ' ~ respectively, but it found to be 4 X 3 . 9 2 1 5 7 X 1 0 ~ and must be remembered that the fundamental frequency may be a sub-multiple of 3.92 157 X 1 0 ~ ~It. is quite evident that this indeed is the case since no very strong absorption band is exhibited at that frequency. I t is true that this frequency lies in the region covered by the A band of the solar spectrum, the corresponding wave-length being 7650 A, but the A bands are only observed with very great thicknesses of oxygen, that is to say the absorptive power is very small. In the infra-red region between 15p and IP osygen only shows two very weak absorption bands at about 4 . 7 ~and 3 . 2 p , and therefore the fundamental frequency must lie on the long wave-length side of 15p. The frequencies of these two absorption bands must themselves therefore be int,egral multiples of the true infra-red frequency. Now the frequency 3 .92 I 5 7 X 10'~ /6 = 6.5359 X 1013 corresponds to the wave-length 4 . 5 9 , ~ and since the acC. Cuthhertson: Proc. Roy. Soc., 83, 151 (1909).
666
E . C . C. BALT A N D R . A . MORTON
curacy of Coblentz's measurements cannot be guaranteed within 0 . 5 % there is little doubt t h a t this is the correct value of the wave-length of the absorption band he found at 4 . 7 ~ . I n order to find the fundamental frequency of which the infra-red frequencies are integral multiples we may take the ratio of t8hese two frequencies corresponding to 4 . 5 9 and ~ 3 . z p which is I .43439. The two smallestinteperswhich havearationear to this are 53 and 37, the ratio of which is I 432, and assuming this to be correct the wave-lengths of the two bands are found t o be 4 . 5 9 and ~ 3 , 2 0 4 ~respectively, which are well within the limits of experimental error. The infra-red fundamental frequency is at once found to be I ,7665 X 1o12 a number which is strikingly close t o the value I 7593 X 1oI2 which was calculatt-d above from the refractivity for infinite wave-length. The value of (n-I), for ozone is found from the disperpion formula t o be o 000488214 and assuming the A4vogadroconstant to be the same as for oxygen the value of the characteristic constant C is found to be 2.10065 X 1o12 which is materially greater than that of oxygen. Although too much stress cannot be laid on the relation between these numbers since the ozone value depends on the accuracy of the absolute determination of the refractivity for one wave-length, it would seem probable that in the ozone molecule an additional valency is called into play. This mould. perhaps lend more support t o the formula 0 = 0 = 0 than to the forinual /'\ . The value of (n-I) oc for 0-0 oxygen is far more trustworthy since the simple Sellmeyer formula which, according to Cuthbertson, expresses the fefractivities of oxygen fairly accurately gives (n-I)= =o.000265308 and C = r . 7 5 6 7 X 1 0 ' ~instead of 0.000265705 and I . 7593 X 1oI2respectively. The evidence afforded by the examples given above establishes some confidence in the correctness of the dispersion formula now brought forward. It may he noted that this formula, if proved to he correct, affords information which is of some importance. In the first place it defines the physical conditions existing in a gas, namely the number and the nature of the phases present and the relative volumes of each. In the second place it gives more accurate values of thc Characteristic absorption bands than can be arrived at by direct measurement. In the third place it leads to an accurate determination of the energy quantum characteristic of a molecule, this quantum being given by thc product of the fundamental frequency of the molecule into the Planck constant. Thus the formula given for chlorine states that this gas a t oo and 760 inm exists as a mixture of two phases of the chlorine molecule, the relative numbers of molecules in each being I O and 3496. Each of these phases is characterised by a+n ultra-violet absorption band, the central wave-lengths of these being 3380.8 A and 926.73 A., respectively. The fundamental molecular frequency of chlorine is I .64325X10'~which corresponds to the wave-length 18 2 5 6 5 ~ and a strong absorpt,ion band should be exhibited at that wave-length. It is now known that chlorine does not exhibit any absorption bands in the infra-
REFRACT1 VITY
667
red at smaller wave-lengths than IS^, but the longer wave region has not yet been examined. The molecular quantum of chlorine will be I 64325 X 1013 x 6.56 X IO-^' = I ,078 X 1 0 - l ~erg per molecule, which is I 6 13 calories per gram molecule. The critical increment of chlorine in any of its reactions should therefore be 1613 calories or some multiple of this. I n view of these deductions from t8hedispersion formula of chlorine it may be noted that these depend within limits on the relative valucs of the refractivities for different wave-lengths and not on the absolute values. The correctness of the value of (n-1); and hence of the constants N1and NSin the formula depend on the latter. Since MI-. and Mrs. Cuthbertson do not claim a greater accuracy than I in 1000 for the absolute refractivity of chlorine which they determined for X = 5460 74, we have not discussed the value of (n-I), for this gas. The relative values are, however, far more accurate and justify the deductions drawn above. The Unic’ersity of Liverpool.
