INDUSTRIAL AND ENGINEERING CHEMISTRY
November. 1931
1241
Some Contributions of Molecular Spectra to Classical Chemistry' Harold C. Urey DEPARTMENT OF CHEMISIRY, COI.UMBIA UNIVERSITY, NEWY O R KN. , Y
The experimental data of molecular spectra and the
HE science of chernis-
h*
+
E = -J ( J 1) theoretical interpretation of these data give, primarily, 8 ~ ~ 1 try d u r i n g t h e past information in regard to the energy states of molecules century and more has where I = moment of inertia and their a priori probabilities. With the aid of the concerned itself to a great exand J = an integral number, theoretical interpretation of these data, dimensions 0,1,2 ,.... tent with the following subof molecules, the masses of the atoms of the molecules, jects: atomic and molecular the binding forces between the atoms, the dissociation I n transitions in which light weiglits, valency and valence energies, and electrical and magnetic moments can is absorbed, J increases by forces. the forms and dimenbe deduced. Data of this kind of a most detailed one unit, and equally spaced sions of molecules, the energy character exist a t the present time and are rapidly frequencies must result, as a n d o t h e r thermodynamic increasing in volume. is observed. properties of substances, the In this paper some phases of the subject are presented. In addition to this rotakinetics of reactions, the deThe methods of deriving the dissociation. energies of tional energy, the molecule tection and e s t i m a t i o n of diatomic molecules are reviewed and the problem of the may have vibrational energy, substances, the synthesis of triatomic molecule is discussed with particular referand the transitions between new c o m p o u n d s , and the ence to C102and SO2. steady states with the emisperiodic law. To all of these sion or absorption of light may subjects, with the exception of the last two, molecular spectra as a new research tool has involve changes in the vibrational energy as well. If this been able to make some valuable contributions. During occurs, the near infra-red rotation-vibration bands are emitted the past fifteen years especially, the data and theory of or absorbed, Also, the molecule may have electronic energy, molecular spectra have increased enormously until a t present and, if changes occur in this as well, the electronic-rotationno adequate review of the entire field could be given in the vibration bands are emitted or absorbed. The derivation space available.2 The present paper will be confined to a of the energy levels is carried out by the aid of theory and the Bohr frequency condition, for there is a unique set of few illustrations. The energy relations and forms of diatomic and triatomic levels which will account for the observed spectrum. molecules, and especially recent work on the C102 and SO1 The energy of a diatomic molecule consists of its electronic, molecules in some detail, will be discussed in this paper. vibrational, and rotational energy. The formula for this energy may be written to a high approximation as:
T
Experimental Data of Molecular Spectra and Their Interpretation
Molecular spectra fall naturally into three groups, on the basis of experimental technic and of theoretical interpretation. These three groups are: (1) the far infra-red or rotation bands; (2) the near infra-red or rotation-vibration bands; and (3) the visible and ultra-violet bands, or the electronic-rotation-vibration bands. The first group consists of more or less regularly spaced monochromatic wave lengths in the region from about 20 to Mop. The spectra of the diatomic hydrogen halides have been extensively studied in this region (7) and it is found that they are very nearly equally spaced in frequency Y = or in wave numbers (7 = These wave lengths are known to be due to light emitted or absorbed in transitions between the rotational levels of the molecule in accordance with the Bohr frequency condition, E' E" y=-
i).
( i),
-
h where E' and E" = two values of energy of the molecule h = Planck's conatant and Y = frequency of light emitted
Theoretically the rotational energy to a high approximation IS given by the formula, 1 Received October 1, 1931. Presented under title of "Molecular Spectra." 2 A number of reviews of the general theory of molecular spectrn are available, though none that could be read i n a short time. A very good elementary review is given by Darrow ( 8 ) . Somewhat more advanced is the treatment given by Ruark and Urey ( Z Z ) , and still more advanced is that of Mulliken ( 2 0 ) . At a former symposium of the AMERICAN CHEMICAL SOCIETY, Darrow ( 9 ) gave a review of the methods of determining dissociation energies of diatomic molecules.
where Y
c = velocity of light N
YO, xuo, u2, e,
a
- = constants
and J = integral numbers, 0, 1, 2,
The vibrational energy, E,, is the second two terms on the right, and the rotational energy, E,, the last three terms. The order of magnitude of these terms is E, >E, >E,. The transitions between levels with the emission or absorption of light give a spectrum consisting of very fine lines, as observed with spectrographic instruments. With low dispersion these lines may not be resolved so that a banded spectrum is observed, whence the name "band spectra." To discuss the methods used for deriving the energy levels from the detailed mass of wave lengths observed would involve too much for the present discussion. It will be assumed that this has been done and that a correct set of energy levels has been obtained. A knowledge of these energy levels makes possible the calculation of the heat capacity, entropy, free energy, etc., of a gas more accurately than they can be measured directly and enables statistical mechanics to give an understanding of the third law of thermodynamics, which could not be secured in any other way. The electronic energy levels of atoms and molecules make it possible to unravel the detailed theory of valence. The dissociation-energy and the detailed potential-energy functions of molecules are especially important in the field of kinetics. The recent discoveries of isotopes of oxygen,
I N D U S T R I A L A N D ENGINEERIXG CHEMISTRY
1242
nitrogen, and carbon are not only interesting discoveries in themselves, but also contribute to the question of correct atomic weights. The analytical applications of molecular spectra may also be quite important in many cases. In this paper the potential-energy functions of molecules and the details of their vibrations will be considered in some detail. !3
I
I'
6
1
I
can then be calculated theoretically in terms of the constants of this assumed power series; and, finally, using the empirical data, it is possible to deduce values for the constants of the assumed potential energy function. This was done by Kratzer (19). [See also (80, 6 2 ) . ] A much simpler formulation has been given by Morse (6) however. He has shown that if the potential energy is V r ro D and a
where
i
Vol. 23, No. 11
+
= -ZDe-a.(r - 7 0 ) De-za(T - ro) = distance between atoms = equilibrium distance
(2)
= constants
quantum mechanics requires that the energy states shall be N
=
Figure 1-Potential-Ener
y and Force Curves of HCI 8 0 )
Dissociation Energy of Diatomic Molecules
It is found that the vibrational energy levels of diatomic molecules follow a formula of the form, =
hc where
x;;'o(zI + + &(a + - and x v-o =- constants
Ygb +
1/2)2
1/2)
+
..
VO(Y
-
+;'/2)
*&
+
I/?)*
(3)
the m's being the masses of the two atoms and c the velocity of light. The potential-energy curve of Figure 1 for the HCl molecule follows this formula. Thus, knowing the values of G and ZG from the experimental data, the potentialenergy curve can be calculated. The significance of the constant, D, is easily seen. If T = a, V from Equation 2 is zero; and if T = T ~ I,. = -D, so that D is the dissociation energy of the molecule and can be calculated from the constants, & and 2%. This calculation will be valid only if the quadratic equation for E holds out to dissociation of
. (1)
YO
c
and v
= velocity of light = an integral number, 0, 1, 2,
... . . .
The important terms in this formula for most molecules are the v l / 2 and (v I / Z ) ~ terms, so that these two terms give good agreement with observations even when v is rather large. For larger values of v it usually happens that no simple power formula can be used to describe the observed energy levels. From these energy levels it is possible to find the forces acting between the two atoms. These forces will be a function of the distance between them; in case a stable molecule is formed by two atoms, this force a t large distances will approach zero but be one of attraction, and a t small distances it will be very large and be one of repulsion, while for some intermediate position the force will be zero. The potential energy, V , which is related to the force through the relation,
+
+
F
=
--SV/Gr
will have a minimum a t the equilibrium position, 6V/6r, will be negative a t smaller distances, and positive a t large distances. These relations are illustrated in Figure 1. While the general form of this curve is known from these general considerations, the exact shape of the curve for any molecule can be secured onlv from the data of molecular spectra and the exact theory of such spectra. A different potential-energy curve can be expected for each electronic state. The potential energy can be written as a power series of the distance between the nuclei: the energy levels
-+Y
Courtesy of McGraw-Hill Book Co.
Figure 2-Energy Levels of Iodine Molecule (22)
the molecule. This is often approximately true, and apprusimate values for the dissociation energy can be secured in this way. I n case the energy does not follow the simple Equation 3, the dissociation energy can still be calculated. The frequency of vibration of the nuclei relative to each other is given by the expression w(n) =
where
IZ
=
Y
+
1 6E - h 6n
(4)
INDUSTRIAL A N D ENGINEERING CHEMISTRY
November, 1931
1243
comes horizontal so that 6E/6n‘ = 0, and the frequency of vibration is zero, which means that the molecule dissociates. The second arrow from the right shows the dissociation energy. This dissociation, however, is into one excited atom and one unexcited atom, and differs from the energy of dissociation of the molecule into two normal atoms by 0.94-volt electrons. The energy of dissociation from the spectrum is 2.5-volt electrons, and, subtracting 0.94, this gives 1.53-volt electrons or 35.3 Kcal., in very good agreement with the thermal value of 36.9 Kcal.
N’
2
+
Figure 3-Plot of E(v:’ l ) - E ( w t ’ ) a g a i n s t w2’ Ti\ o lines drawn through lower points indicate uncertainty in drawing curve through these points.
-4 second method for securing dissociation energies of diatomic molecules is from the phenomenon of predissociation discovered by Henri (16). One of the best examples of this phenomenon occurs in the spectrum of SZ. The individual bands are made up of numbers of nearly monochromatic wave lengths, which are grouped together in a characteristic way. These lines, as they appear on a photograph, are sharp and distinct. In the case of the SJ molecule these lines are sharp only to 2799.1 A., and a t shorter wave lengths the lines become diffuse. This sudden transition from sharp monochromatic wave lengths to the diffuse region is h o w n as predissociation and was 6rst explained by Bonhoeffer and Farkas (4). This explanation c m be understood by referring to Figure 2 and noting that the addition of more vibrational energy levels to the lower electronic state might result, in some cases, to an overlapping of the vibrational states of the two electronic levels. (This is not true for these two electronic states of 12.) If this occurs, the upper set of vibrational and rotational energy levels overlaps a region of discrete energy levels and aho a region of continuous energy levels of the lower electronic state. When this occurs, all the energy levels overlapping the continuous region may be diffuse and continuous, and transitions involving these levels will give diffuse and continuous spectra. The structure of the levels is not com-
assuming that classical mechanics holds for the molecule, an assumption which is suffisiently good for present purposes. When this frequency becomes equal to zero, the molecule is dissociated, since the time of one oscillation is infinite. The energy of dissociation is then equal to
where no = value of
(Y
+-
I/*)
a t which w ( n ) becomes equal to zero
If an empirical formula for the energy levels can be constructed from the data, this integration can be performed analytically, or if necessary it can be carried out graphically. This method was first used by Hulthen (16), Franck ( l a ) and Dymond (11), and was discussed and applied to a number of molecules by Birge and Sponer (1, 2). I n many cases the method has given good values for the dissociation energies of diatomic molecules, though i t must be used critically. Unless the vibrational energy levels :%reknown nearly to the dissociation limit, the method cannot be used with confidence, for extrapolations over large distances give unreliable results. Moreover, the dissociation may be into excited atoms, so that care must be taken to avoid errors because of this possibility. The relations which have been discussed are illustrated by Figure 2, which is a diagram of the observed spectrum of I2 (at the bottom) and the transitions which give these bands. The levels numbered from 0 to 38 in the upper part of the diagram are the vibrational levels of the higher electronic state, while only two levels, 0 and 1, of the lower electronic state are shown. The arrow heads form a plot of energy plotted upward against vibrational quantum number to the right, so that they trace a parabola approximately. This curve a t Y ’ S 66 be-
E X C T l P ST4TE
Figure 4-Normal
Coordinates of ClOn Molecule
pletely washed out but only made diffuse, so that bands are still observed but with the h e structure due to the individual monochromatic wave lengths destroyed. The energy level a t which this diffuseness first appears is therefore a maximum value for the energy of dissociation. Its appearance may be due to a dissociation into excited atoms, and care must be used in deducing dissociation energies in this way. The energy of dissociation of Sz into two normal S atoms is determined in this way in 101.5 Kcal., and this value is a very precise one. C102 and SO2 Molecules
After this review of the principal facts of the methods for determining the dissociation energies of diatomic molecules, a discussion of the triatomic molecules, ClOz and SOZ, may be made. The data necessary for the discussion of the Cl02 molecule have been secured by Miss Helen Johnston (17) and the writer, and will be published soon. The discussion of the SO2 molecule is based on calculations made with the help of Donald MacGillnrrp.
INDUSTRIAL AND ENGINEERING CHEMISTRY
1244
Again the internal energy may be roughly separated into electronic, vibration, and rotational energy. I n the case of both molecules, two electronic steady states, which will be
= 719.34(uzt
+
- 2.817(02' + l/#
l/~)
Vol. 23, No. 11
+ terms in
vi'
(7)
and from V'Z = 15, to larger values of v ' ~by a formula of the This is illussame form, but different values of ;;;1 and x& trated by Figure 3, which shows a plot, 8E/8uz1,against t 2 ' . If the energy is a quadratic function of (v2' l/2), the plot should be a straight line, while the figure shows that t h e plot consists of two straight lines. Such an irregularity makes the determination of dissociation energies by extrapolating such curves to a point, for which 6E/6vz' = 0, still more uncertain unless the levels are known nearly to the dissociation point. The writer believes this discontinuity is due to a change in the mode of vibration of the molecule and will return to it later. Predissociation is observed in the bands of C102. Finkelnberg and Schumacher ( I d ) , who have studied these bands under highest dispersion, conclude that i t first appears in bands coming from the level with VI' = 0, 02' = 10. These authors, as well as Goodeve and Stein (14), conclude that this indicates dissociation into a C10 molecule and a normal 0 atom. Further evidence in support of this view has been secured in the present author's study of this spectrum. Bjerrum (3) considered two types of forces for describing the vibrations of a triatomic molecule with two like atoms: (1) the forces may be assumed to act along the lines joining the atoms; (2) the forces act along the chemical bonds and perpendicular to them. He called these the assumptions of central forces and of valency forces, respectively, and these terms will be used here. He calculated the vibration frequencies for infinitesimal amplitudes for each of these types of forces and applied them to the COz molecule. Neither of these two assumptions appears to be superior to the other in the case of the normal state of CO1, though other evidence indicates that the second is correct.
+
. L'.
012
5
I
I
XI
Po
Figure 5-Vibrational
I
I
40
30 IF-
I
1
JD
€0
Levels of Excited Electronic State of Clot
referred to as the normal and excited states, are to be considered. I n the case of these triatomic molecules there will be three vibrational degrees of freedom, three of rotation, and three of translation, and again only the vibrational degrees of freedom are to be treated. The vibrational energy will now depend on three vibrational quantum numbers, vl, V Z , and v3, and the vibrational energy should be similar t o that of the diatomic molecule,
-
EE
=
-Y
h
+
1/21
+
x171(v1
+
l/#
+
bwo
I
f
0 plus higher terms containing cross products and higher powers of the vibrational quantum numAv 00 Y bers. While in the case of diatomic molecules eFigure 6-Potential Energy Curve for C l o t Excited State the quadratic terms are negative [the only known exception is in the case of LiH (21)1, these terms The frequencies (these are approximately the constants, are usua!ly positive in the case of these triatomic molecules (64, 25). Thus no estimate of dissociation energies could ;;; and E, of Equation 6) are functions in each case of be made by using only the linear and quadratic terms, for three constants, two force constants, and the half angle a t the levels would never converge so that 6E/8n would never the apex of the isosceles triangle. For central forces these be zero. I n estimating the dissociation energies by the frequencies in terms of the force constants are: method described for diatomic molecules, great care must be taken, or completely erroneous results will be secured. 1 Even when the quadratic terms in the energy are negative, they are usually too small to give a correct approximation 1 kl 4- 2k2 kl cos2 e (8) to the dissociation energy, so that the use of the linear and = 2R +M quadratic terms in the energy, neglecting the others, gives too large values for the dissociation energies. - 2klkn mPM t + @?.??)' M I n addition to these difficulties, there is another irregularity where ki = force constant for a harmonic force acting bein the formulas for the energy levels, which is known to occur tween the unlike atom and one of the like atoms in the excited energy states of ClOz ( I d ) , SO2 (as), and kl = constant for the force acting between two like CZHz (18). The nature of this irregularity is well illustrated atoms by the ClOz vibrational levels of the excited electronic state. 8 = half angle a t the apex Up to the level with vz = 15, the energy can be represented and m and M = masses of like atoms and the unlike atom, reby the formula, spectively
$
T[
d(?!&?&
November, 1931
I X D U S T R I A L A N D ENGINEERIhlG CHEMISTRY Table I-Data
857.7
727.0
528.8
c
38 69
8.41
7.96
21.81
for Normal and Excited States
Normal state
719.3
(Calcd.) 382.1
(Calcd.) 10.32
(Calcd.)
-0.80
-6.06
-
130'
d
304.8
1245
37.50
(Calcd.) -2.80
-5.88
.*
(Obsd.) -4.60
(Obsd.) -3.26
-0.80
-6.37
k
Excited state
I f three observed frequencies, v l , vZ, and v3, are observed, they can be substituted in these formulas, and ICl, k2, and 8 can be solved. The equation for 8 is quadratic in cos28, so that there are two possible vahes of 8. I n addition, the three observed frequencies may be assigned in three ways to these formulas and, therefore, six values for 8 are possible. In the case of the normal state of CIOz and the excited state of S02, all these possible solutions give a complex value for cos2 8 or a value which does not lie between 0 and 1 and thus give imaginary angles. This formula, for purposes of the present discussion, may be discarded.
Figure ?-E(v'
+ 1) - E(v') Plotted against v'
that of the third two is most extensive and the precision is probably as great as that secured by the others. For the normal state the third authors have observed five of the possible six quantities, and for the excited state four of the possible six quantities. Of the nine possible values of 8 for the normal state only three are real, and of these three only one leads to agreement with all five observed quantities. Again, only one value of 8 for the excited state leads to agreement with the four observed quantities and the intensity distribution of the bands. Table I shows the agreement secured, the assignment of the observed frequencies to each of the formulas of Equation 9, and the values of kl, IC2, and 8 required for the normal and excited states. The normal modes of vibration can be calculated by wellknown method8 (3, l o ) and are shown in Figure 4. The arrows, by direction and magnitude, show the simultaneous relative displacements of the atoms for each vibration frequency. It is evident that a t large amplitudes the mode of vibration, 7 = 304.8, will result in the dissociation of the molecule into C10 and 0. There are sufficient energy levels of this normal mode of vibration to extrapolate to dissociation, using only the quadratic formula. This value is not precise
(25)
The assumption of valency forces gives the following formulas 'or the frequencies, kz being the force constant perpendicular to the bond and the other symbols having the same meaning as before:
1
Jb
8-
Figure 8-Potential Energy Curves for SO, Excited State
Again the substitution of the observed frequencies can be made and the equations solved for IC,, k ~ and , 0. This gives a cubic equation in cos2 8 and thus nine values of 8 are possible, but only real roots lying between 0 and 1 give real angles, and others may be ignored. Another method for determining the constants is available in the case of C102, for the spectra due to the molecules containing C136and those containing C13' are different. The difference can be predicted by using the two values of M in the equations for the frequencies, since kl, h,and 8 should not be different for the two molecules. Thus, all together, six quantities can be observed-three frequencies and three isotope effects. From these it is desired to determine three unknown constants. It is evident that three of the possible six quantities must be observed in order to solve for the unknown constants and, if more than three observed quantities are known, all must be calculable from the same values of kl, k2, and 0. These checks have been used in the work on Cl02. The available data on the Cl02 bands consist of the work of Goodeve and Stein ( I 4 ) , of Finkelnberg and Schumacher ( l a ) ,and of Urey and Johnston. That of the second two authors was secured with the highest resolving power;
but, together with the predissociation limit, fixes the energy of dissociation into C10 and 0 with certainty. Figure 5 shows a plot of the energy levels of the excited electronic state and the extrapolation of this mode of vibration (z' = 304.8) to dissociation. The successive dots in each vertical row are the energies for increasing values of 02. The successive columns are the energies for increasing values of vI. The levels with 03 = 0 have not been observed. The extrapolated value for dissociation is 28,968 crn.-' above the normal state. The predissociation value is 27,162 cm.-l (Finkelnberg and Schumacher's value interpreted in terms of the present author's energy-level diagram). The latter value is the more precise but the present author's extrapolation shows that the dissociation is into C10 and 0. Quantities which can be calculated when this quantity and others, secured in a similar way, are known are illustrated in the following table: REACTION
(b (b
++ cc +
(a) (b) (c) (d) d) d)
AE
+
ClOa --f CIO 0 ClOz --f l/zClz 4- 01 20 '/?Clz CI ClOz CI 4- 2 0 CIO --f 0 C1
01
+ + +
+
METHOD OF DETERMINING AB
Kcal. 77.2 23.6 118.0 28.7
170.2 93.0
Predissociation Thermal Predissociation Band spectra
Returning to a discussion of the discontinuity in the energylevel formula associated with the frequency, ;;;I = 719.3 (see Figures 3 and 4), it is seen that the unusual intensity
1246
INDUSTRIAL AND ENGINEERING CHEMISTRY
distribution in the observed bands can be explained by the marked difference in the shape of the molecule in its two steady states. The Franck-Condon (5) principle requires that, during an electron transition, the position and velocities of the nuclei shall not change. The velocities are small in the normal state and thus, in a transition due to the absorption of light, the molecule arrives in the higher electronic state with its nuclei a t rest and an angle of 130' a t the apex of the isosceles triangle. The equilibrium angle is 37.5", and thus it is far away from equilibrium. A wide-amplitude oscillation of a flopping kind results, the oscillation being between the 130' angle and an angle somewhat less than 37.5'. This means, in the language of quantum transitions, that the quantum number for this mode of vibration must change by large amounts. The most intense band of the entire spectrum is that for which v2 changes from 0 in the normal state to 11 in the excited state. The twelfth state is the one, then, for which the oscillation is between 130" and
