The determination of molecular param- eters from spectral data forms a highly satisfactory undergraduate physical chemistry experiment [see, for example, reference (I)], but unfortunately requires the availability of a good infrared spectrometer. The pur- pose of this article is to provide enough data for a typical case to allow "dry-labbing." We feel that this approach is acceptable since the collection of data is not the main purpose of such an experiment.'

Usually such experiments involve diatomic molecules since they constitute the simplest possible case. How- ever, the extension to linear triatomic molecules io- volves no real difficulty and does allow the introduction of the ideas of isotopic substitution and degenerate vibrations. A very convenient molecule of this type is hydrogen cyanide, HCN. A mixture of it and deuterium cyanide, DCN, can readily be made by dropping concentrated sulfuric acid diluted with heavy water, DzO, onto sodium cyanide and collecting the evolved gas.

Robefi Linle Harvard University

Cambridge, Massachusetts

T h e Infrared Spectra

Molecular Structure and Thermodynamic Properties of HCN and DCN

Figures 1-3 show three infrared spectra of a gaseous mixture of HCN and DCN (prepared as outlined above) ; Figure 1 is a general survey and Figures 2 and 3 show two of the bands in greater detail. The fre- quencies of the individual rotational lines shown in Figures 2 and 3 were taken from accurate data in the chemical literature (2) .

These spectra contain sufficient information (provided we neglect anharmonicity effects) to calculate the n~olecular structure and thermodynamic properties of the molecules. Molecular Structure

The equations connecting the frequencies of the individual rotational lines in a vibrational band with the rotational quantum number J are (1 ,S): P branch i. = s.~,. - Jn(B' + B") + (JX)l(R' - B") Q branch i = i.,~.,, (forbidden for stretching vibrations in

linear molecules) R branch i = i . j . . , + 2.5' + J"(3B' - B") + (JX)'(B' - B") where i; is in cm-l, i~,. is the classical vibration fre- quency of the molecule, and B' and B" are the rotational

EDITORS NOTE: See, for example, the article by Boer and Jordan in the February issue of THIS JOURNAL, 42, 76 (1965), which provided densitometer tracings of alkali halide X-ray powder photographs. Some comments on the justification of this kind of publication were included in the Editorially Speaking page of the same issue.

Setting B' = B" is an approximation; this can been seen by plotting the frequencies of the lines against rn. A curve is ob- tained instead of a straight line.

constants for the upper (') and lower (") vibrat,ional states respectively. We also have that:

where I' and I" are the moments of inertia in the upper and lower vibrational states.

From the data in Figures 2 and 3 the values of B' and B" (and hence I' and I") can be determined by the method of combination differences ( 1 , 5, 4). A simple discussion of the basis of this method is given by Barrou- (5). Alternatively B' and B" can be assumed to have the same value; the equations given above then reduce to:

P branch i Q branch i R branch i

If we now define m = -J for the P branch m = 0 for the Q branch m = J + 1 for the R branch

the three equations reduce to the same e q u a t i ~ n . ~

where m is merely the ordinal number assigned to the rotational line, counting out from the center of the band, negative in the P branch and positive in the R branch. I t is these numbers that are inscribed on Figures 2 and 3.

The value of B can then be obtained simply by aver- aging the separations of the rotational lines, and from this an average value of the moment of inertia can be calculated.

The relation of the moment of inertia of a linear triatomic molecule to the bond lengths and atomic masses of the molecule is given by (6) :

where the bond lengths and atomic masses are as shown below:

By substituting values for I and ml, mz, and ma we get an equation in rl and rz: in order to determine 1.1 and r2 independently we must use the data for the isotopic molecule. This gives us two simultaneous equations in rl and rz which can be readily solved. Thermodynamic Properties

The calculation of thermodynamic data from spectro- scopic observations is treated in a number of sources (7-9) and so here we will only list the essential equations

2 / Journal of Chemical Mucation

as they refer to a linear triatomic molecule. In addi- tion, by calculating only the specific heat at constant volume and the absolute entropy we avoid complica- tions due to the zero-point energy. For the calcula- tions we need the rotational constants and the funda- mental vibrational frequencies of the molecules. In hydrogen and deuterium cyanides the vibrational modes are :

H C N a n

8-8------ (The + and - signify 0 ---- 0 - + -

w motion at right angles + t,o the plane of the paper.)

a-

C

:? -----a "8

H N

The fundamental freque~~cy v, is absent from the spec- trum of the gas mixture shown in Figure 1. The ex- planation is that for these molecules, undergoing this particular vibration, the changes in dipole moment are so small that the absorptions are too weak to be de- tected. This means that v1 must be determined from the combination band vl + u,:

Rotational Contributions (C,).,t = R cal/mole/'K

Vibrational Contributions

where 81 = hvl /k , 8~ = hv&, 8~ = h v d k and v is in sec-I or u = cs where s is in em-'. Translational Contributions

The Calculation The calculation of the thermodynamic functions can

be carried out either ab initio or with the aid of a table of Einstein functions (8, 10). If the former method is

used the calculations should be set out in the form of a table so that the computations can be followed through in a logical sequence.

The exercise outlined above has been done by stu- dents in the first year chemistry course at Harvard University. Using the approximate method to deter- mine the rotational constants, students obtained values for the bond lengths within about 2% of the accepted values in the literature. Using a table of Einstein functions, each student calculated the thermodynamic functions at a particular (assigned) temperature and plotted his results on a communal graph to show the temperature variation of the functions.

The students results were also checked using an IBM 1620 computer. Copies of the program and reprints of this article are available from Professor L. I

Volume 43, Number I , Jonuory 1966 / 5

6 / Journal o f Chemicol Education