Molecular Structure and Thermodynamic
Robert Little Harvard University Cambridge, Massachusetts
Properties of HCN and DCN
The determination of molecular parama highly satisfactory undergraduate physical chemistry experiment [see, for example, reference (1)], but unfortunately requires the availability of a good infrared spectrometer. The purpose of this article is to provide enough data for a typical case to allow "dry-labbing.” We feel that this
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eters from spectral data forms
approach is acceptable since the collection of data is not the main purpose of such an experiment.1 Usually such experiments involve diatomic molecules since they constitute the simplest possible case. However, the extension to linear triatomic molecules involves 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, INO, onto sodium cyanide and collecting the evolved gas. The Infrared Spectra
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 frequencies 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 molecular structure and thermodynamic properties of the molecules. Mo/ecu/ar Sfrucfure
The equations connecting the frequencies of the
individual rotational lines in a vibrational band with the rotational quantum number J are (1,3): P branch Q branch R branch
i
-
v
=
y,,ut,
v
=
p„,aa5
-
J"(B' + B") + (J’y(B’
B") (forbidden for stretching vibrations in
linear molecules) + IB' + J"(3B’
-
B") +
-
_
-
B")
where V is in cnr1, PciM! is the classical vibration frequency of the molecule, and Br 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. 2 B" is an approximation; this can been seen by Setting B' plotting the frequencies of the lines against m. A curve is obtained instead of a straight line. 1
=
2
/
Journal of Chemical Education
constants for the upper (') and lower (") vibrational states respectively. We also have that:
where /' and J" 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 /' and I") can be determined by the method of combination differences (1, 3, 4.). A simple discussion of the basis of this method is given by Barrow (5). Alternatively B' and B" can be assumed to have the same value; the equations given above then reduce to: P branch CJ branch R branch
If
wc now
v
=
P
—
Pciass
—
Pcilaa
=
P
2BJ"
+ 2B(J” + 1)
define m m m
=
for the P branch for the Q branch
—J
=
0
=
J +
1
for the R branch
the three equations reduce to the same equation.2 ihiass + ‘2Bm i>
=
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. It is these numbers that are inscribed on Figures 2 and 3. The value of B can then be obtained simply by averaging the separations of the rotational lines, and from this an average value of the moment of inertia can be
where
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): j
_
mim2ria J- m3m\rd
+
2mrmsrir2
mi
+
m,z
+
+
where the bond lengths and atomic below: mi
m2
n
mira3r22
+ myniyr^
nia
masses
are
as
shown
m3 r2
o-o-o
By substituting values for I and m,i, m2, and m3 we get an equation in ri and r2: in order to determine and r2 independently we must use the data for the isotopic molecule. This gives us two simultaneous equations in ri and r2 which can be readily solved. Thermodynamic Properties
The calculation of thermodynamic data from spectroscopic observations is treated in a number of sources (7-9) and so here we will only list the essential equations
they refer to a linear triatomic molecule. In addition, by calculating only the specific heat at constant volume and the absolute entropy we avoid complications due to the zero-point energy. For the calculations we need the rotational constants and the fundamental vibrational frequencies of the molecules. In hydrogen and deuterium cyanides the vibrational modes
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 students in the first year chemistry course at Harvard University. Using the approximate method to deter-
as
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. Iv. Nash of the Department of Chemistry, Harvard. This calculation could readily be extended to other linear molecules for which sufficient data is available in the literature [for example, C2H2 (11) and C2HD (18)]. However, because of the difficulties of isotopic substitution and resolution, such data are not always available. For this reason hydrogen and deuterium cyanides are particularly satisfactory for the purpose.
are:
HCN ,-
V\
9-6--9^ o-o--o +
(The + and signify motion at right angles to the plane of the paper.) —
>V2 +
“
)
/T
Chap. 4. (5) Barrow, G. M., “The Structure of Molecules,” W. A. Benjamin, Inc., New York, 1963, pp. 99-105. (6) Townes, C. H,, and Schawlow, A, L., "Microwave Spectroscopy,” McGraw-Hill Book Co., New York,
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