A triatomic molecule can have a linear configuration, as does CO 2 (Figure a), o
ID: 1316358 • Letter: A
Question
A triatomic molecule can have a linear configuration, as does CO2 (Figure a), or it can be nonlinear, like H2O (Figure b). Suppose the temperature of a gas of triatomic molecules is sufficiently low that vibrational motion is negligible.
(a) What is the molar specific heat at constant volume, expressed as a multiple of the universal gas constant (R) if the molecules are linear?
Eint/nT =
(b) What is the molar specific heat at constant volume, expressed as a multiple of the universal gas constant (R) if the molecules are nonlinear?
Eint/nT =
At high temperatures, a triatomic molecule has two modes of vibration, and each contributes
R
to the molar specific heat for its kinetic energy and another
R
for its potential energy. (c) Identify the high-temperature molar specific heat at constant volume for a triatomic ideal gas of the linear molecules. (Use the following as necessary: R.)
Eint/nT =
(d) Identify the high-temperature molar specific heat at constant volume for a triatomic ideal gas of the nonlinear molecules. (Use the following as necessary: R.)
Eint/nT =
(e) Explain how specific heat data can be used to determine whether a triatomic molecule is linear or nonlinear.
Explanation / Answer
(a) for linear molecule, number of degrees of freedom= 3n-2= 3(3)-2=7
energy associated with all the 7 degrees of freedom of one mole of gas=E= 7(1/2 RT)=7/2 RT
Cv=E/T
Cv= (7/2 RT)/T
Cv= 7/2 R.
(b) for non linear molecule, number of degrees of freedom= 3n-3= 3(3)-2=6
energy associated with all the 7 degrees of freedom of one mole of gas=E= 6(1/2 RT)=3RT
Cv=E/T
Cv= (3 RT)/T
Cv= 3 R.
(c) contribution of vibrational modes per degree of freedom for linear molecule= 1/2RT+1/2RT= RT
total energy for linear molecule= RT+ 7/2 RT=9/2 RT
Cv= E/T= 9/2 RT/T
Cv= 9/2R.
(d)contribution of vibrational modes per degree of freedom for non linearmolecule= 1/2RT+1/2RT= RT
total energy for non linear molecule= RT+ 3RT=4RT
Cv= E/T= 4RT/T
Cv= 4R.
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