Statistische Thermodynamica 2014 Werkcollege 5 Exercise 13 Torsions in ethane

Statistische Thermodynamica 2014 Werkcollege 5
Exercise 13
Torsions in ethane
Consider the ideal gas ethane, C2 H6 (g). We consider the rotation of the methyl groups around the C-C bond
(torsion). The molecule’s torsion energy, t (φ), can in a classical model be described by a cosine (see figure).
The difference between the eclipsed (φ = 0◦ ) and staggered (φ = 60◦ ) conformations is 12 kJ/mol.
a) Calculate the temperature at which the ratio eclipsed : staggered is 40%, i.e.,
neclipsed
nstaggered
= 0.40
b) Give an expression for t (φ).
c) Give an expression for the torsion-partition function. For that we approximate the partition sum over all
discrete torsion states i with an integral and change the integration variable to the torsion angle φ. In
di
.
doing so it will be necessary to introduce a density of states per interval of torsion angle ρ(φ) = dφ
d) The expression for the torsion-partition function found in c. cannot be solved analytically. Simplify the
partition function by replacing the cosine by a triangle, i.e., by drawing straight lines between the maxima
and minima of the cosine. In this approximation ρ(φ) simplifies to a constant C. Calculate the torsion
partition function at T = 300 K in terms of C.
e) Calculate
R the mean total energy at T = 300 K for the approximation used in d.
Hint: dxx exp(x) = x exp(x) − exp(x)
12
Torsion energy εt(φ)
10
8
6
4
2
0
Exercise 14
0
60
120
180
240
Torsion angle φ [deg]
300
360
Speed of sound
21
The heat capacity ratio of a gas determines the speed of sound cs in the gas by cs = γRT
, where γ = CP /CV
M
and M is the molar mass of the gas. CP and CV are molar heat capacities.
For most gases, at room temperature, one can use the high temperature approximation for the rotational
contribution to the partition function, while at that temperature the contribution of the vibrations can be
neglected.
Use the statistical thermodynamic expression for CV to find expressions for the speed of sound considering only
translations and rotations, in the high temperature approximation, for a perfect gas of
a) diatomic molecules
b) linear triatomic molecules
c) non-linear triatomic molecules
d) Give an estimate of the speed of sound in air at 25 ◦ C
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