CH273
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CH273
SECTION A
1. Answer ALL parts.
(a) What is the relation between the phase space and the partition function?
[10 %]
(b) The translational, rotational and vibrational canonical partition functions for a diatomic molecule can be derived using the energy levels of three different quantum mechanical systems: which ones? List the main approximation we have employed when deriving the vibrational canonical partition function.
[10 %]
(c) The following equation can be solved numerically to e.g. investigate the extent of the ionic atmosphere
around biomolecular species:
Explain why this equation cannot be solved analytically an(1)dφ(desc)ribe how the 2nd Debye-Huckel approximation can provide a workaround. Remember that fcorrespond to the electrostatic potential.
[10 %] (d)
(i.) Demonstrate that the vibrational partition function for a diatomic molecule can be written, when measuring the vibrational energy levels relative to the bottom of the internuclear potential well, as:
e −βhv/2
qVib . = 1 − e−βhv
Remember that the vibrational energy levels can be e −βhvn = 1 − e(1)−βhv written as: and that: En = ✓n + ◆ h⌫
[20 %]
(ii.) Modify the expression of this partition function assuming that we are now measuring the vibrational energy levels relative to the vibrational ground state.
[20 %]
(iii.) Using the expression you have obtained in (ii.), calculate the vibrational partition function of the HBr molecule (vibrational frequency = 7.68 . 1013 Hz) at 2000 K.
[30 %]
2. Answer ALL parts.
(a) Define the concept of entropy according to Boltzmann.
[10 %]
(b) Consider a system containing 100 particles: 1, 2, …, 100. What is the meaning of the following
marginal probability?
P (1 ,2) (r1 , r2 )
Note that r1 and r2 correspond to the positions of two particles 1 and 2.
[10 %]
(c) The 1st Debye-Huckel approximation assumes that the potential of mean force is entirely electrostatic in nature - which means that any other interactions are ignored. This is usually a sensible approximation when dealing with ions in dilute solutions: why? Hint: the attractive part of Lennard-Jones potential scales as r--6, where r is the distance between two ions.
[ 10 %]
(d) Consider the 16O2 molecule. The electronic wave function corresponding to its ground state is antysimmetric with respect to the exchange of the two 16O nuclei. Knowing that the latter have nuclear spin I = 0, and that the degeneracy of the nuclear wave function is in this case:
a. (I+1) (2I+1) for symmmetric (ortho-) spin states, and
b. I (2I+1) for antisymmmetric (para-) spin states
(i.) Illustrate the reason why the 16O2 molecule cannot exist in symmetric rotational states.
[20 % ]
(ii.) Write down the expression for the nuclear/rotational partition function of the 16O2 molecule.
Remember that the rotational partition function for a heteronuclear diatomic molecule can be written as:
qe(.)ro. “ ÿ p2J ` 1q¨ e ´βhcBJpJ+1q
J=0
[25 %] (iii.)
Calculate the value of the nuclear/rotational partition function of the 16O2 molecule using the expression obtained in (ii.), up to the third excited rotational state. Note that in this case:
βhcB = 0.075
[25 %]
2022-08-18