Question

1. Consider a three level system in which the energies are equally spaced (by energy ε); each of the levels has certain (nonzero) degeneracy g .

A. Write down the general expression for the average energy and the partition function of the system.

B. Compute the occupations for ε = kT, when (i) all the states are singly degenerate and (ii) when the degeneracies are g0 = 1, g1 = 1, g2 = 3. Here gj represents the degeneracy of the jth state.

C. Suppose that ε = 2kcal·mol−1 and the degeneracies are g0 = 1, g1 = 1, g2 = 1000. At what temperature T will the occupation probability for finding the system in energy state 0 and 2 be the same? What will be the occupation of state 1 at this temperature?

Answer #1

Consider a system of distinguishable particles with five states
with energies 0, ε, ε, ε, and 2ε (degeneracy of the states has to
be determined from the given energy levels). Consider ε = 1 eV (see
table for personalized parameters) and particles are in equilibrium
at temperature T such that kT =0.5 eV: (i) Find the degeneracy of
the energy levels and partition function of the system. (iii) What
is the energy (in eV) of N = 100 (see table)...

Find the average energy and the heat capacity at constant volume
for a two-state system. Take the two energies of the system to be ±
ε/2. (b) Show that the result for CV is proportional to 1/T2 for kT
>>ε , and that it is of the form CV ≈ (constant/T2 e ^(−ε/kT)
at low temperature, kT << ε

Consider a molecule having three energy levels as follows:
State
Energy (cm−1)
degeneracy
1
0
1
2
500.
3
3
1500.
5
part a) What is the value of the partition function when
T = 360 K ?
part B)What is the value of the partition function when
T = 3600K ?

The first excited vibrational energy level of ditomic chlorine
(Cl2) is 558 cm−1 above the ground state.
Wavenumbers, the units in which vibrational frequencies are usually
recorded, are effectively units of energy, with
1cm−1=1.986445×10−23J. If every vibrational
energy level is equally spaced, and has a degeneracy of 1, sum over
the lowest 4 vibrational levels to obtain a vibrational partition
function for chlorine.
A) Determine the average molar vibrational energy
<Em.vib> for chlorine at 298 K.
B) Determine the population...

Consider a system of N distinguishable atoms, each of which can
be in only one of two states: the lowest energy state with energy
0, and an excited state with energy ɛ > 0. When there are n
atoms in the excited state (and N-1 atoms in the lowest state), the
total energy is U = nɛ.
1. Calculate the entropy S/k = ln(Ω(n)) and find the value of n for
which it is maximum.
2. Find an expression for...

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