Question

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 temperature T as a function of n, N, and ɛ.

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)...

Consider a system composed of two distinguishable free
particles, each of which can have energy E = nε, for n = 0, 1, 2,
.... List the microstates with energy E = 3ε.

Consider a system composed of a very large number (N
>>1) of indistinguishable atoms,
each of which has only two energy levels 0 and . In thermodynamic
equilibrium, there
are n particles each with energy and the total energy of the
system is E. (10 points)
a) Write and equation for the number of microstates in terms of
N, E, and .
b) Use the Stirling approximation to write an equation for the
entropy S.
E 1 and...

A molecule has three degenerate excited vibrational states, each
with excitation energy ? above the ground state.
a) At temperature T, what is the ratio between the number of
molecules in (all of) these vibrational states and the number in
the ground state?
b) At very high T, what is this ratio?
c) Assume you have N distinguishable molecules of this type. Use
the free energy to compute entropy S/k of the system at temperature
T
d) Compute the number...

A. Consider a hydrogen atom with one electron and quantized
energy levels. The lowest energy level (n = 1) is the ground state,
with energy -13.6 eV. There are four states corresponding to the
next lowest energy (n = 2), each with energy-3.4 eV. For the
questions below, consider one of these four states, called one of
the first excited states.
2. Assume that this hydrogen atom is present in a gas at room
temperature (T ~ 300 K, kBT...

A system consists of N = 106 particles that can occupy two
energy levels: a nondegenerate ground state and a three-fold
degenerate excited state, which is at an energy of 0.25 eV above
the ground state. At a temperature of 960 K, find the number of
particles in the ground state and in the excited state.

Derive the Sacker-Tetrode Equation which is the entropy
expression for N indistinguishable ideal gas atoms of total energy
U in a container of volume V.

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...

Assume you have a system that can have one of six distinct
energy states E1 E2 E3 E4 E5 E6 what is the entropy of an ensemble
of five systems where one system is in energy state E2 three
systems are in energy state E3 and one system is in energy state
E6.
Choose the correct answer
S= Kb ln(120)
S= Kb ln(24)
S= Kb ln(20)
S= Kb ln(5)
S= Kb ln(2)
S=0

Consider a molecule that can be in one of two different
conformation states A or B. These states are two different
arrangements of the atoms: e.g., in state B, one part of the
molecule could be rotated about a bond with respect to the rest of
the molecule. Assume the energies of states A and B are 4e-21 and
8e-21 J respectively. At room temperature, T = 298 K, what is the
relative likelihood of the molecule being found in...

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