Chem 120B Spring 2012
Problem Set 6 Solutions
1. Chemical potentials and particle density uctuations: the grand canonical ensemble.
(a) We can derive the probability either from the microcanonical (N, V, E ) or the canonical (N, V, T )
enssemble.
Microcan
YOUR NAME HERE
October 7, 2014
Chem 120B Midterm #1
Denitions and Useful Formulas:
Inverse temperature: = 1/kB T , kB = 1.38 1023 J/K
Boltzmann distribution and canonical partition functions:
eE
,
Q
P () =
Q=
PA
QA
=
PB
QB
eE ,
Equilibrium averages:
E =
Chem 120B Spring 2013
Problem Set 7 Solutions
1. (a) The temperature of coexistence is a function of both p and x, T (p, x) = T0 (p) + T (p, x) where
T0 (p) is the coexisting temperature for pure liquid water. Use a Taylor series to express T as a
linear
Chemistry 120B Problem Set 4
due March 1, 2013
1. Einstein crystal. Albert Einstein pictured a low temperature crystal as if each atom was vibrating
independently of all the others, so that the net energy of a N atom microstate is
N
N
(x + y + z ) +
[x n
Chemistry 120B Problem Set 8
due April 17, 2013
k1
k2
k1
k2
1. Consider a threestate kinetic scheme 1 2 3. That is to say,
dc1
= k1 c1 + k1 c2
dt
dc2
= k1 c1 k1 c2 + k2 c3 k2 c2
dt
dc3
= k2 c3 + k2 c2
dt
where the net probability is normalized, c1 + c2 +
Chem120B
Problem Set 5
Due: October 2, 2015
1. Rotations about a covalent bond, such as the carboncarbon bond in ethane, are hindered by forces
between other chemical groups attached to the bonded atoms. The potential energy U due to these
forces typical
Chem 120B
1.
Problem Set 4 Solutions
Due: October 2, 2013
(i) At what temperature does the vibrational transition energy of O2 , 1580 cm1 , equal
kB T ? Wavenumbers (cm1 ) are used as a unit of energy, with conversion factor cm1 =
2 1023 J . Using this co
Chem 120B Spring 2012
Problem Set 4 Solutions
1. Einstein Crystal
(a) Determine the average energy of the Einstein crystal as a function of N , temperature T , and the
fundamental frequencies x , y , and z .
Q = exp
N
(x + y + z )
2
nx1 =0
N
exp
nxN =0
Chemistry 120B Problem Set 7
due April 3, 2013
1. In this problem you will derive an expression for the eect of solute concentration on the temperature of
phase equilibrium, and then use what you have derived to predict the freezing temperature and boilin
Chem 120B
Problem Set 7
Due: October 30, 2013
1. Salt water is plentiful on earth; natural supplies of fresh water, on the other hand, are dwindling while
mankinds demands grow. One appealing solution to this imbalance is to remove the salt from ocean
wat
Chem 120B Spring 2013
Problem Set 3 Solutions
1. Ideal Polymer Chain
Since M is xed through the entire process, note that the energy depends only on temperature. Because the equilibrated system starts and ends at the same temperature, we therefore know th
Problem Set 1 Solutions
Chem 120B, UC Berkeley
September 12, 2013
1. NL particles on the left side of the box, N particles total.
i
ii
iii
The number of particles is the same as the number of times each student ips his or her coin (like
we did in class),
#2
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#4
iii) In summary, the feature of a probability distribution that leads to large values of S is that the probability is spread out over a large range of values. This is consistent with the result from the next problem set, where it is deter
Chem120B
Problem Set 4
Due: February 22, 2008
1. Problems 1837, and 1839 in Simon and McQuarrie's Physical Chemistry. 2. Rotations about a covalent bond, such as the carboncarbon bond in ethane, are hindered by forces between other chemical gro
Chem 120B
Problem Set 1
Due: February 1, 2008
1. (i) Problem 166 in McQuarrie & Simon's Physical Chemistry textbook. (ii) Let N be the number of molecules contained in the ultrahigh vacuum chamber of Problem 166. How large are typical fluctuati
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Chem 120B
Problem Set 4
Due: October 2, 2013
1. In lecture, we considered the rotational and vibrational partition functions for an oxygen molecule at
room temperature. We considered the nature of the partition function to count accessible states.
(i) At
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Chem 120B
Problem Set 1
Due: September 13, 2013
1. In our rst lecture, we simulated the measurement of NL , the number of particles on the lefthand
side of a sealed box of N randomly distributed particles by ipping coins.
(i) Explain this analogy: in the
Chem 120B
Problem Set 2
Due: September 18, 2013
1. In this problem, you will consider the temperature dependence of characteristic parameters of the
MaxwellBoltzmann distribution of speeds.
(i) As discussed in lecture, the probability to nd a molecule at
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Chem 120B
Problem Set 3
Due: September 25, 2013
1. In thermodynamics temperature is dened by the derivative
(
)
1
S
=
T
E N,V
(i) A system at temperature T and energy E0 absorbs a small amount of energy E . By how much
does its entropy change? Include in
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Chem 120B
Problem Set 1
Due: September 2, 2016
A2
of a fluctuating quantity A can be written
1. In class we showed that the mean squared deviation
2
in terms of its mean A and the mean of its square A :
A2 = A2 A2 ,
where A is the deviation of A away fro
Chem 120B
Problem Set 2
Due: September 9, 2016
1. A proton in a magnetic field has two spin states, one aligned with the field B and the other antialigned. We will refer to them as up and down, respectively. The energies of these spin states are
Edown = B