Chem 120B
PSET 1 solutions
1. (a) Knowing that the square of number is positive, so that hA2 i 0, we see that
hA2 i hAi2 0
= hA2 i hAi2
(b) More generally,
hABi = h(A hAi)(B hBi)i
= hABi hAihBi
(c) If
NAME:
SID:
Chem 135, Spring 2015, Problem Set 6
Due in class, Tuesday, April 7, 2015
Problem Set 6
Question 1. Provide a detailed, arrowpushing mechanism for the following transformations. These
conc
Problem Set 2 Solutions
Luning(Chris) Zhao
(Dated: September 9, 2016)
I.
QUESTION 1
(i)From Boltzmann distribution, we know that:
pup = e~B/2 ,
pdown = e~B/2
thus,
B
pup /pdown = eB = e kB T
(ii) With
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,
#2
#3
#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 r
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 liqu
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 en
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
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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
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,
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 at
Chem 120B Problem Set 11 Solutions
Chenlu Xie
November 30th, 2015
Problem 1
(i)
For a dilute solution, the osmotic pressure should be
= , = , =
M is the molecular weight of the enzyme, and is the so
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 ea
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
Chem 120B Problem Set 9 Solutions
Jon Weisberg
Nov 6, 2015
Problem 1
i)
We want to plot (0) = (0) (water) (0) (octanol) as a function of T for the six peptide solutes, as
well as (0) /T versus 1/T . L
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 ,
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
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
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
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 chambe
PS6 Solutions
Chem 120B
Oct. 13, 2017
1
Work on an Ideal Gas
(i)
N kB T
V
Z V2
=
pext dV
pext =
wrev
(1)
(2)
V1
Z V2
N kB T
dV
V
V1
V2
= N kB T ln ( )
V1
= N kB T ln ()
=
(3)
(4)
(5)
(ii)
E = w + qrev
Chem120B
Problem Set 7
Due: October 20, 2017
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
PS3 Solutions
Chem 120B
Sep. 15, 2017
1
Atmospheric Probability and Density
(i)
E(h) = mgh and E(0) = 0
p(h) eE(h)
and p(0) e0
p(h)
emgh
mgh
= 0 = emgh = e kB T
p(0)
e
The hint is useful in understan
Problem Set 1 Solutions
September 3, 2017
Quick Notes
Recall that by taking an average (or expectation value) of some variable hAi we are sampling the value of A many
times (say T times), summing all
UC Berkeley Chemistry 120B, Fall 2017: Problem Set 2 Solutions
PROBLEM 1: SPINS IN A MAGNETIC FIELD
i) We learned in class that the probability of finding the system in a state is proportional to eE ,
YOUR NAME HERE
October 3, 2017
Chemistry 120B Mid Term #1
Definitions and Formulas
Inverse temperature: = 1/kB T ,
kB = 1.38 1023 J/K
Boltzmann distribution and canonical partition functions:
p() =
eE
Chem 120B
Problem Set 1
Due: September 1, 2017
hA2 i
1. In class we showed that the mean squared deviation
of a fluctuating quantity A can be written
2
in terms of its mean hAi and the mean of its squ
Chem 120B
Problem Set 2
Due: September 8, 2017
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, respec
Chem120B
Problem Set 4
Due: September 22, 2017
1. In lecture we estimated the partition function for a quantum mechanical particle in a box using the
approximation
Z
X
2 2
2
h2 n2 /8mL2
Q=
dx eh x /8