3374/6351 Updates
Dec 5
I've extended the due date for the last set by a day.
Suggestions for Reif 6.9:
Part (a):
You can find a discussion of Gauss' Law in any introductory
physics text, such as Serway or Halliday and Resnick, as well
as a discussion of the connection between the electric field
and potential. If you don't have an intro text handy, you
can get one in the library or borrow one from me.
Don't worry about statvolts; use the usual mks units for
voltage (as found in the standard intro texts).
Part (b):
You may express the result in terms of the electric charge density
at one particular value of r. (R seems easiest, but r0 is fine, too.)
(Or alternatively, you could assume you know the total charge Q
and normalize it this way, but it's a little more work.)
To understand how to specify the configuration (referred to by the
subscript r in equations such as 6.2.8 and 6.2.10) and its energy Er
for a classical system in this and similar problems, you might
find it useful to review the section on ideal gases in section
6.3 of Reif. For classical systems specified by continuous parameters,
sums as in 6.2.10 become integrals.
Some suggestions for 6.11:
Cultural note: This potential provides an accurate description of the
Van der Waals force, which can bind electrically neutral atoms. It's
a leftover electric attraction due to an average displacement of the
electron clouds surrounding the nuclei, and is much weaker than the
Coulomb attraction holding the electrons to the individual atoms.
Take into account both the kinetic and potential energies for the
atoms. Including the kinetic shouldn't be much more complicated
than considering the potential alone. You may treat this as a
one-dimensional problem (with position and momentum along the
x-axis only).
Power series = Taylor series.
You might find it convenient to compute the average of (x - xmin) rather
than the average of x directly. (Here xmin is the location of the minimum
for the potential.)
You may need to expand U further than second order; that is, beyond the
quadratic term.
The discussion in appx A.6 is useful for setting up the integrals,
particularly around eqns A.6.12 and A.6.13. Appx A.4, which we discussed
in class, should help in evaluating them.
The purpose for a Taylor series expanding a function of x around some value
x0 is to give a function that is accurate near x0 but simpler and easier
to work with. Note in appx A.6 how Reif uses this idea to approximate a
part of the integrand. The integrand only contributes appreciably when
its argument is near 0, so he's content to get an approximation for part
of his function near 0; it doesn't matter what happens away from that point.
Prob. 6.11 is nearly identical to this example.
I believe that the result in the back of the book is correct
(to leading order in small T).
Suggestions for 7.11:
When Reif asks for Cv at T=300 K, I don't think he means a numerical
value, just a form for Cv(T) that's a good approx in that region.
In any case, that's all I'm asking for. (If you wanted to get a numerical
value, you'd need info on the two omegas; in practice, I think people
work backwards, fitting Theta_e (the Einstein temp) to data, and using
that as one way to estimate the omegas.) Specifically, I'd like you to
take advantage of the obvious simplification for your result when T=300 K.
Incidently, when you give an approximation for a quantity that's becoming
small, you want to say more than it's approximately zero; that doesn't tell
very much. What is useful is to give the form of the function in that limit,
which tells how fast it approaches zero.
Some final comments:
I'll be around all the way through finals to answer questions on these
problems.
If there are homework sets from earlier in the semester that
you weren't able to hand in, let me remind you that I'll take
them up to the end of the course for a 25% credit, and you can
use the solutions to prepare them. (This way I can get you to
at least look through the solutions on those sets.) If you're
not sure which sets you didn't hand in, if any, just ask.
Thank you all for your hard work this past semester, and good luck
on your finals.
Nov 21
For problem 5.14, the beauty of the approach we've taken in our
discussion of thermodynamics is its generality. Although we've
concentrated on V as the only external parameter and p as the
generalized force associated with V, the results apply to any external
parameter and its force. For example, to apply these to a spin system
in a magnetic field, you can simply swap the magnetic field B for V, and
magnetizaion M for p. Relations such as dE = TdS - pdV become TdS - MdB,
and so on. All the Maxwell Relns, for example, still hold with the
appropriate substitutions. For this problem, once you've identified the
external parameter and its generalized force, you should be able to treat
it just as we treated the ideal gas or the Van der Waal gas. (Be careful
about sign, though. For p and V, p wants to act such that V increases.
The tension in this problem will act in the other direction with respect
to its parameter (the length), and so it's -tension that's the generalized
force.)
There may be a typo in your text for the solution to 5.14(e).
I get C_L(T,L) = b T - a T (L - L_0)^2. (This might have been
corrected in your edition.)
For part (f), Reif wants you to compute S(T,L) starting from S(T0,L0),
but using the other obvious path than the one you used in (c) to
show that you get the same answer.
For 5.18(c), don't worry if you end up with a general expression
for (dT/dV)_E that depends on T (or, if integrated, an expression
for Tf - Ti in terms of an integral that depends on T). When you
apply it to the Van der Waal's gas, this dependence disappears.
Also, don't worry if you can't figure out how to use (b); I couldn't,
either. (But if you can, please let me know.)
For problem 5.25: One of the great things about stat mech/thermo
is that you can actually give a precise answer to questions like
this one (and in this case, fairly simply). Things to remember:
(a) From our fundamental postulate, the probability of finding
our system in a particular state is proportional to the number
of states for which that can occur (ie Omega).
(b) S = k Log(Omega)
(c) Think of the weight and paddlewheel as a single degree of freedom
connected to the heat reservoir; that is, don't think of the weight
and wheel as something that could absorb heat, and don't worry about
accounting for the states of the molecules that make up the weight plus
wheel. Think of their state as completely specified by the weight's
height: there's no ambiguity about what they're doing, therefore no
contribution to S. So you only need to consider states available for
the reservoir. We expect that when the weight pulls E out of the reservoir,
there are less states available to it; so Omega, and therefore the probability,
decreases.
(d) We can use the macro laws to figure out how Omega compares for
the weight at different heights if we can think of some quasi-static
process that gets the reservoir from the same initial to final states.
(e) Getting the probability relative to some reference state is sufficient;
you can get the full expression by normalizing to one.
(f) The length of these hints is probably 3 times the length of the
solution
Note that the model for a gasoline engine used in 5.26 is also discussed
in the Van Ness reading I posted some time back. Also, the material in
Van Ness regarding QS processes is directly relevant for prob 5.5 from
last week. Specifically, it answers just what you'd have to do to make
raising the piston a QS process, and how that could keep dS = 0. Take a
look at the solutions to see a discussion of this.
Nov 19
We have a bit of extra time before the next set is due,
and this week's problems are a little more involved than
average, so I'll take the set due today anytime through
tomorrow (Thursday) without penalty.
Nov 18
Just a reminder that we'll have two lectures tomorrow:
10-10:50am and the usual lecture at 11-11:50am.
Nov 14
For 5.1 (b), you can assume that Cv and Cp are constant
for an ideal gas. Section 5.4 should be helpful for this
part (though there are probably other ways to do this).
For 5.5, this is clearly not a quasistatic process. Think
carefully before using any results for adiabatic expansion.
For problem 5.7, because many of the quantities change with
height z, you can't treat the whole atmosphere as if it's
an ideal gas in equilibrium. For example, there isn't a single
pressure; it changes with z. But if you treat this as you would
approximate an integral (where you treat the function as approximately
const over some range z to z+dz), you can treat these quantities
as approx constant. In particular, you can break up the atmosphere
into small cubes of height dz sitting one on top of the other.
(It doesn't matter how wide you make them, since nothing changes
parallel to the ground.) You can treat each cube as if it contained
an ideal gas at a const p and V. p will go from p to p+dp from
one cube to the next on top of it. Now if each cube stays where it
is, what can you say about all the forces on it?
Cultural note: Ch 5 is supposed to give you a flavor of the historical
approach to thermodynamics. If you'd like to learn more history,
there's a link to a timeline history of stat mech and thermo from about 1500
to today, among other useful resources under Web References on our home page.
Nov 12
For 4.2(a), I obtained Tf = 283 K = 10 C, which might be useful
(and maybe even correct) if you want to check your result.
Nov 6
For 4.1, you can treat the specific heat of water as approximately
constant over the range 0C to 100C. (Don't forget to convert to
absolute temperature unless you're only considering differences.)
For water, the specific heat at fixed pressure is 1 calorie per
gram per degree C (or K); that is, 1 cal/g C. (This is how the
calorie is defined.) Also, 1 cal = 4.186 J.
For 4.2, I've never been able to figure out why knowing
the specific gravity is useful. If you can think of
why it's relevant, please let me know.
Oct 31
For 3.4, remember from class that you can only use
statistical reasoning (and the results derived from it),
for quasi-static processes, and this process is definitely
not QS for the small system. But, as we'll discuss,
if you can imagine a QS process (or sequence of them)
that gets you to the same final equilibrium state,
you can calculate things like the change in S for
the imaginary process, and get the right answer.
Don't assume that S for the combined system will
increase to get a relation; this is what you're proving.
For the problem testing Reif 3.8.9, it's very useful
to note that H always comes in a particular combination
with E in Omega. I recommend judicious use of the chain rule.
You should be able to derive the result without actually taking
a derivative of Omega (though it should work if you do; it's
just messy). If you use the chain rule, you can do it in a
few lines.
I've included a section from Van Ness on Quasi-Static or Reversible Processes
in the reading for the next problem set. (You'll need the same password
you use for homework solutions.) You can also find the link under Course
References on our class home page. Van Ness is a well-written, short, and
very inexpensive Dover publication that focuses on just a few important
and sometimes confusing topics in Thermo. At $5.00, it's a pretty good
investment.
Oct 27
Since 3.2 and 3.3 rely on the results of 2.4, it may
help to look at the posted solution for that problem.
For 3.2:
For the starting form for Omega, you can just use the result
for 2.4b that Reif was looking for; ie, just use the result
that comes from the Stirling approx up to N! = N log N - N.
(It's not completely consistent to do this, but that's ok.)
There are some simple relations that connect inverse
hyperbolic functions to logs. I'll list a few below.
These might help in simplifying an expression in this
week's problems.
asinh(x) = sinh^(-1)(x) = log( x + (x^2 + 1)^(1/2) )
acosh(x) = log( x +- (x^2 - 1)^(1/2) )
atanh(x) = 1/2 log(|(1+x)/(1-x)|)
acoth(x) = 1/2 log(|(x+1)/(x-1)|)
You can save yourself a great deal of work in 3.3 by
writing Omega_total = Omega*Omega' explicitly as a
function of E. You should find, by completing
the square in the exponent, that the product of two
Gaussians is also a Gaussian, which is a very good
thing.
Oct 17
Surprisingly enough, part (but only part) of the random walk
distribution may have some relevance for problem 2.4.
2.7(a) is meant to be a simple question (as far as I
can tell). Just think of the connection between force
and work, and between work and the change of E.
For 2.7(b), you can get the relation between the average
pressure and energy by relating them both to the averages
for n_x^2, etc. As I mentioned in the assignment, you won't
actually need to work out these averages for n_x^2 etc to
get this.
For those students who were not afflicted with English units
as children:
1 ton = 2000 lbs
1 lb = 4.448 N
The answer in the back of the text for 2.11(a) should read
Q = 18,800 joules.
Grading:
I've had a couple requests to help translate the grades on the
homeworks into letter grades. Very roughly, the correspondence
will be
0.87 - 1.0 A
0.72 - 0.87 B
0.55 - 0.72 C
0.40 - 0.55 D
Here the number is the fraction of points given on homework
out of the total possible (with +'s or -'s added to grades
depending on whether you're near the top or bottom, as you'd
expect).
Oct 8
For 2.3: the area for an ellipse is Pi a b, where a and b are half
the lengths of the major and minor axes. (Note this goes to Pi r^2
for a circle.) The circumference is 2 Pi Sqrt((a^2 + b^2)/2),
but for our homework, I recommend doing the same thing we'll do
in class: work out Phi(E) (the number of states with energy from
0 to E), then use Omega(E) = Phi(E+dE) - Phi(E), which is approx
equal to Phi'(E) dE for small dE.
The last example discussed in Reif sect. 1.8 is mathematically
very similar to something you'll need to consider in problem
2.3a: the connection between a probability as the function of
a vector's polar angle vs. as a function of its x-component.
You might find this helpful, though there are of course several
ways to solve this.
For the three spin-1/2 problem, when I ask for a probability
distribution, I want the probability for each possible
value.
The last problem in the current set is simple, but you may
find the instructions obscure. I thought it might help to
give some parts of the solutions so you can check that you
understand the question:
----------------------------------------
(a) N=6
one state has 1 particle in n=3, 3 particles in n=1,
and the rest in n=0 (M particles total)
another state has 1 particle in n=4, 1 particle in n=2,
and the rest in n=0 (M particles total)
There are 11 different states
(b) N=5
one state has 1 particle in n=4 and 1 particle in n=1
(2 particles total)
another has state has 1 particle in n=4, 1 particle in n=1,
and 1 particle in n=0 (3 particles total)
There are 6 different states
(c) N=4, M=3
one state has the first particle in n=3, the second in n=1,
the third in n=0. There are 6 rearrangements of this case,
and each one counts, because the particles are distinguishable.
There are 15 different states
----------------------------------------
I recommend listing the possibilities in an organized table.
Sept 30
For problem 1 on the exam, short answers (ie without explanations)
are fine.
For problem 2, recall that in the process of converting the binomial
probability P(m) into a continuous distribution, we first replaced
P with a Gaussian approximation, then (trivially) changed the
normalization so that it would be appropriate for a density. It's
that first part that's relevant.
For problem 3, your answer should be in terms of N. I didn't tell
you N, but everything else you'll need is calculable from the info
I gave.
Note that for problem 4, I'm asking for an approx that's
good near 1/2, which is not where the function peaks.
So it's a little different (and simpler) than the previous hw
problem, but more like the first example we did in lecture.
As in those problems, you are still approximating it in a
region where it changes rapidly. Again, your result will be
a function of N.
Sept 24
For problem 19, section 1.9 is relevant.
For prob 1.15, I found the problem's statement to be somewhat
ambiguous. I interpret it by thinking of myself as one of
the callers, and my attempt to call tests whether there is
a line free; that is, whether there are currently 2000 or more
people trying to call at the moment. Having the minimum number
of lines M will mean I fail 1% of the time or less. (If you find
this more confusing than Reif, just ignore it.)
You can make direct use of results from section 1.9 for 1.15
as well. First determine explicitly what would correspond to the
random variables s_i and determine the values each can take,
and then what would correspond to the total x.
A little more on Prob 15:
To get started, there are a couple quantities you should
think about. The first is some variable associated with the
individual callers which tells you whether they're on the phone
or not. The second is the total number trying to make a call
at any one time, which you'll want to compare to the number
of lines available. Both of these are random variables,
and the latter should be in some sense the sum of the
former.
There's enough information to immediately write down a
distribution for the first quantity, and we've discussed in
detail in lecture (see also section 1.9) how it's related to
the distribution for the second.
Sept 19
I've posted solutions for set 2.
Sept 18
Oops - I had solutions for the first set posted last Monday, but
forgot to put the link in. If you try to look at solutions after
you get your homework back and don't see anything there, please
send me an email. The solutions for the second set will (I hope)
be visible tomorrow.
Sept 17
For your series for ( x(1-x) )^N, leave everything (including the
estimate of its range of validity) in terms of N, since you don't
have a specific value for it. Just assume it's a lot bigger than 1.
(If you'd like to plot your result to see how you're doing, N of about
10 or so would be plenty large.)
To estimate the range, you could either compare the quadratic term
to the leading term, or the quartic to quadratic (which would require
two more derivatives). The results are roughly comparable.
For part (b) of the diffusion equation problem, substitute rho
into the equation, and figure out what D would need to be in terms
of constants in rho (such as v, l, etc) to get it to work. It may
look intimidating, but as long as you know what a partial derivative
is, this problem is fairly simple.
Part (a) you should be able to do in a line or two; there's no
calculation needed.
For problem 1.29 (for the grads), there are probably lots of
ways to do the sin integrals at the end (and all are probably
tedious). To do them myself, I converted them to integrals of
exponentials and related them to the dirac delta function.
Please do the integrals by hand, though you can use Mathematica
or an equivalent to check them.
Sept 11
A student in a previous class, Martina McNeely, pointed
out a site where you can try out the Let's Make a Deal problem
experimentally. The difference between choices is significant
enough that you should see it work after a handful of attempts.
If you find other interesting web sites related to stat mech,
please let me know.
I've made a few additions to our homework rules. Please take a look.
Aug 29
Grads taking this course should look through Reif to determine
whether they've already had an adequate course in this material.
If so, we should discuss other options.
The first problems are posted. Please first review our homework rules.
Some suggestions:
The reading from Appx 1 should be useful for problem 5.
When computing probabilities, it's sometimes easier to compute
the probability that what you're interested in didn't occur,
and then subtract that from 1.
"ace" is a synonym for 1
For prob 5(c), it might be helpful to think about what happens
to the sum of a geometric series (Appx 1) when you take a
derivative with respect to the variable being summed; that
is, with respect to x if you're summing x^n.
For Prob 1.3, the discussion in Reif around equation 1.2.5
might be useful. (There it's used for the random walk,
which we'll be discussing soon in lecture, but the reasoning
is applicable; see the box on the following page.)
Aug 20
I'll post hints for current problems, answers to questions, and
other useful information here.
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