| Topic: |
Science > Physics |
| User: |
"kitty5656" |
| Date: |
19 Feb 2005 11:33:42 PM |
| Object: |
entrpy |
Hi,
i'm having trouble with a thermal physics problem relating to the
partition function and i was wondering if anyone could help me out.
the problem is as follows:
(a) Consider a molecule which has energy levels En=c|n| , where n is a
vector with integer components. Compute the partition function for a
one-dimensional ideal gas of such molecules. Show that the average
energy (assuming large temperature compared to the difference between
adjacent energy levels) gives U=N(kB)T.
(b) Take c=0.5eV. For a lab-sized box of gas of our molecules,
estimate the temperature (in Kelvins) at which the assumption of part
(a) breaks down.
(c) Find the entropy. Figure out if this expression breaks down at low
temperature. Are the two breakdown temperatures of (b) and (c)
similar? Why or why not?
----------------------------------
For part b), I guess the assumption is that the temperature, t (in
fundamental units), is much greater than the change in energy levels,
(E(n+1) - E(n)). As i understand, this assumption was used so that the
summation in the partition function can be approximated by an
integral.
Let T be temperature in kelvins and kB be the boltzman constant. I
got,
t >> E(n+1) - E(n)
t >> c|n+1| - c|n|
t >> c
t >> 0.5eV
(kB)T >> 0.5eV
T >> [0.5eV * 1.602 *10^-19 J/eV] / kB
T >> 5802.26 Kelvins
Is my approach for doing this problem correct?
----------------------------------------
I'm mainly stuck on part c). let s = entropy (in fundamental units), F
= free energy, t = temperature (in fundamental units) and Z =
partition function. let N = # of gas molecules in the box.
I found the entropy as follows:
from part a), i found Z to be 1/N! * (t/c)^N
s = - dF/dt
= - d/dt [-t ln(z)]
= d/dt [t ln(1/N! * (t/c)^N)]
= d/dt t[ln(t^N) - ln(c^N) - N ln(N) + N] --> by stirling's approx.
.
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