| Topic: |
Science > Physics |
| User: |
"Texas Longhorns" |
| Date: |
15 Nov 2004 05:51:33 PM |
| Object: |
Specific Heat Problem [HELP NEEDED] |
Can anyone help me with this? Any help would be greatly appreciated. Thank
you.
"Consider an assembly of N identical and indipendent atomic systems each
having just two discrete energly levels at energies zero and Eo (epsilon 0).
Assume that the systems are identifiable and that there is a single quantum
state associated with each level. Find (a) the internal energy of the
system, (b) the heat capacity of the system, and (c) approxiamte expressions
for the heat capacity in the high and low-temperature limits. Plot the heat
capacity as a function of temperature.
For the assembly of systems described above, show that the heat capacity
reaches a maximum value at a temperature To defined by (x tanh x = 1) where
x = Eo/2kT
Also show that the heat capacity at this temperature can be written as
Cv=Nk(xo^2-1) where xo = Eo/2kTo
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| User: "Bjoern Feuerbacher" |
|
| Title: Re: Specific Heat Problem [HELP NEEDED] |
16 Nov 2004 03:54:38 AM |
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|
Texas Longhorns wrote:
Can anyone help me with this? Any help would be greatly appreciated. Thank
you.
Well, what work have you done so far, and where exactly is your
problem with the assigment?
"Consider an assembly of N identical and indipendent atomic systems each
having just two discrete energly levels at energies zero and Eo (epsilon 0).
Assume that the systems are identifiable and that there is a single quantum
state associated with each level. Find (a) the internal energy of the
system, (b) the heat capacity of the system, and (c) approxiamte expressions
for the heat capacity in the high and low-temperature limits. Plot the heat
capacity as a function of temperature.
For the assembly of systems described above, show that the heat capacity
reaches a maximum value at a temperature To defined by (x tanh x = 1) where
x = Eo/2kT
Also show that the heat capacity at this temperature can be written as
Cv=Nk(xo^2-1) where xo = Eo/2kTo
You did not mention a heat bath above, but since you talk about
the temperature of the system, probably there is one. So you simply
have to use the state sum Z = sum_states exp(-beta H),
where as usual H is the Hamiltonian of the system and beta = 1/(kT),
with k = Boltzmann's constant and T = temperature.
After calculating Z (hint: the hardest part is to find out what
all the possible states are, and how one can sum about them in
a sensible way), you can get the energy and the heat capacity simply
by taking derivatives.
Bye,
Bjoern
.
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