Entropy of ensemble of two level systems

  • #1

Homework Statement


The fundamental equation of a system of [itex]\tidle{N}[/itex] atoms each of which can exist an atomic state with energy [itex]e_u[/itex] or in atomic state [itex]e_d[/itex] (and in no other state) is
[tex]
F= - \tilde{N} k_B T \log ( e^{-\beta e_u} + e^{-\beta e_d} )
[/tex]
Here [itex]k_B[/itex] is Boltzmann's constant [itex]\beta = 1/k_BT[/itex]. Show that the fundamental equation of this system, in entropy representation, is
[tex]
S = NR \log \left (\frac{1+Y^{e_d/e_u}}{Y^Y} \right)
[/tex]
where
[tex]
Y= \frac{U-\tilde{N}e_u}{\tilde{N}e_d-U}
[/tex]

Hint: introduce [itex] \beta =1/k_BT[/itex] and show that [itex]U = F + \beta \partial F / \partial \beta = \partial ( \beta F) /\partial \beta [/itex]. Also, for definiteness, assume [itex] e_u < e_d[/itex] and note that [itex]\tilde{N}k_B = NR[/itex].


Homework Equations


[tex]
S = -\partial F / \partial T = -(\partial F /\partial \beta )(\partial \beta /\partial T) = (\partial F / \partial \beta)( \beta /T)
[/tex]

The Attempt at a Solution


I showed the hint but will not write out the solution here. Then we have
[tex]
U = \partial ( \beta F) /\partial \beta = \tilde{N} \frac{e_u e^{-\beta e_u} + e_d e^{-\beta e_d}}{ e^{-\beta e_u} + e^{-\beta e_d}}[/tex]
from which I get [itex]
e^{\beta( e_u - e_d)} = Y [/itex].

Also,
[tex]
S = \tilde{N}k_B \log ( e^{-\beta e_u} + e^{-\beta e_d} ) +\tilde{N} \beta k_B \frac{e_u e^{-\beta e_u} + e_d e^{-\beta e_d}}{ e^{-\beta e_u} + e^{-\beta e_d}}
[/tex]
which I rewrote as
[tex]
NR\log (1+Y) - NR\beta e_u + NR \beta \frac{e_u +e_d Y}{1+Y} = NR \log (1+Y)+ NR \beta \frac{Y(e_d-e_u)}{1+Y} = NR \log (1+Y) - NR \frac{Y}{Y+1} \log Y = NR \log \left ( \frac{1+Y}{Y^{Y/Y+1}} \right)
[/tex]
which is not (at least obviously) the same, but I cannot see my mistake. Help would be appreciated.
 

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