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Average number of particles/subsystems in a state

  1. Oct 15, 2015 #1
    1. The problem statement, all variables and given/known data
    A system in thermal equilibrium at temperature T consists of a large number mimetex.gif of subsystems, each of which can exist only in two states of energy mimetex.gif and mimetex.gif , where mimetex.gif . In the expressions that follow, k is the Boltzmann constant.

    For a system at temperature T, the average number of subsystems in the state of energy mimetex.gif is given by

    1. 2$.gif
    2. kT}}$.gif
    3. kT}$.gif
    4. kT}}$.gif
    5. kT}}{2}$.gif

    2. Relevant equations
    Probability of a system to be in a system-microstate of total energy ##E_R##,
    ##P_R = \frac{e^{-\beta E_R}} {\sum_{R} e^{-\beta E_R}} ##

    3. The attempt at a solution
    We have the constraint ##\sum_r n_r= N_0 ## and ## \sum_r n_r \epsilon_r = E_R##
    Where, r labels the single particle states.
    Therefore, the average number of particles in the sth 1 particle state,
    ##\langle n_s\rangle = \frac{\sum_R n_s e^{-\beta (\sum_r n_r \epsilon_r)}}{\sum_{R} e^{-\beta (\sum_r \epsilon_r)}}##

    To proceed one needs the nature of the particles.
    For example,
    ## \langle n_s\rangle = \frac{1}{e^{\beta \epsilon_s} -1}## for Photons
    ## \langle n_s\rangle = \frac{1}{e^{(\beta \epsilon_s -\mu)}+1}## for FD statistics etc.

    How do i proceed without further info? The question seems to conflate states of the total system with the subsystem states. I think the question is problematic and ambiguous. Anyhow, the supplied 'correct' answer is option (B).
     
  2. jcsd
  3. Oct 17, 2015 #2
    The given options only makes sense if by 'system' they mean ensemble and by 'subsystem' they mean members of the ensemble.
    Then, ##P(E_1)= \cfrac{e^{-\beta E_1}}{e^{-\beta E_1}+e^{-\beta E_2}}##
    Again, ##P(E_1)= \cfrac{N_1}{N_0}. ## Therefore, ##N_1 = N_0\left(\cfrac{e^{-\beta E_1}}{e^{-\beta E_1}+e^{-\beta E_2}}\right)##
    And hence, ##N_1 = N_0\left(\cfrac{1}{1+e^{-\beta \epsilon}}\right)## , where ## E_2-E_1= \epsilon## .

    But, if they're talking about a single system, then the options don't make sense to me. But, this question appeared in the GRE, so they aren't likely to make such errors. So, what am i missing?
     
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