(Solved):   The Boltzmann distribution, which gives the probability \( p_{i} \) of a system at tempera ...

 

The Boltzmann distribution, which gives the probability \( p_{i} \) of a system at temperature \( T \) being in its \( i \) th microstate, is \[ p_{i}=\frac{c^{-E_{i} / k_{H I} T}}{Z} \] where \( k_{B} \) is Boltzmann's constant and \( E_{i} \) the energy of the \( i \) th state. The partition function \( Z \) is given by \[ Z(T)=\sum_{i} g_{i} \exp \left(-\frac{E_{i}}{k_{B} T}\right) . \] A model system has 1 microstate of energy zero, 3 microstates of energy \( \Delta(>0), 5 \) microstates of energy \( 2 \Delta \), and no other microstates. It is held at temperature \( T \). Q.3(a) [7.33 Marks] Determine the partition function \( Z \) of the model system, and hence the probabilities \( p_{1}, p_{2}, \ldots, p_{9} \) of the system being in each one of its 9 states. Q.3(b) [7 Marks] Show that the mean energy of the system is \[ \bar{E}=\Delta\left(\frac{3 x+10 x^{2}}{1+3 x+5 x^{2}}\right) . \] where \( x=e^{-\Delta / k_{B} T} \). Q.3(c) [7 Marks] Determine the limiting values of the mean energy at high and low temperatures, and interpret your results physically. Q.3(d) [12 Marks] Use the Gibbs formula for the entropy of a general system, \( S=-k_{B} \Sigma p_{i} \ln p_{i} \), explicitly, to show that the entropy of the model system described above is given by \[ S=\frac{\Delta}{T}\left(\frac{3 x+10 x^{2}}{1+3 x+5 x^{2}}\right)+k_{B} \ln \left(1+3 x+5 x^{2}\right) . \]