- Homework Statement:
Please tell me whether my a,b,c,d,e parts are right and teach me how to solve f and g
- Relevant Equations:
- Attached below
Equations that might be helpful:
a) (N_max)!/(n!*(N_max-n)!) i.e. N_max C n
b) Total Z = sum n=0 to N_max [(N_max C n) e^(buN)] = (1+e^(bu))^N_max
Individual Z = 1+e^(bu*1) = (1+e^(bu))
so individual Z^N_max = total Z
c) Now, I use Z to represent the total Z,
By equation 6.14, N = KT d(InZ)/du = kTN_max b/(1+e^bu)= N_max / (1+e^-bu) (because b=1/kT)
After some manipulation, bu=In(N/ (N_max-N))
d) N decreases, so -bu increases, so bu decreases.
e) using the grand potential. S=-uN/T + k/T*lnZ = -uN/T + kNln(1+e^bu)=-uN/T+kNln(1+N/(N_max-N)) = -uN/T+kNln(N_max/(N_max-N))
when N tends to zero it becomes 0, when N tends to N_max it becomes infinity (which I think is a bit strange), when N tends to N_max/2, it tends to N_max(kln2-u/2T).
f) I try to write down the grand partition function but I don't know how to model it and not sure if it is helpful... Can someone tell me what to do...?