ClearAll["Global`*"] 
kB=1.38064852×10^(-23)* m^2* kg/s^2/K;
Troom=300*K;
hbar=6.626176/(2*Pi)*10^(-34)*kg*m^2/s

(* problem 1 *********************************************************8*)

me=9.10938356*10^(-31)*kg
mcarrier=0.04*me
nquant=(mcarrier*kB*Troom/2/Pi/hbar^2)^(3/2)

(* so for original n<< nquant, as needed for ieal gas treatment, but not true for maximally doped *)

(* problem 2 *****
the mass times the centrifugal acceleration Omega^2*L is the force, times l is energy
**************************************)

Solve[Exp[-M*Omega^2*L*l/kB/T]==0.01,Omega]
kB=1.38064852×10^(-23)* m^2* kg/s^2/K;
T=300 K;
L=1*m;
l=0.05*m;
mN=1.67*10^(-27)*kg;
M=10^6*mN;

Assuming[m>0&&s>0&&kg>0&&K>0,Simplify[Omega=((2.145966026289347` (Sqrt[kB]) (Sqrt[T]))/((Sqrt[L])(Sqrt[l])(Sqrt[M])))]]

(* centrifuge acceleration about 20 times g=9.8 m/s^2 *)

gcentri=(((15.11408281900868`)/s))^2*L

(* problem 3 *******************************************)

(* triangular cycle, (a) V0,2T0, p=R 2T0/V0
(b) 2V0,2T0, p=R 2T0/V0  ; (c) V0,T0, p=R T0/V0 ; 
the line bc is p=const isobar, so in the plot in p-V plot bc is horizontal
S=R*log(V*T^(3/2))+const so Sa=Rlog(V0*(2T0)^(3/2)+const
Sb=Rlog(2V0*(2T0)^(3/2)+const, Sc=Rlog(V0*(T0)^(3/2)+const

Heat in is heat on (ab)=integral T dS=(2T0)*(Sb-Sa)=(2T_0)R*log[2]
Work  on (ab)=integral p dV=(2T0 R) integral  dV/V=(2T0 R)log(2)
work on cycle is work on (ab) -(bc) since no work on (ca) 

Work on cycle (ab-bc)=2*T0*R*log(2)-(RT0/V0)*V0
eta=work on cycle/heat *)

eta=1-(R*T0/V0)*V0/(2*T0*R*Log[2])

1-(1/(2 Log[2.]))

(* problem 4 *******************************************)

(* Maxwell relation (partial S/partial V)_T=N*KB/V
(partial p/partial T)_V= (partial (N*kB*T/V)/partial T)=N*KB/V, the same *)

Derivatives in Jacobian

(partial T/partial p)_V=V/N/KB;
(partial T/partial V)_p=p/N/KB;
(partial S/partial p)_V=N*kB*(3/2)/p;
(partial S/partial V)_p=N*kB*(5/2)/V;

-(partial T/partial p)_V*(partial S/partial V)_p+(partial T/partial V)_p*(partial S/partial p)_V=-5/2+3/2=-1