Solve Difficult ThermalDynamics Questions

[Lil Frank]

Following questions are from my final. I found them pretty difficult. I hope someone help me with them. Thank you.

1 Please give a clear argument by using the concept of entropy to explain that the heat will always flow from the high temperature to the low temperature objects if there is no external work.

2 Please construct the plots of P versus V, T versus S, and S versus E_internal (a) for the isothermal expansion and isobaric expansion thermodynamic process.
(b) for the adiabatic expansion thermodynamic process.(Please point out the initial and final state on your curve.)

3 For adiabatic processes in an ideal gas, show that
(a)the bulk modulus is given by
dp
B = -V ——— =γP
dV And therefore
(b)the speed of the sound in the gas is v=√γp/ρ =√γRT/M

4 (This one is the most difficult one, can anyone help me with it?)
(a)Derive the entropy chang: ΔS=Sf-Si=nRln(Vf/Vi)+nCvln(Tf/Ti) for all reversible processes that take the ideal gas from state i to state f.
(b) Please use this relation to calculate the change in the entropy for a free expansion process from V to 4V. Please also give the reason that you may do in this way.
(c)Derive this increase of entropy with statistical mechanics(using the Boltsmann's entropy S=klnW,where k is the Boltsmann's constant,W the multiplicity of the confriguration).Before doing this,please give explain.
(Hint:lnN!=N(lnN)-N,while N is large)

 

[malawi_glenn]

Here we don't help if you don't provide us with attempt to solutions and some relations/forumulas that you know.