relating microstates of entropy to gibbs for dT=0

[Tor Fredrik]

 

 

The text above derives gibbs energy to only be a function of entropy. Since I can't get math signs working I have to upload my question in an attachment:

 

[Tor Fredrik]

I could come up with one reasoning:

 

 

Can anyone come up with a bigger picture for why dS_sur is given by -dH/T? I know that dH=dU+pdv+Vdp=dQ-pdV+pdV+Vdp if dp=0 dH=dQ. The problem is why this Q is the Q of the surroundings?

[Tor Fredrik]

I believe I have made some progress:

 

 

But in an equilibrium which I want to calculate with an ideal gas approximation. One uses standard chemical potential. Does anyone know how I can find the numerical value of the chemical potential for an ideal gas?