Stat Mech- How does heat vary with temperature over an arbitrary pathway

mandir08

1. The problem statement, all variables and given/known data

You are told that you have 1 mole of an ideal gas with heat capacity at constant volume being 1.5R and you send it over an arbitrary path where dq/dt|pathway= 2R. In the end, the volume of the gas doubles, so figure out by what factor the temperature must change. Assume that the process is reversible


3. The attempt at a solution


du=dq +dw= dq -p*dv
dq=du + p*dv
dq/dt|path =du/dt |path + (d/dt* (P)* dv + P dv/dt|path)

Saying that P= -dF/dv|t,n cause it to become

dq/dt|path =du/dt |path + (0+ P dv/dt|path)

dq/dt|path=2R

u=Cv*T=NRC *T (I am not sure I can apply equation of state u=Cv*T
du/dt=Cv=1.5R =NRC

2R =du/dt |path + (0+ P dv/dt|path)
2R =1.5R + (0+ P dv/dt|path)=RC + (0+ P dv/dt)
.5R=P* dv/dt|path
P/.5R=dt/dv|path

V*P/.5R=T such that T2=V2*P2/.5R=2V1*P2/.5R

However because I am sure that I did this wrong because I can get the same result knowing that PV=NRT such that T2=2*(P2/P1) T1

The hint at the end about the process being reversible makes it that Cv=(du/dt)|v=T(ds/dt)|v

I am kind of lost at this point and not really sure how to proceed. Anyone want to help point me in the right direction?