IRREVERSIBLE adiabatic work

1. The problem statement, all variables and given/known data
So I came down with something and missed some class and am making it up from others notes which arent entirely clear plus the long weekend for turkey day and Im really hoping someone can help me make some sense of this...

I am trying to come up with the derivations for irreversible adiabatic expansions/work, ill try to explain as much as possible and hopefully it will be sufficient for someone to see what im getting at give me a hand






2. Relevant equations

The notes begin with adiabatic reversible work
Du=C_v,mdT
dU=dq+dW -->q=0 for adiabatic so dU=dW
p_ex=P_f=nRT/V
∴nC_v,mdT=-nRTdV/V

the notes then integrate each side wrt T & V end then exponentiate each side to arrive at v₁T₁^c=v₂T₂^c where c =C_v,m/R

they then get sloppy to where i cant make anything of it and arrive at P_iv_i^{[itex]\gamma[/itex]}P_fv_f^{[itex]\gamma[/itex]} where[itex]\gamma[/itex]=Cpm/Cvm


Then there is a huge note that says DO NOT USE THIS IF WORK IS IRREV and that the instructor wants us to derive similar for irreversible work Hint: (it changes where dU=dw)

3. The attempt at a solution

I tried working on this all morning and didnt make any progress...integrating dV/V to get the ln(vf/vi) is how we were taught to do reversible work and for non-reversible it was just w=-P_exΔV

I dont see any way this could turn into similar expressions relate T,P&V

Someone please help! If i see a derivation a couple times they make sense but I can never figure this out solo.

Thank you so much