Adiabatic expansion with temperature-dependent gamma

[usrename]

Homework Statement

Two moles of a gas, whose initial temperature and pressure are 200ºC and 20 atm, undergo an adiabatic expansion until the temperature reaches 100ºC. Calculate the pressure and the volume at the end of the process if the constant pressure molar heat capacity follows this formula:

Relevant Equations

c_p= a + bT + cT²

The statement does not say whether the process is reversible or not, but I suppose the only way to solve the problem is by thinking it actually is.
I tried using the formula for reversible adiabatic processes, i.e. PV^γ = constant. First, I calculated the initial volume with the ideal gas law. Then, I tried to find /gamma as the quotient between c_p and c_v. If I take c_v to be c_p - R, then the value of γ depends on temperature.
I think the way to do is by calculating P₀V₀^γ₀ using the value of γ at the initial temperature and then using γ_F, i.e. the value of γ at the final temperature, on the other side of the equation P_FV_F^γ_F. However, I am not sure if I can use the formula this way.
Is this approach correct?

 

[haruspex]

Since $\gamma$ is variable, I think you need to take a calculus view of $PV^\gamma=c$. I.e. consider a small change in volume, and what that does to the temperature.