A $\mathrm{5.00\ L}$ sample of $\ce{CO2}$ at $800 \ \mathrm{kPa}$ underwent a one-step (irreversible) adiabatic expansion against a constant external pressure of $100\ \mathrm{kPa}$. The initial temperature of the gas was $300\ \mathrm{K}$.
An alternative path between the initial and final states consists of a reversible isothermal expansion from $5.00\ \mathrm{L}$ to the final volume $V$, followed by (reversible) constant volume cooling to the final temperature $T$.
a) Give equations (in terms of $V$ and $T$, the final volume and temperature of the gas) for $\Delta U$, $Q$ and $W$ for all three processes.
b) Briefly state why $\Delta U$ is the same for both paths.
c) Hence or otherwise calculate the final volume and temperature of the gas.
My Attempt
Part A
I can do this part. I am pretty sure I got the correct answers except for $\Delta U$ for the cooling process. Could you please check if my answers are correct.
Adiabatic expansion: $\Delta U = 33.33(T-300)$, $Q = 0$ and $W = 33.33(T-300)$
Isothermal expansion: $\Delta U = 0$, $Q = 4000\ \ln\frac{V}{5}$ and $W = -4000\ \ln\frac{V}{5}$
Cooling: $\Delta U = 33.33(T-300)$, $Q = 33.33(T-300)$ and $W = 0$
Part B
That is just because internal energy is a state function, independent of the path taken.
Part C
What confuses me here is the word 'hence', which implies that I need to use the fact that internal energy are equal for the two process. However I have no idea how to use this to find the final temperature and volume. I suspect that I might gotten the expression for $\Delta U$ wrong for the cooling process.