Calculating Final Temperature and Volume of Adiabatic Expansion
The final volume V and final temperature T can be determined exclusively from the information for the irreversible path if we assume that the gas continues to expand until it equilibrates at a final uniform pressure of $P=P_{ext}=100$kPa and a final uniform temperature of T.  Let $T_0$ be the initial temperature (300 K), $P_0$ be the initial pressure (800 kPa), and $V_0$ be the initial volume (5 L).  Then the number of moles is $$n=\frac{P_0V_0}{RT_0}$$
The work done on the surroundingsis given by:$$W=P_{ext}(V-V_0)$$
The change in internal energy is $$\Delta U=nC_v(T-T_0)$$
So, from the first law,$$nC_v(T-T_0)=-P_{ext}(V-V_0)$$
The final volume is given by: $$V=\frac{nRT}{P_{ext}}$$
These equations are sufficient to solve uniquely for the final temperature T and the final volume V exclusively in terms of the input data values.