Work done by a gas during an adiabatic process involving a state chang

1. The problem statement, all variables and given/known data
The problem involves a container holding a gas at high pressure. The container is opened to the environment, where the gas will cool down, producing a liquid or solid, and I want to find the work done by the gas throughout this process.

2. Relevant equations
PV^γ = constant
U = Cv (ΔT)

3. The attempt at a solution
The process that I would do is find the new values of the P, V after the expansion using PV^γ = constant, find out the new temperature, and use the change in internal energy associated with the temperature to find the work, as in an adiabatic process the heat flow is 0. However, with the phase change, I’m not really sure on how to best do this.

Currently, I’m thinking that the best way to do this would be to do the adiabatic expansion until the sublimation or boiling point, calculate the work as above, then calculate the work by determining the change in volume from the gaseous to the liquid or solid phase, then using -PΔV to determine the work due to phase change, with P being the atmospheric pressure.

Can anyone tell me if I’m in the right direction, or if there is something I’m missing?

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