Decompressed Air Temperature Change

I have a jet of compressed air that is impinging on a plastic disc. I know the pressure and temperature of the compressed air, and the pressure after being released through a nozzle.
The Compressed air conditions are as follows:

1. Pressure: 931 kPa (135 psia)
2. Temperature 0°C

final condition:

1. Pressure: 101 kPa.

I need to find the estimated temperature of the air as it impinges on the plastic plate.

I am making the following assumptions:

1. It is adiabatic – there is no heat transfer to the jet of air
2. Pressure is constant once released – that is I am not considering the effect of the velocity on the pressure and even though I know that there will be pressure recovery on the face of the disc, I am assuming that is negligible in comparison to the pressure drop.
3. No work was done during the expansion.

My question is, what is the temperature of the air that has been released? I have tried to use the ideal gas law PV=nRT, but I am getting stuck on the fact that I don’t know the final volume.

The assumption I can make is that the mass is constant. Therefore the next step is to consider that energy is constant and that the same energy that existed per unit mass in the compressed state exists in the expanded state, except that it is now dispersed into the larger volume that I don’t know how to calculate.

Even with a master’s degree in engineering (from long ago), this situation is leaving me stymied! Any help?

Thanks,

John