# Thermodynamic process, gas expansion

## Homework Statement

1kg air with the pressure 10^6 Pa and temperature 398K expands until the volume is 5 times its' initial value. The expansion occurs so that in every moment the absorbed heat energy (dQ) is 1/4 of the work done by the gas. Find the final pressure. 1kmol of air has the mass 29kg, $$C_v=\frac{5}{2}R$$

The attempt at a solution
Made numerous but it eventually stopped being about physics ended up being a desperate plug and chug. More interested in learning where I'm going wrong with my general line of thought.

I'm ruling out that the process is isochoric or isobaric, the former by definition and the latter because the book barely has any trick questions and I also looked at the answer.
And it's not adiabatic because dQ=/=0. That leaves isothermic but as $$dQ = C_v \nu dT$$ it can't be isothermic either.
And if it's none of those I simply don't know how to calculate the work, and as the work depends on path I can hardly chose an arbitrary one.

The only step forward I can take is $$dU=\frac{3}{4}dW$$, as dQ = 1/4 dW. I thought of somehow calculating change in internal energy as it doesn't depend on path, but it does depend on temperature and I don't have dT (nor do I know how to get it).

Any input would be appreciated, thanks.

Last edited: