First post, sorry for the poor formatting.
Consider a non adiabatic system closed to mass exchange where an ideal gas traverses along a reversible path from state: to . What is the work done by/to the system? (Without resorting to statistical thermo)
First law is We cannot simplify this. The internal energy of an ideal gas is a function of temperature hence remains. The system is not adiabatic hence remains. Finally, the system is not isochoric, hence remains.
The ideal gas equation for a system closed to mass exchange is
Because the path is reversible we may use:
However, we must deal with an implicit function and without an additional equation to parameterize it, prevents us from analytically integrating. (I believe?)
We cannot dissect the path into isobaric, isotherm steps because work is not a state function.
My best idea is that if we knew the exact path, we could solve for the work numerically.
Internal energy for an ideal gas is a function of temperature; because it is a state function we can choose to take a constant volume path between our initial and final temperature.
What remains is to solve for the heat and by the first law we may obtain the work.
For a reversible path we have the relation:
Which may be rearranged to give:
Again, if we know the exact path and resort to numerical methods, we should be able to solve this by using: