Ideal gas law for a gas with nonuniform temperature

Question

Consider a piston-cylinder assembly with an axial temperature distribution $T(x)$ in the gas.

The entire system, however, has a uniform pressure. Can I still apply the ideal gas law $PV = nRT(x)$?

Answer 1

If you have such a system, then, assuming the pressure P is uniform, you can calculate the local molar volume as $v(x)=\frac{RT(x)}{P}$. The total number of moles of gas in the cylinder is: $$n=A\int_0^L{\frac{dx}{v(x)}}=\frac{PA}{R}\int_0^L{\frac{dx}{T(x)}}\tag{1}$$If we call $\bar{T}$ the "effective average temperature," we can also write $$n=\frac{PV}{R\bar{T}}=\frac{PAL}{R\bar{T}}\tag{2}$$Combining Eqns. 1 and 2, we obtain:$$\frac{1}{\bar{T}}=\frac{1}{L}\int_0^L{\frac{dx}{T(x)}}\tag{3}$$From this we see that the effective average temperature (for use with the ideal gas law) is equal to the harmonic mean of the temperature variation over the length of the cylinder.

If we want to extend this to an adiabatic reversible expansion, it can be done in a similar way by considering the gas in each differential length along the cylinder, using the original axial coordinates as a set of material coordinates. But, I highly doubt the value of such a calculation because the axial temperature variations to begin with will result in transient heat conduction during any process.