# Heating a gas in an insulating cylinder

Let's say I have an ideal gas in a cylinder with adiabatic walls. This cylinder has a piston (thermally insulating as well) in between dividing the volume in half. If I have a heating coil in one portion, causing the gas to expand, what happens to the pressure and temperature of this gas? I feel like this process could be isothermal (if we assume the gas is heated slowly), but I can't be sure. Hence may I know what happens to the temperature of the gas? Thanks!

• If you have a heating coil, then the process is not adiabatic. You can consider the gas as being in a container with a non-trivial topology: its walls are the cylinder's walls and the surface of the coil. You can transform this problem into an equivalent one with two gases separated by a mobile, adiabatic partition. One gas can only undergo adiabatic processes, the other is heated instead. Jul 9, 2020 at 7:58
• But the question is a little ambiguous: are you asking about the final equilibrium state? or about the non-equilibrium state after a time $t$? In either case some initial conditions (initial pressure and volumes) and boundary conditions (heating rate) are necessary. Jul 9, 2020 at 8:00
• The temperature will increase in both chambers. In the one with the coil, it will increase due to the heating, although not as much as if the piston were immovable. In the one without the coil, it will increase because the gas is being adiabatically compressed. For a frictionless piston, the pressure on both sides of the piston will be the same as long as the heating rate is slow. One can, of course, easily model this process. Jul 9, 2020 at 11:58
• @pglpm sorry for the ambiguity, let us say the initial temperatures, pressure and volumes are given. Given that pressure increases and I am interested only in the final state. @/chetmiller I realised that it would be adiabatic expansion on the other side which would be the key to analyzing this. I agree that pressure would be equal on both sides, thank you! Jul 9, 2020 at 14:16

The forces acting on piston are weight $$mg$$ downwards, $$P_oA$$ downwards and $$PA$$ upwards.
$$P_o$$ is the atmospheric pressure and P is the pressure of gas.
Considering slow expansion of gas we have, $$mg+P_oA=PA$$