I am self-studying thermodynamics, and was reading up Carnot heat engine (Yunus Cengel - thermodynamics book). So, the experiment in the textbook is set up such that, we have an adiabatic piston-cylinder device and the insulation at the cylinder head can be removed to transfer energy from thermal energy reservoir(s) to supply/absorb heat depending upon the where you are in the cycle. It begins with reversible isothermal process by connecting the cylinder to a heat source at temperature T_H. The gas expands, which is also at temperature T_h, and does work on the piston and temperature of the gas drops. This where I have some doubts.
Now, why does the gas expand if both the hot reservoir and system is in thermal equilibrium (same temp. T_h)?. Secondly, from the ideal gas equation Pv = RT, isn't volume and temperature proportional to each other, i.e., as the volume of the gas increases, shouldn't the temperature increase as well?
Thank you for taking the time to read my question.