Thermodynamics – Relationship Between Volume and Temperature of an Ideal Gas

carnot-cyclethermodynamics

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.

Best Answer

Now, why does the gas expand if both the hot reservoir and system is in thermal equilibrium (same temp. T_h)?

Because during the entire expansion process the external pressure is infinitesimally reduced allowing the gas to expand and do work,

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?

As the volume increases the pressure decreases so that the temperature remains constant. This is done by slowly reducing the external pressure and keeping it in equilibrium with the gas pressure so that $PV$=constant.

The following sequence may be helpful:

  1. External pressure is reduced infinitesimally
  2. Infinitesimal reduction in external pressure results in gas undergoing infinitesimal expansion (thus doing infinitesimal amount of work)
  3. Infinitesimal expansion results in an infinitesimal decrease in gas temperature.
  4. Infinitesimal decrease in gas temperature results in infinitesimal temperature difference between gas and surroundings.
  5. Infinitesimal difference in temperature results in infinitesimal addition of heat, restoring the temperature of the gas to that of its surroundings.
  6. Repeat steps 1-5 until final equilibrium state is reached.

Net result is temperature is constant, total work done by gas equals total heat added to gas and since, for an ideal gas, internal energy depends only on temperature, the change in internal energy is zero per the first law.

See my answer in the following link which includes a diagram with the description of the above steps:Work done by a gas in an isothermal process

Hope this helps.

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