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As far as I understand, in thermodynamics an isothermal process is a process (a state change) during which the temperature of the system remains constant.

If I consider an ideal gas, the work obtained from (or put into) the system during an isothermal expansion (or compression) from volume $V_1$ to volume $V_2$ is $W = n R T \ln \frac{V_2}{V_1}$.

Considering this equation alone, I'm puzzled, as $\ln$ as a function is unbounded. This equation alone means that by letting my gas expand to arbitrarily large volume, I can get out arbitrarily large amounts of work. As this obviously cannot be the case, there must be something that forbids the assumption of expansion to arbitrarily large volume. What is this something?

Please understand I see there would be technical obstacles (e. g. how would you build a machine that can handle arbitrarily large volumes?) but I am not interested in these. I'm asking for a theoretical reason that restricts expansion, or a characterisation of the set of possible isothermal processes of a system (an ideal gas) in a given state.

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    $\begingroup$ As long as you keep adding heat, the volume can continue to increase, albeit at lower pressure. So you can continue to do work as long as you add more heat. $\endgroup$
    – Chet Miller
    Jun 24, 2018 at 16:33
  • $\begingroup$ I see. Is it correct to expand this argument to saying that isothermal expansion of an ideal gas is possible only by adding heat? So that an isothermal expansion of an isolated system is impossible? (Is it correct to say that for an ideal gas, constant temperature means constant internal energy, so the energy for any work done must come from the outside?) $\endgroup$
    – MHvM
    Jun 24, 2018 at 16:50
  • $\begingroup$ Yes. That is correct (neglecting changes in potential and kinetic energy). $\endgroup$
    – Chet Miller
    Jun 24, 2018 at 19:41

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From the first law ∆U=Q-W for a closed system (which could be a tight fitting piston/cylinder so that the mass is constant) where ∆U is the change in internal energy, Q is the heat added to the system and W is the work done by the system on the surroundings (such as the expansion of the gas in the cylinder against the piston). Also, for an ideal gas ∆U= Cv ∆T for any process, where Cv is the specific heat at constant volume. Since ∆T = 0 for an isothermal process, then ∆U= 0 and Q = W. Bottom line, the work done by the gas expanding against the piston equals the heat added at constant temperature of the gas. The work is not unbounded. It is limited by the heat added (as well as by second law considerations if the system is intended to operate in a closed operates in a closed thermodynamic cycle). Hope this helps.

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