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I feel a little bit dumb asking this, but what's the difference between Adiabatic and Isothermal?

Isothermal means: keeping constant temperature

Adiabatic means: No heat leaving or entering system

How are these different?

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    $\begingroup$ This is very simple. You will get answer in any text book. $\endgroup$
    – Rajendra Pd
    Commented Sep 11, 2017 at 5:59
  • $\begingroup$ If you do work on the gas, its temperature can change even without adding any heat. Have you ever pumped up a bicycle tire and felt the pump getting hot? $\endgroup$
    – Chet Miller
    Commented Sep 11, 2017 at 13:02

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The main point: Don't equate heat with temperature. There are more types of energy than thermal and more types of energy transfer than heat.

Isothermal: A process that happens at constant temperature.

Adiabatic: A process that happens without heat transfer help from the surroundings.

If you pop a champagne bottle, then the gas outflow is so fast that it practically has no time to exchange heat with the surrounding air. It just cools down rapidly, and that's why you see a white mist which is small ice crystals forming. This is an adiabatic but non-isothermal process.

In general, remember that heat does not imply temperature. And vice versa. Heat could easily be converted into something else than thermal energy, and thermal energy can be gained in other ways than from heat (for example from work).

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The total change in internal energy of a (gas) system is given by: $$dU=TdS-pdV$$ where $TdS=dQ$ is the heat entering the system and $-pdV$ is the work done on the system. From this you can see that the heat transfer is constant at constant entropy ($S$). It is therefore possible to let heat enter the system by increasing $S$ whilst keeping $T$ constant. (Here $dU$ etc represents an infinitesimal change in $U$ etc.)

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I believe your confusion lies in the colloquial definition of heat. In thermodynamics, the heat ($Q$) refers to the energy ($E$) of the system that isn't available to do work ($W$) i.e. it has units of $\mathrm{Joules}$.

The temperature of a system is a defined quantity. It is the amount of heat the system increases by per unit of entropy ($S$), i.e:

$$ T = \frac{\mathrm{d}Q}{\mathrm{d}S}\;\;\;\;\ \langle \mathrm{Kelvins} \rangle $$

Your next question probably is what is entropy? Briefly, it's the number of ways you can have your system at a particular energy. This is a strictly positive number.

Qualitatively, the temperature scales how much energy can be transferred via the system in a given ensemble of states.

EDIT, oops better actually answer the question:

The temperature and heat of a system are related, but can evolve independently and non-linearly with energy thanks to the joys of entropy.

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