# Why is entropy of gas reamains constant during adiabatic expansion?

In adiabatic compression, we add energy to the gas via piston and decrease volume. Now the increase in energy and decrease in volume will cause increase and decrease in entropy respectively, but it turns out(according to the books) that the entropy of gas remains unchanged. How to prove it mathematically? Although I know that energy given in adiabatic process is zero but the energy is given indirectly via piston and that confuses me.

We measure change in entropy according to the following integral from state 1 to 2:

$$dS = \int_1^2 \frac{dQ}{T}$$

For adiabatic change $$dQ = 0$$ from 1 to 2

Therefore the integral becomes

$$dS = \int_1^2 \frac{0}{T}$$ Since $$dS=0$$, no entropy change from 1 to 2.

What might be confusing you is whether the Piston itself is able to conduct heat. When we define an adiabatic system, by definition the walls and the piston itself are not able to transfer energy in the form of thermal energy (heat). But only in terms of mechanical energy (work).

We can expect the piston to be built of a thermally insulating material although this doesn't happen practically.