# [Physics] Why does increasing the volume in which a gas can move increase its entropy

entropystatistical mechanicsthermodynamics

Let's say we have a box with a non-permeable wall separating the box in half. There is gas on the other side of the wall. Now we remove the wall so that the gas can diffuse to the other half of the box.

It is said that the entropy of the gas increases because the molecules now have more room to move, and therefore there are more states that the gas can be in. I can understand this well.

But the change in entropy is also defined as follows:

$$\displaystyle \Delta S = \frac{Q}{T}$$

Where $$T$$ is the temperature of the gas and $$Q$$ is the change in heat of the system. But if we look at this definition, why did the entropy change for the gas inside the box? By just removing the wall, the kinetic energy of the molecules does not change, therefore the temperature does not either. We also didn't add any heat to the system, so $$Q$$ is zero as well. So why did the entropy change?

This is a classic example of making sure you know what your equations actually mean. You are thinking along the lines of $$Q=T\Delta S$$ Where your change in entropy determines the heat exchange. This is not the right way to view the equation.
The actual meaning of the equation is "if you have reversible heat flow $$Q$$ at temperature $$T$$ for a reversible process, then you have a change in entropy $$\Delta S$$". Since there is no heat flow, this equation does not tell you anything useful. All you can say is there is no entropy change due to a heat flow.