Suppose Q and W represent the amount of heat supplied to an ideal...

For an adiabatic process, there is no exchange of heat between the system and its surroundings.

Therefore, the heat supplied to the gas is zero, which means $Q = 0$.

From the first law of thermodynamics, $Q = \Delta U + W$.

Since $Q = 0$, we have $W = -\Delta U$.

For a general adiabatic process involving expansion or compression, the work done $W \neq 0$.

Hence, $Q \neq W$.