Isochoric Process

An isochoric process (also called an isovolumic process) occurs at constant volume, so $\Delta V = 0$. Two familiar examples are heating a gas that is sealed inside a rigid, fixed-volume container, and the diffusion of a gas within a closed chamber whose volume cannot change.

Since the volume never changes, the system does no work at all:

Applying the First Law, $Q = \Delta U + W$, with $W = 0$:

And, as in the isobaric case, the change in internal energy for a fixed volume-heat-capacity is

The physical picture is simple and direct: since the system does no work, every joule of heat added in an isochoric process goes entirely into raising the system's internal energy (and hence its temperature) — none of it is 'diverted' into doing work, unlike in an isobaric process. The temperature of the system does genuinely change ($\Delta T \ne 0$) in an isochoric process — only the volume is held fixed. …