Isochoric Process Calculator

Boundary work is defined as $$W = \int P\,dV$$. When volume is constant, $$dV = 0$$, so the integral evaluates to zero regardless of pressure. Physically, the rigid walls prevent the gas from expanding, so no displacement occurs against external pressure and no mechanical work is performed.

According to Gay-Lussac's law, pressure is directly proportional to absolute temperature at constant volume: $$P \propto T$$, or $$P_1/T_1 = P_2/T_2$$. Doubling the absolute temperature doubles the pressure. This is why sealed containers can become dangerously pressurized when heated.

$$C_v$$ is the molar heat capacity at constant volume — the heat needed to raise 1 mol by 1 K without volume change. $$C_p$$ is the heat capacity at constant pressure. For ideal gases, $$C_p - C_v = R$$. At constant volume, all heat goes to internal energy; at constant pressure, some heat also does expansion work, requiring more total heat.

The Otto cycle (gasoline engines) models combustion as isochoric heat addition — the spark ignites the fuel-air mixture so rapidly that the piston barely moves, approximating constant volume. The exhaust blowdown is also modeled as isochoric heat rejection. This distinguishes the Otto cycle from the Diesel cycle, which uses isobaric heat addition.

A bomb calorimeter is a thick-walled sealed container used to measure the heat of combustion of a sample. Because the vessel is rigid and sealed, the reaction occurs at constant volume. The measured heat equals $$\Delta U$$ (internal energy change) rather than $$\Delta H$$ (enthalpy change), though the two are related by $$\Delta H = \Delta U + \Delta(nRT)$$.