Heating a gas at constant volume

If a gas is enclosed in a rigid container (fixed volume) and is heated, the piston does not move. No displacement means no work: neither does the gas do work ($L_\text{gas}=0$), nor does the atmosphere compress it ($L_\text{atm}=0$). The energy balance reduces to its simplest possible form.

Constant volume — all the heat becomes internal energy

$\ev{\frac{f}{2}nR(T_B - T_A) = Q}$

The physical meaning is direct: at constant volume all the heat supplied goes into increasing the internal energy of the gas, i.e. into making its molecules move faster. None of it is “wasted” pushing away the atmosphere, because the gas does not expand. Note that $Q$ does not depend on the volume of the container: it depends only on the amount of gas $n$, on its degrees of freedom $f$ and on the temperature change.

This is the reference case against which to compare the others: it defines the heat capacity at constant volume $C_v = \frac{f}{2}R$. The worked exercise on nitrogen in a rigid cylinder shows the full numerical calculation: Nitrogen heated in a rigid cylinder.

Connections

Topics: Termodinamica Concepts: Primo principio della termodinamica · Energia interna · Trasformazioni termodinamiche Objects: Gas ideale

Related exercises: Worked example — reversible vs irreversible isothermal expansion · Nitrogen heated in a rigid cylinder · Same temperature, different heat