The first law and heat capacities

For a closed system at rest, neglecting changes in bulk kinetic and gravitational energy, write $\Delta U=Q+W_{\mathrm{on}}$. Heat Q is positive into the system, and work on it is positive into the system. Boundary expansion work is therefore negative. The law applies between equilibrated endpoints even when the intermediate state cannot be assigned a single pressure or temperature. Internal energy excludes the separately tracked motion or elevation of the apparatus as a whole.

A cyclic device returns to its original internal state, so its net internal-energy change is zero. Net heat then balances minus net work on the device. An adiabatic process has zero heat exchange, not necessarily zero temperature change: compression can increase internal energy by positive work input. Conversely, ideal-gas free expansion has both heat and work zero and leaves temperature unchanged because that model’s internal energy depends only on temperature. These are different mechanisms, despite both being adiabatic.

At fixed volume and amount, pressure-volume work vanishes. Reversible heating then gives $C_V=(\partial U/\partial T)_{V,\nu}$. At fixed pressure define enthalpy $H=U+pV$; its differential is $dH=dU+p\,dV+V\,dp$. Along a constant-pressure path with only pressure-volume work, the first law gives heat equal to the enthalpy change. Thus $C_p=(\partial H/\partial T)_{p,\nu}$. Both are total sample capacities in joules per kelvin.

Molar capacities divide these total quantities by the amount in moles; mass-specific capacities divide by mass. A numerical comparison is meaningless until the normalization agrees. Capacities can depend on temperature and on the equilibrium state, so multiplying one constant value by a large temperature interval is an approximation unless constancy is part of the model. Phase changes also require latent heat and cannot generally be represented by a finite smooth single-phase capacity across the transition.

For an ideal gas, $H(T)=U(T)+\nu RT$. Differentiating with respect to temperature gives $C_p=C_V+\nu R$. The extra term is the energy that accompanies expansion during constant-pressure heating. It does not require that the gas be monatomic. A monatomic classical ideal gas further has $U=3\nu RT/2$ and therefore a specific capacity ratio, whereas molecular rotations or vibrations change U and the capacities without changing the ideal-gas difference.

For a nonideal fluid, internal energy can depend on volume as well as temperature, and the difference is determined by expansion and compressibility responses rather than simply the gas constant. Later Maxwell relations provide that general identity. Even here the first law diagnoses common errors: isothermal does not mean zero heat, and adiabatic does not mean zero work. Decide which coordinate or transfer is held fixed before substituting a capacity formula into the energy balance.