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Internal energy $U$ is the total microscopic kinetic and potential energy of all molecules. For an ideal gas with no intermolecular forces, $U$ depends only on temperature: $U = nC_vT = (f/2)nRT$, where $f$ is the number of degrees of freedom (3 for monatomic, 5 for diatomic at moderate temperatures, 6 for polyatomic). This temperature-only dependence is a defining property of ideal gases.

The First Law of Thermodynamics states $\Delta Q = \Delta U + \Delta W$: heat added to a system equals the increase in internal energy plus the work done by the system. Sign convention: $Q > 0$ when absorbed, $W > 0$ when gas expands. This is energy conservation applied to thermodynamic systems.

A critical insight: $\Delta U = nC_v\Delta T$ holds for every process of an ideal gas — not just constant-volume processes. This is because $U$ depends only on $T$. The subscript $v$ in $C_v$ refers to how the quantity was originally measured, not a restriction on its use. Students frequently lose marks by applying $\Delta U = nC_v\Delta T$ only to isochoric processes when it is universally valid for ideal gases.