= Internal energy
{title2=$U$}
{wiki}
Internal energy $U$ is the <energy> stored in the microscopic <degrees of freedom> of a <thermodynamic system>: random molecular <kinetic energy>, internal molecular excitations, and interaction <potential energy>. Its value requires a specified system and an <energy> reference. In <thermodynamic equilibrium>, <internal energy> is a <state function>; it records the present state, without remembering whether a previous change came from stirring, compression or <heat> transfer. This makes <internal energy> the common accounting quantity connecting mechanical experiments to thermal ones.<University Physics Volume 2 by Samuel J. Ling and others/3.2 Work, Heat, and Internal Energy>[§3.2, “Internal Energy”, equation 3.5 and the discussion of molecular motion]{ref}
<Heat> and <work> describe <energy> crossing the system boundary during a process. They are not separately stored contents of the system. For a <closed system> whose bulk <kinetic energy> and external <potential energy> have unchanged endpoint values, the <first law of thermodynamics> gives
$$
\Delta U=Q_{\rm in}+W_{\rm on}=Q_{\rm in}-W_{\rm out}.
$$
Here $Q_{\rm in}$ is positive for incoming <heat>, and $W_{\rm on}=-W_{\rm out}$ is positive for net <work> done on the system. Two routes between the same equilibrium states have the same $\Delta U$, although their <heat> and <work> can differ. The infinitesimal notation $dU=\delta Q_{\rm in}+\delta W_{\rm on}$ distinguishes a <state function> change from path-dependent transfers.<Statistical Physics by David Tong/4 Classical Thermodynamics>[§4.2, equation 4.156 and “Quasi-Static Processes”: state energy versus heat and work; Tong uses work positive inward]{ref}
For a calculated example, take one <mole> of a classical <monatomic gas> in the <ideal gas> regime, at fixed <volume (thermodynamics)>, from $300$ to $310\,\mathrm K$. With stipulated <molar gas constant> $R=8.314\,\mathrm{J\,mol^{-1}\,K^{-1}}$, its <internal energy> change is $3nR\Delta T/2=124.71\,\mathrm J$. A purely thermal route supplies $Q_{\rm in}=124.71\,\mathrm J$ with $W_{\rm on}=0$. An ideal insulated stirring route supplies $W_{\rm on}=124.71\,\mathrm J$ with $Q_{\rm in}=0$, measured between equilibrium endpoints after the motion dies away. The same stored increment has two different transfer histories; neither history defines a separate kind of stored <energy>.
\Image[/internal-energy-storage-and-transfers.png]
{title=<Internal energy> is stored; <heat> and <work> are ways to change it}
{id=internal-energy-storage-and-transfers}
{description=Original schematic and calculated <ideal gas> example with the stipulated parameters above; no experimental data. The two routes end at the same <temperature> and <volume (thermodynamics)>. Microscopic <kinetic energy> and interaction <potential energy> belong inside $U$; whole-body motion and external elevation are accounted for separately under the stated convention.}
The familiar <internal energy of an ideal gas> is a useful special case. Neglect intermolecular interactions, hold the <amount of substance> $n$ and composition fixed, and suppose that $f$ independent quadratic terms per particle obey the classical <equipartition theorem>. Choosing the constant reference contribution as zero gives
$$
U=\frac f2nRT,\qquad C_V=\frac f2nR.
$$
Here $T$ is absolute <temperature>, $R$ is the <molar gas constant>, and $C_V$ is the whole sample's <heat capacity at constant volume>. A <monatomic gas> has $f=3$ translational terms. A rigid diatomic <molecule> has $f=5$ when its two rotational terms are active and vibration is frozen; activating one <harmonic oscillator> vibration adds two terms, one <kinetic energy> and one <potential energy>, giving $f=7$. The counting is of quadratic contributions, so a vibrational coordinate is not just one contribution.<Statistical Physics by David Tong/2 Classical Gases>[§§2.2.1 and 2.4, equations 2.57–2.58 and the rotation/vibration energy counting]{ref}
This constant-$f$ formula applies only over a regime with those same active terms, classical translational statistics, nonrelativistic motion and unchanged composition. Quantum freezing, <quantum degeneracy>, <chemical reactions> or appreciable intermolecular interactions invalidate that simple counting. More generally, at fixed amount and composition an <ideal gas> can still have $U=U(T)$ with varying <heat capacity at constant volume>, so use $\Delta U=\int_{T_1}^{T_2}C_V(T)\,dT$. A nonzero constant reference adds $U_*$ to $fnRT/2$ without altering $\Delta U$.
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