= Internal energy of an ideal gas
For a fixed <amount of substance> and composition of classical <ideal gas>, <internal energy> depends only on <temperature>: $U=U(T,n)$, with no independent <volume (thermodynamics)> dependence. This does not require constant <heat capacity at constant volume>. Between equilibrium endpoints,
$$
dU=C_V(T)\,dT=nC_{V,\mathrm m}(T)\,dT,
\qquad U(T_2)-U(T_1)=n\int_{T_1}^{T_2}C_{V,\mathrm m}(T)\,dT.
$$
Here $C_V$ is total <heat capacity at constant volume>, $C_{V,\mathrm m}$ is <molar heat capacity>, and $U/m$ for sample <mass> $m$ is <specific internal energy>, whose change is $\int_{T_1}^{T_2}c_V(T)\,dT$. These endpoint relations apply even when the intervening process is not at fixed <volume (thermodynamics)>; they do not say that transferred <heat> always equals $\Delta U$. With constant $C_V$, the safe statement is $U(T)=U(T_*)+C_V(T-T_*)$. Writing $U=C_VT$ additionally chooses the reference <energy>; it does not prove that a physical <gas> remains classical and ideal down to zero <temperature>. https://www.feynmanlectures.caltech.edu/I_45.html#Ch45-S1[Richard Feynman, Robert Leighton and Matthew Sands, §45-1: internal-energy derivatives and heat capacity]{ref}
A thermodynamic proof outline uses the <first law of thermodynamics> in equilibrium form, $dU=T\,dS-p\,dV$, and the <Maxwell relation> $(\partial S/\partial V)_T=(\partial p/\partial T)_V$:
$$
\left(\frac{\partial U}{\partial V}\right)_T
=T\left(\frac{\partial p}{\partial T}\right)_V-p
=T\frac{nR}{V}-\frac{nRT}{V}=0.
$$
The last step uses the <ideal gas law> and the <molar gas constant> $R$, with fixed composition and <amount of substance>. Microscopically, separated <ideal gas> particles store no intermolecular interaction <energy>; their translational <kinetic energy> and any internal molecular <energy> are determined by <temperature>. Negligible intermolecular interactions still allow a <molecule> to have internal vibrational <potential energy>.
An original analytical model makes the distinction between an <ideal gas> and constant <heat capacity> concrete. Assume classical translation, two fully active rotational terms, and one independent <quantum harmonic oscillator> per diatomic <molecule>, with fixed <angular frequency> $\omega$. Define the vibrational <temperature> scale $\Theta=\hbar\omega/k_B$, using the reduced <Planck constant> $\hbar=h/(2\pi)$ and <Boltzmann constant> $k_B$. Summing its thermal occupations gives
$$
\frac{U(T)-U_*}{nR}=\frac52T+\frac{\Theta}{e^{\Theta/T}-1},\qquad
\frac{C_{V,\mathrm m}(T)}{R}=\frac52+\frac{x^2e^{-x}}{(1-e^{-x})^2},\qquad x=\frac{\Theta}{T}.
$$
The constant $U_*$ includes the chosen reference and <zero-point energy>, whose <temperature> derivative vanishes. When $T\ll\Theta$ the vibrational contribution to <heat capacity> is exponentially small; when $T\gg\Theta$ it tends to $R$ per <mole>. This model assumes rotations remain active throughout the interval and excludes dissociation, electronic excitation and <real gas> corrections. Its low-$T/\Theta$ edge is not a model of rotational freezing or of an actual substance at absolute zero.<Statistical Physics by David Tong/3 Quantum Gases>[§3.4: quantum vibrational spectrum, thermal occupations and zero-point energy]{ref}
\Image[/heat-capacity-ratio-vibrational-model.png]
{title=Temperature-dependent <internal energy of an ideal gas>, <molar heat capacities> and <heat capacity ratio> in a vibrational model}
{description=Analytical illustration with active rotations and one <quantum harmonic oscillator> per <molecule>. Left: <Mayer's relation> keeps the capacity difference equal to $R$. Middle: <heat capacity ratio> falls from $7/5$ toward $9/7$. Right: the <internal energy> change from $T=0.2\Theta$ is an <integral> of varying <heat capacity at constant volume>; the dashed line suppresses the vibrational contribution. No measured gas curve or material-specific vibrational frequency is implied.}
For this stipulated model, heating from $T_1=0.2\Theta$ to $T_2=\Theta$ gives $\Delta U/(nR\Theta)\simeq2.57519$. Holding the frozen-vibration value $C_{V,\mathrm m}=5R/2$ throughout would give $2$, underestimating the required <internal energy> increase by about $22.3\%$ of the model's full value. The <integral> measures all thermal storage activated along the interval. At fixed <volume (thermodynamics)> with only <heat> input, this is also the supplied <heat>; at fixed <pressure>, expansion <work> must be added through <Mayer's relation>.
<Joule's free-expansion experiment> tests the independence from <volume (thermodynamics)> in a different way: an insulated rigid vessel allows <gas> into an evacuated compartment, so no external <work> is delivered and ideally no <heat> is exchanged. The <first law of thermodynamics> gives $\Delta U=0$; positive $C_V$ then predicts equal initial and final equilibrium <temperatures> for the <ideal gas>. The apparatus, thermal-capacity controls and limitations of this inference are described in <Joule's free-expansion experiment> and <temperature sensitivity of a free-expansion experiment>. This endpoint test differs from the reversible <adiabatic expansion> of a loaded piston, which delivers <work> and lowers <internal energy>.
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