= Heat capacity ratio
{title2=$\gamma$}
{wiki}
= Specific-heat ratio
{synonym}
= Adiabatic exponent
{synonym}
= Adiabatic index
{synonym}
The heat capacity ratio compares <heat capacity at constant pressure> $C_p$ with <heat capacity at constant volume> $C_V$ at the same equilibrium state and fixed composition:
$$
\gamma=\frac{C_p}{C_V}
=\frac{C_{p,\mathrm m}}{C_{V,\mathrm m}}
=\frac{c_p}{c_V}.
$$
Here $C_p,C_V$ refer to the whole sample, in $\mathrm{J\,K^{-1}}$; <molar heat capacities> $C_{p,\mathrm m}=C_p/n$ and $C_{V,\mathrm m}=C_V/n$ are per <mole>; <specific heat capacities> $c_p=C_p/m$ and $c_V=C_V/m$ are per unit <mass>. The <amount of substance> $n$ or <mass> $m$ cancels only when both capacities use the same normalization. Thus <heat capacity ratio> is dimensionless and independent of sample size. For a fixed-composition <ideal gas> with positive finite $C_V$, <Mayer's relation> gives
$$
\gamma(T)=1+\frac{nR}{C_V(T)}=1+\frac{R}{C_{V,\mathrm m}(T)}>1,
$$
where $R$ is the <molar gas constant>. More ways to store <internal energy> make a given <temperature> increase require more <heat>, bringing <heat capacity ratio> closer to unity. https://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node18.html[MIT Unified Engineering, Specific heats: definitions and ideal-gas relations]{ref}
The <equipartition theorem> predicts $C_{V,\mathrm m}=fR/2$ when $f$ independent quadratic terms in molecular <energy> are thermally active and their number stays fixed; consequently $\gamma=1+2/f$. A classical <monatomic gas> with no appreciable electronic excitation has $f=3$, giving $C_{V,\mathrm m}=3R/2$ and $\gamma=5/3$. For an <ideal gas> of diatomic <molecules> with three translational and two rotational terms active, but its vibrational <normal mode> effectively frozen, $f=5$ gives $C_{V,\mathrm m}=5R/2$ and $\gamma=7/5$. A fully classical harmonic vibration adds both <kinetic energy> and <potential energy>, giving $f=7$ and $\gamma=9/7$. These are model regimes, not universal constants of every <gas>: rotational freezing, electronic excitation, and changing composition invalidate the simple count.<Statistical Physics by David Tong/2 Classical Gases>[§§2.2.1 and 2.4: equipartition and diatomic gases]{ref}
The mechanical significance of <heat capacity ratio> follows from the <first law of thermodynamics>. For a fixed amount of <ideal gas> in a reversible <adiabatic process>, $C_V(T)\,dT=-p\,dV$. Combining this with the <ideal gas law> gives the local relation
$$
\left(\frac{\partial\ln p}{\partial\ln V}\right)_S=-\gamma(T),\qquad
\int_{T_1}^{T_2}\frac{C_V(T)}{T}\,dT+nR\ln\frac{V_2}{V_1}=0.
$$
The <partial derivative> holds at fixed <entropy> and composition even when <heat capacity ratio> varies with <temperature>. The finite-path formulas $pV^\gamma=\mathrm{constant}$ and $TV^{\gamma-1}=\mathrm{constant}$ require the additional approximation of constant <heat capacity ratio>. A rapid compression also needs internal equilibration to approximate this path; rapid motion alone does not ensure reversibility. The same local response gives <isentropic compressibility> $1/(\gamma p)$ and, for small sound disturbances in this <ideal gas> regime, <speed of sound> $a=\sqrt{\gamma p/\rho}$, where $\rho$ is <mass density>. Thus thermal storage controls mechanical stiffness, sound propagation and the measurements in <Rüchardt experiment> and <Clément-Desormes experiment>.
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