= Isothermal compressibility
{title2=$\kappa_T=-V^{-1}(\partial V/\partial P)_{T,N}$}
Isothermal compressibility is the <compressibility> measured at fixed <temperature>, particle number and composition. A slowly compressed sealed piston in contact with a <thermal reservoir> can approach this constraint. For an <ideal gas>, differentiating $V=Nk_BT/P$ at fixed <temperature> gives $(\partial V/\partial P)_{T,N}=-V/P$, hence $\kappa_T=1/P$. A small <pressure> increase $\Delta P$ therefore gives $\Delta V/V\simeq-\Delta P/P$. For a finite change, use the whole <isothermal process> curve $V_2/V_1=P_1/P_2$ instead.
Take one <mole> of classical <monatomic gas>, with stipulated <molar gas constant> $R=8.314\,\mathrm{J\,mol^{-1}\,K^{-1}}$, initially at $T_0=300\,\mathrm K$ and $P_0=100\,\mathrm{kPa}$. The initial <volume (thermodynamics)> is $V_0=nRT_0/P_0=0.024942\,\mathrm{m^3}$. Compressing to $V_0/2$ gives
$$
\begin{aligned}
\text{Isothermal:}\quad &P_2=2P_0=200\,\mathrm{kPa},\quad T_2=300\,\mathrm K,\\
\text{Reversible adiabatic:}\quad &P_2=2^{5/3}P_0\simeq317.48\,\mathrm{kPa},\quad
T_2=2^{2/3}T_0\simeq476.22\,\mathrm K.
\end{aligned}
$$
The <isothermal compression> releases <heat> to the <thermal reservoir>; the reversible <adiabatic compression> retains the input <work> as <internal energy>. Reversing either curve gives the corresponding expansion, with <adiabatic expansion> cooling the <gas>.
\Image[/gas-isothermal-adiabatic-compression.png]
{title=An <ideal gas> resists reversible <adiabatic compression> more strongly than <isothermal compression>}
{description=Left: two <pressure>-<volume (thermodynamics)> curves through the same initial state; arrows point toward compression. Right: the corresponding <temperature> changes, including cooling during reversible <adiabatic expansion>. The dotted line marks half the initial <volume (thermodynamics)>. One <mole> of classical <monatomic gas>, with $\gamma=5/3$ and the stipulated $R$ above.}
The equilibrium curvature of <Gibbs free energy> provides another measurement of <isothermal compressibility>:
$$
\kappa_T=-\frac1V\left(\frac{\partial^2G}{\partial P^2}\right)_{T,N}.
$$
This follows from $V=(\partial G/\partial P)_{T,N}$, with $G$ denoting <Gibbs free energy>. Within a stable single <thermodynamic phase>, increasing <pressure> reduces <volume (thermodynamics)> and makes this curvature negative. At <phase coexistence>, changing the <phase fraction> can change the <volume (thermodynamics)> at constant <temperature> and <pressure>; the usual finite single-phase <isothermal compressibility> then need not exist. Near a <critical point>, it can diverge. https://www.feynmanlectures.caltech.edu/I_45.html#Ch45-S3[Feynman, Leighton and Sands, section 45-3: compression and phase coexistence]{ref}
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