= Latent heat of fusion
{title2=$L$}
An ice bath can absorb <heat> while its <temperature> barely changes: the <heat> converts <ice> into <water>. The <latent heat of fusion> quantifies this <melting> budget. For a pure substance at its equilibrium <melting> <temperature> $T_m(p)$ and specified <pressure> $p$, define it here per unit <mass>:
$$
L(p)=h_l(T_m(p),p)-h_s(T_m(p),p),\qquad h=H/m.
$$
Here $h_l$ and $h_s$ are the <specific enthalpies> of the coexisting bulk <liquid> and <solid>, and $H$ is <enthalpy>. For ordinary <melting>, $L>0$, in $\mathrm{J\,kg^{-1}}$; an amount $m$ has a total fusion <enthalpy> $mL$ in <joules>. A molar fusion <enthalpy> instead has units $\mathrm{J\,mol^{-1}}$ and equals $ML$, where $M=m/n$ is <mass> divided by <amount of substance> in $\mathrm{kg\,mol^{-1}}$. Always check the normalization before inserting a tabulated value into an <energy> balance.<University Physics Volume 2 by Samuel J. Ling and others/1.5 Phase Changes>[§1.5: fusion and specific latent heat]{ref} https://goldbook.iupac.org/terms/view/12214[IUPAC Gold Book, mass divided by amount of substance]{ref}
For a <closed system> converting completely from <solid> to <liquid> at fixed <pressure>, with only <pressure>-<volume (thermodynamics)> <work> and negligible changes in bulk <kinetic energy> and <potential energy>, the <first law of thermodynamics> gives $Q_p=\Delta H=mL$. Both endpoints are at $T_m(p)$: warming initially colder <ice> or subsequently warming the resulting <water> requires additional <sensible heat>. Since $H=U+pV$, the corresponding <specific internal energy> jump is $\Delta u=L-p\Delta v$, where $v=V/m$; $L$ includes the <work> associated with the change in <volume (thermodynamics)>. <Heat> is transferred <energy>, whereas <enthalpy> is a <state function>. https://davidtong.org/teaching/statistical-physics/statmechhtml/S4#S4.SS4.SSS1[David Tong, Statistical Physics, §4.4.1 and the first law]{ref}
\Image[/fusion-latent-heat-plateau.png]
{title=During equilibrium <melting>, added <heat> changes the <phase fraction> while <temperature> stays fixed}
{description=An original idealized ice-inspired model, not measured data: $m=0.100\,\mathrm{kg}$, $T_m=273.15\,\mathrm K$, constant <specific heat capacity at constant pressure> in each phase, $c_s=2100$ and $c_l=4180\,\mathrm{J\,kg^{-1}\,K^{-1}}$, and $L=334000\,\mathrm{J\,kg^{-1}}$. The initial <solid> is $10\,\mathrm K$ below $T_m$; the final <liquid> is $10\,\mathrm K$ above it. The horizontal axis is net sample <enthalpy> input, excluding the container.}
Let $x=m_l/m$ be the <liquid> <phase fraction> by <mass>, rather than by <volume (thermodynamics)>. Neglecting interfacial contributions to <enthalpy>, equilibrium <phase coexistence> at fixed <pressure> gives
$$
h=(1-x)h_s+xh_l=h_s+xL,\qquad Q_p=mL(x_2-x_1),\qquad T=T_m(p).
$$
In the model above, reaching $T_m$ takes $2.10\,\mathrm{kJ}$ of <sensible heat>; the plateau then absorbs $33.40\,\mathrm{kJ}$, ending at a cumulative $35.50\,\mathrm{kJ}$. At $18.80\,\mathrm{kJ}$, half the <mass> is <liquid> and the <temperature> is still $T_m$. The final warming takes another $4.18\,\mathrm{kJ}$. The rounded $L$ and $c_l$ are inspired by the introductory <ice-melting calorimetry> values in Case Western's demonstration; constant properties and exact equilibrium throughout are stipulated model assumptions. https://case.edu/artsci/phys/courses/demos/spec.htm[Case Western Reserve University, specific heat and heat of fusion demonstration]{ref}
Within one <thermodynamic phase>, <specific heat capacity at constant pressure> is $c_p=(\partial h/\partial T)_p$, in $\mathrm{J\,kg^{-1}\,K^{-1}}$; a sample's <heat capacity at constant pressure> is $C_p=mc_p$, in $\mathrm{J\,K^{-1}}$. The plateau is a finite <enthalpy> change at zero <temperature> change through varying $x$, so neither a finite single-phase $c_p$ nor $Q=mc_p\Delta T$ represents it. The <heating curve> needs separate <sensible heat> intervals and a fusion contribution. https://goldbook.iupac.org/terms/view/H02753[IUPAC Gold Book, heat capacity]{ref}
The plateau assumes a pure substance, a stable bulk <solid>-<liquid> <phase coexistence> pair, fixed <pressure>, and sufficiently slow, uniform heating to remain near <thermodynamic equilibrium>. The total <mass> and composition must remain fixed, with negligible evaporation and no other <phase transition>. A fast heater can produce <temperature> gradients, so a local <thermometer> need not read the equilibrium $T_m$. Neither $T_m$ nor $L$ is universal across <pressures> or compositions: <common-pressure melting-point slope> describes the pressure dependence of $T_m$, while mixtures can melt over a <temperature> range as their coexisting compositions change. Their <liquidus> must be consulted instead of assuming a pure-substance plateau. https://www2.tulane.edu/~sanelson/eens211/2compphasdiag.pdf#page=3[Stephen A. Nelson, Tulane, Two Component Phase Diagrams, pp. 2–4: melting ranges and liquidus]{ref}
This makes <ice> useful as a <temperature> buffer: under these conditions it can absorb $m_sL$ before the remaining <solid> <mass> $m_s$ disappears, after which the <water> warms. If a stipulated constant net input <power> $P$ goes entirely into <melting>, that buffer lasts $m_sL/P$. Reversing the equilibrium conversion releases the same fusion <enthalpy> during <freezing>. The beautiful thermodynamic connection is $L=T_m(s_l-s_s)$, with $s$ the <specific entropy> and $T_m$ absolute <temperature>: equal <Gibbs free energies> at <phase coexistence> allow different <enthalpies> and <entropies>. The nearly stationary <thermometer> hides a finite, measurable change of state. https://davidtong.org/teaching/statistical-physics/statmechhtml/S5[Tong, Statistical Physics, §5.1.2: coexistence and latent heat]{ref}
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