The enthalpy of fusion of a substance, also known as (latent) heat of fusion is the change in its enthalpy resulting from providing energy, typically heat, to a specific quantity of the substance to change its state from a solid to a liquid, at constant pressure. For example, when melting 1 kg of ice (at 0 °C under a wide range of pressures), 333.55 kJ of energy is absorbed with no temperature change. The heat of solidification (when a substance changes from liquid to solid) is equal and opposite.

This energy includes the contribution required to make room for any associated change in volume by displacing its environment against ambient pressure. The temperature at which the phase transition occurs is the melting point or the freezing point, according to context. By convention, the pressure is assumed to be 1 atm (101.325 kPa) unless otherwise specified.

Overview

The 'enthalpy' of fusion is a latent heat, because during melting the heat energy needed to change the substance from solid to liquid at atmospheric pressure is latent heat of fusion, as the temperature remains constant during the process. The latent heat of fusion is the enthalpy change of any amount of substance when it melts. When the heat of fusion is referenced to a unit of mass, it is usually called the specific heat of fusion, while the molar heat of fusion refers to the enthalpy change per amount of substance in moles.

The liquid phase has a higher internal energy than the solid phase. This means energy must be supplied to a solid in order to melt it and energy is released from a liquid when it freezes, because the molecules in the liquid experience weaker intermolecular forces and so have a higher potential energy (a kind of bond-dissociation energy for intermolecular forces).

When liquid water is cooled, its temperature falls steadily until it drops just below the line of freezing point at 0 °C. The temperature then remains constant at the freezing point while the water crystallizes. Once the water is completely frozen, its temperature continues to fall.

The enthalpy of fusion is almost always a positive quantity; helium is the only known exception.[1] Helium-3 has a negative enthalpy of fusion at temperatures below 0.3 K. Helium-4 also has a very slightly negative enthalpy of fusion below 0.77 K (−272.380 °C). This means that, at appropriate constant pressures, these substances freeze with the addition of heat.[2] In the case of 4He, this pressure range is between 24.992 and 25.00 atm (2,533 kPa).[3]

Standard enthalpy change of fusion of period three

Standard enthalpy change of fusion of period two of the periodic table of elements

Substance Heat of fusion

(cal/g) (J/g)

water 79.72 333.55

methane 13.96 58.99

propane 19.11 79.96

glycerol 47.95 200.62

formic acid 66.05 276.35

acetic acid 45.90 192.09

acetone 23.42 97.99

benzene 30.45 127.40

myristic acid 47.49 198.70

palmitic acid 39.18 163.93

sodium acetate 63–69 264–289[4]

stearic acid 47.54 198.91

gallium 19.2 80.4

paraffin wax (C25H52) 47.8-52.6 200–220

These values are mostly from the CRC Handbook of Chemistry and Physics, 62nd edition. The conversion between cal/g and J/g in the above table uses the thermochemical calorie (calth) = 4.184 joules rather than the International Steam Table calorie (calINT) = 4.1868 joules.

Examples

A) To heat 1 kg (1.00 liter) of water from 283.15 K to 303.15 K (10 °C to 30 °C) requires 83.6 kJ. However, to melt ice also requires energy. We can treat these two processes independently; thus, to heat 1 kg of ice from 273.15 K to water at 293.15 K (0 °C to 20 °C) requires:

(1) 333.55 J/g (heat of fusion of ice) = 333.55 kJ/kg = 333.55 kJ for 1 kg of ice to melt

PLUS

(2) 4.18 J/(g·K) × 20K = 4.18 kJ/(kg·K) × 20K = 83.6 kJ for 1 kg of water to increase in temperature by 20 K

= 417.15 kJ

From these figures it can be seen that one part ice at 0 °C will cool almost exactly 4 parts water from 20 °C to 0 °C.

B) Silicon has a heat of fusion of 50.21 kJ/mol. 50 kW of power can supply the energy required to melt about 100 kg of silicon in one hour, after it is brought to the melting point temperature:

50 kW = 50kJ/s = 180000kJ/h

180000kJ/h * (1 mol Si)/50.21kJ * 28gSi/(mol Si) * 1kgSi/1000gSi = 100.4kg/h

Solubility prediction

The heat of fusion can also be used to predict solubility for solids in liquids. Provided an ideal solution is obtained the mole fraction ( x 2 ) {\displaystyle (x_{2})} (x_{2}) of solute at saturation is a function of the heat of fusion, the melting point of the solid \( (T_{{{\mathit {fus}}}}) \) and the temperature (T) of the solution:

\( \ln x_{2}=-{\frac {\Delta H_{{{\mathit {fus}}}}^{\circ }}{R}}\left({\frac {1}{T}}-{\frac {1}{T_{{{\mathit {fus}}}}}}\right) \)

Here, R is the gas constant. For example, the solubility of paracetamol in water at 298 K is predicted to be:

\( x_{2}=\exp {\left(-{\frac {28100{\mbox{ J mol}}^{{-1}}}{8.314{\mbox{ J K}}^{{-1}}{\mbox{ mol}}^{{-1}}}}\left({\frac {1}{298}}-{\frac {1}{442}}\right)\right)}=0.0248 \)

This equals to a solubility in grams per liter of:

\( {\displaystyle {\frac {0.0248*{\frac {1000{\mbox{ g}}}{18.0153{\mbox{ mol}}^{-1}}}}{1-0.0248}}*151.17{\mbox{ mol}}^{-1}=213.4} \)

which is a deviation from the real solubility (240 g/L) of 11%. This error can be reduced when an additional heat capacity parameter is taken into account.[5]

Proof

At equilibrium the chemical potentials for the pure solvent and pure solid are identical:

\( \mu _{{solid}}^{\circ }=\mu _{{solution}}^{\circ }\, \)

or

\( \mu _{{solid}}^{\circ }=\mu _{{liquid}}^{\circ }+RT\ln X_{2}\, \)

with \( R\, \( the gas constant and \( T\, \) the temperature.

Rearranging gives:

\( RT\ln X_{2}=-(\mu _{{liquid}}^{\circ }-\mu _{{solid}}^{\circ })\, \)

and since

\( {\displaystyle \Delta G_{\mathit {fus}}^{\circ }=\mu _{liquid}^{\circ }-\mu _{solid}^{\circ }\,} \)

the heat of fusion being the difference in chemical potential between the pure liquid and the pure solid, it follows that

\( RT\ln X_{2}=-(\Delta G_{{{\mathit {fus}}}}^{\circ })\, \)

Application of the Gibbs–Helmholtz equation:

\( \left({\frac {\partial ({\frac {\Delta G_{{{\mathit {fus}}}}^{\circ }}{T}})}{\partial T}}\right)_{{p\,}}=-{\frac {\Delta H_{{{\mathit {fus}}}}^{\circ }}{T^{2}}} \)

ultimately gives:

\( \left({\frac {\partial (\ln X_{2})}{\partial T}}\right)={\frac {\Delta H_{{{\mathit {fus}}}}^{\circ }}{RT^{2}}} \)

or:

\( \partial \ln X_{2}={\frac {\Delta H_{{{\mathit {fus}}}}^{\circ }}{RT^{2}}}*\delta T \)

and with integration:

\( {\displaystyle \int _{X_{2}=1}^{X_{2}=x_{2}}\delta \ln X_{2}=\ln x_{2}=\int _{T_{\mathit {fus}}}^{T}{\frac {\Delta H_{\mathit {fus}}^{\circ }}{RT^{2}}}*\Delta T} \)

the end result is obtained:

\( \ln x_{2}=-{\frac {\Delta H_{{{\mathit {fus}}}}^{\circ }}{R}}\left({\frac {1}{T}}-{\frac {1}{T_{{{\mathit {fus}}}}}}\right) \) \)

See also

Heat of vaporization

Heat capacity

Thermodynamic databases for pure substances

Joback method (Estimation of the heat of fusion from molecular structure)

Latent heat

Lattice energy

Heat of dilution

Notes

Atkins & Jones 2008, p. 236.

Ott & Boerio-Goates 2000, pp. 92–93.

Hoffer, J. K.; Gardner, W. R.; Waterfield, C. G.; Phillips, N. E. (April 1976). "Thermodynamic properties of 4He. II. The bcc phase and the P-T and VT phase diagrams below 2 K". Journal of Low Temperature Physics. 23 (1): 63–102. Bibcode:1976JLTP...23...63H. doi:10.1007/BF00117245.

Ibrahim Dincer and Marc A. Rosen. Thermal Energy Storage: Systems and Applications, page 155

Measurement and Prediction of Solubility of Paracetamol in Water-Isopropanol Solution. Part 2. Prediction H. Hojjati and S. Rohani Org. Process Res. Dev.; 2006; 10(6) pp 1110–1118; (Article) doi:10.1021/op060074g

References

Atkins, Peter; Jones, Loretta (2008), Chemical Principles: The Quest for Insight (4th ed.), W. H. Freeman and Company, p. 236, ISBN 0-7167-7355-4

Ott, BJ. Bevan; Boerio-Goates, Juliana (2000), Chemical Thermodynamics: Advanced Applications, Academic Press, ISBN 0-12-530985-6

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States of matter (list)

State

Solid Liquid Gas / Vapor Plasma

Phase change

Low energy

Bose–Einstein condensate Fermionic condensate Degenerate matter Quantum Hall Rydberg matter Rydberg polaron Strange matter Superfluid Supersolid Photonic matter

High energy

QCD matter Lattice QCD Quark–gluon plasma Color-glass condensate Supercritical fluid

Other states

Colloid Glass Crystal Liquid crystal Time crystal Quantum spin liquid Exotic matter Programmable matter Dark matter Antimatter Magnetically ordered

Antiferromagnet Ferrimagnet Ferromagnet String-net liquid Superglass

Transitions

Boiling Boiling point Condensation Critical line Critical point Crystallization Deposition Evaporation Flash evaporation Freezing Chemical ionization Ionization Lambda point Melting Melting point Recombination Regelation Saturated fluid Sublimation Supercooling Triple point Vaporization Vitrification

Quantities

Enthalpy of fusion Enthalpy of sublimation Enthalpy of vaporization Latent heat Latent internal energy Trouton's ratio Volatility

Concepts

Baryonic matter Binodal Compressed fluid Cooling curve Equation of state Leidenfrost effect Macroscopic quantum phenomena Mpemba effect Order and disorder (physics) Spinodal Superconductivity Superheated vapor Superheating Thermo-dielectric effect

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