Enthalpy of fusion

A log-log plot of the enthalpies of melting and boiling versus the melting and boiling temperatures for the pure elements. The linear relationship between the enthalpy of melting and the melting temperature is known as Richard's rule.

Enthalpies of melting and boiling for pure elements versus temperatures of transition, demonstrating Trouton's rule.

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. 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. By convention, the pressure is assumed to be 1 atm (101.325 kPa) unless otherwise specified.

The 'enthalpy' of fusion is a latent heat, because during melting the introduction of heat cannot be observed as a temperature change, 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)	Heat of fusion (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 (cal_th) = 4.184 joules rather than the International Steam Table calorie (cal_INT) = 4.1868 joules.

Examples

1) 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. 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

Thus one part ice at 0 °C will cool almost exactly 4 parts water from 20 °C to 0 °C.

2) 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 $(x_{2})$ 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:

${\frac {0.0248*{\frac {1000{\mbox{ g}}}{18.053{\mbox{ mol}}^{{-1}}}}}{1-0.0248}}*151.17{\mbox{ mol}}^{{-1}}=213.4$

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 }\,

\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

\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:

\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)

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

States of matter (list)
State	Solid Liquid Gas / Vapor Plasma
Low energy	Bose–Einstein condensate Fermionic condensate Degenerate matter Quantum Hall Rydberg matter Strange matter Superfluid Supersolid Photonic matter
High energy	QCD matter Lattice QCD Quark–gluon plasma Supercritical fluid
Other states	Colloid Glass Liquid crystal Quantum spin liquid 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	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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