# Enthalpy of fusion

The **enthalpy of fusion** (symbol: **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \Delta{}H_{fus}**
), also known as the **heat of fusion** or **specific melting heat**, is the amount of thermal energy which must be absorbed or evolved for 1 mole of a substance to change states from a solid to a liquid or vice versa. It is also called the **latent heat of fusion** or the **enthalpy change of fusion**, and the temperature at which it occurs is called the melting point.

When you withdraw thermal energy from a liquid or solid, the temperature falls. When you add heat energy the temperature rises. However, at the transition point between solid and liquid (the melting point), extra energy is required (the heat of fusion). To go from liquid to solid, the molecules of a substance must become more ordered. For them to maintain the order of a solid, extra heat must be withdrawn. In the other direction, to create the disorder from the solid crystal to liquid, extra heat must be added.

The heat of fusion can be observed if you measure the temperature of water as it freezes. If you plunge a closed container of room temperature water into a very cold environment (say −20 °C), you will see the temperature fall steadily until it drops just below the freezing point (0 °C). The temperature then rebounds and holds steady while the water crystallizes. Once completely frozen, the temperature will fall steadily again.

The temperature stops falling at (or just below) the freezing point due to the heat of fusion. The energy of the heat of fusion must be withdrawn (the liquid must turn to solid) before the temperature can continue to fall.

The units of heat of fusion are usually expressed as:

- joules per mole (the SI units)
- calories per gram (old metric units now little used, except for a different, larger calorie used in nutritional contexts)
- British thermal units per pound or Btu per pound-mole

- Note: These are not the calories found in food. The calories found in food are more properly known as kilocalories—equal to 1000 calories. 1000 calories = 1 kilocalorie = 1 food calorie. Food calories are sometimes abbreviated as kcal as if small calories were being used, while calories are abbreviated as cal. Another distinguishing method, though often confusing, uses capitalisation. A Calorie is a food calorie, or 1000 calories. So 1 Cal = 1000 cal.

## Reference values of common substances

Substance | Heat of fusion (cal/g) | Heat of fusion (kJ/kg) |
---|---|---|

water | 79.72 | 333.55 |

methane | 13.96 | 58.41 |

ethane | 22.73 | 95.10 |

propane | 19.11 | 79.96 |

methanol | 23.70 | 99.16 |

ethanol | 26.05 | 108.99 |

glycerol | 47.95 | 200.62 |

formic acid | 66.05 | 276.35 |

acetic acid | 45.91 | 192.09 |

acetone | 23.42 | 97.99 |

benzene | 30.45 | 127.40 |

myristic acid | 47.49 | 198.70 |

palmitic acid | 39.18 | 163.93 |

stearic acid | 47.54 | 198.91 |

These values are from the CRC **Handbook of Chemistry and Physics**, 62nd edition. The conversion between cal/g and kJ/kg 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.

## Applications

To heat one kilogram (about 1 litre) of water from 10 °C to 30 °C requires 20 kcal.

However, to melt ice and raise the resulting water temperature 20 °C requires extra energy. To heat ice from 0 °C to water at 20 °C requires:

- (1) 80 Cal/g (heat of fusion of ice) = 80 kcal for 1 kg
- PLUS
- (2) 1 cal/(g·°C) = 20 kcal for 1 kg to go up 20 °C
- = 100 kcal

## 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 of solute at saturation is a function of the heat of fusion, the melting point of the solid and the temperature of the solution:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \ln x_2 = - \frac {\Delta H^\circ_{fus}}{R} \left(\frac{1}{T}- \frac{1}{T_{fus}}\right)**

For example the solubility of paracetamol in water at 298 K is predicted to be:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \ln x_2 = - \frac {28100 \mbox{ J mol}^{-1}} {8.314 \mbox{ J K}^{-1} \mbox{ mol}^{-1}}\left(\frac{1}{298}- \frac{1}{442}\right) = 0.0248**

This equals to a solubility in grams per liter of:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \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 ^{[1]}

### Proof

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

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \mu^\circ_{solid} = \mu^\circ_{solution}\,**

or

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \mu^\circ_{solid} = \mu^\circ_{liquid} + RTlnX_2\,**

with **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): R\,**
the gas constant and **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): T\,**
the temperature.

Rearranging gives:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): RTlnX_2 = - (\mu^\circ_{liquid} - \mu^\circ_{solid})\,**

and since

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \Delta G^\circ_{fus} = - (\mu^\circ_{liquid} - \mu^\circ_{solid})\,**

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

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): RTlnX_2 = - ( \Delta G^\circ_{fus})\,**

Application of the Gibbs-Helmholtz equation:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \left( \frac{\partial ( \frac{\Delta G^\circ_{fus} } {T} ) } {\partial T} \right)_{p\,} = \frac {\Delta H^\circ_{fus}} {T^2}**

ultimately gives:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \left( \frac{\partial ( lnX_2 } {\partial T} \right) = \frac {\Delta H^\circ_{fus}} {RT^2}**

or:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \partial lnX_2 = \frac {\Delta H^\circ_{fus}} {RT^2}*\delta T**

and with integration:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \int^{x_2=x_2}_{x_2 = 1} \delta lnX_2 = ln x_2 = \int_{T_fus}^T \frac {\Delta H^\circ_{fus}} {RT^2}*\Delta T**

the end result is obtained:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): ln x_2 = - \frac {\Delta H^\circ_{fus}} {R}\left(\frac{1}{T}- \frac{1}{T_{fus}}\right)**

## See also

- Heat of vaporization
- Heat capacity
- Specific heat capacity
- Thermodynamic databases for pure substances

## References

- ↑
*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

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