What is the energy required to melt 1kg of ice at?

This worked example problem demonstrates how to calculate the energy required to raise the temperature of a sample that includes changes in phase. This problem finds the energy required to turn cold ice into hot steam.

What is the heat in Joules required to convert 25 grams of -10 °C ice into 150 °C steam?Useful information:heat of fusion of water = 334 J/gheat of vaporization of water = 2257 J/gspecific heat of ice = 2.09 J/g·°Cspecific heat of water = 4.18 J/g·°Cspecific heat of steam = 2.09 J/g·°C

The total energy required is the sum of the energy to heat the -10 °C ice to 0 °C ice, melting the 0 °C ice into 0 °C water, heating the water to 100 °C, converting 100 °C water to 100 °C steam and heating the steam to 150 °C. To get the final value, first calculate the individual energy values and then add them up.

Step 1:

Find the heat required to raise the temperature of ice from -10 °C to 0 °C. Use the formula:

q = mcΔT

where

  • q = heat energy
  • m = mass
  • c = specific heat
  • ΔT = change in temperature

In this problem:

  • q = ?
  • m = 25 g
  • c = (2.09 J/g·°C
  • ΔT = 0 °C - -10 °C (Remember, when you subtract a negative number, it is the same as adding a positive number.)

Plug in the values and solve for q:

q = (25 g)x(2.09 J/g·°C)[(0 °C - -10 °C)]q = (25 g)x(2.09 J/g·°C)x(10 °C)

q = 522.5 J


The heat required to raise the temperature of ice from -10 °C to 0 °C = 522.5 J


Step 2:

Find the heat required to convert 0 °C ice to 0 °C water.


Use the formula for heat:

q = m·ΔHf

where

  • q = heat energy
  • m = mass
  • ΔHf = heat of fusion

For this problem:

  • q = ?
  • m = 25 g
  • ΔHf = 334 J/g

Plugging in the values gives the value for q:

q = (25 g)x(334 J/g)
q = 8350 J

The heat required to convert 0 °C ice to 0 °C water = 8350 J


Step 3:

Find the heat required to raise the temperature of 0 °C water to 100 °C water.q = mcΔTq = (25 g)x(4.18 J/g·°C)[(100 °C - 0 °C)]q = (25 g)x(4.18 J/g·°C)x(100 °C)q = 10450 JThe heat required to raise the temperature of 0 °C water to 100 °C water = 10450 J

Step 4:

Find the heat required to convert 100 °C water to 100 °C steam.
q = m·ΔHvwhere

q = heat energy

m = mass

ΔHv = heat of vaporization

q = (25 g)x(2257 J/g)q = 56425 J

The heat required to convert 100 °C water to 100 °C steam = 56425

Step 5:

Find the heat required to convert 100 °C steam to 150 °C steamq = mcΔTq = (25 g)x(2.09 J/g·°C)[(150 °C - 100 °C)]q = (25 g)x(2.09 J/g·°C)x(50 °C)q = 2612.5 J

The heat required to convert 100 °C steam to 150 °C steam = 2612.5

Step 6:

Find total heat energy. In this final step, put together all of the answers from the previous calculations to cover the entire temperature range.


HeatTotal = HeatStep 1 + HeatStep 2 + HeatStep 3 + HeatStep 4 + HeatStep 5
HeatTotal = 522.5 J + 8350 J + 10450 J + 56425 J + 2612.5 J
HeatTotal = 78360 J

Answer:

The heat required to convert 25 grams of -10 °C ice into 150 °C steam is 78360 J or 78.36 kJ.

  • Atkins, Peter and Loretta Jones (2008). Chemical Principles: The Quest for Insight (4th ed.). W. H. Freeman and Company. p. 236. ISBN 0-7167-7355-4.
  • Ge, Xinlei; Wang, Xidong (2009). "Calculations of Freezing Point Depression, Boiling Point Elevation, Vapor Pressure and Enthalpies of Vaporization of Electrolyte Solutions by a Modified Three-Characteristic Parameter Correlation Model". Journal of Solution Chemistry. 38 (9): 1097–1117. doi:10.1007/s10953-009-9433-0
  • Ott, B.J. Bevan and Juliana Boerio-Goates (2000) Chemical Thermodynamics: Advanced Applications. Academic Press. ISBN 0-12-530985-6.
  • Young, Francis W.; Sears, Mark W.; Zemansky, Hugh D. (1982). University Physics (6th ed.). Reading, Mass.: Addison-Wesley. ISBN 978-0-201-07199-3.

In thermodynamics, 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.

What is the energy required to melt 1kg of ice at?

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

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.

The 'enthalpy' of fusion is a latent heat, because, while 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.

  • To heat 1 kg of liquid water from 0 °C to 20 °C requires 83.6 kJ (see below). However, heating 0 °C ice to 20 °C requires additional energy to melt the ice. 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) × 20 K = 4.18 kJ/(kg⋅K) × 20 K = 83.6 kJ for 1 kg of water to increase in temperature by 20 K (1 + 2) 333.55 kJ + 83.6 kJ = 417.15 kJ for 1 kg of ice to increase in temperature by 20 K 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.
  • 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: 50 kW = 50kJ/s = 180000kJ/h 180000kJ/h × (1 mol Si)/50.21kJ × 28gSi/(mol Si) × 1kgSi/1000gSi = 100.4kg/h

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})}   of solute at saturation is a function of the heat of fusion, the melting point of the solid ( T fus ) {\displaystyle (T_{\text{fus}})}   and the temperature ( T ) {\displaystyle (T)}   of the solution:

ln ⁡ x 2 = − Δ H fus ∘ R ( 1 T − 1 T fus ) {\displaystyle \ln x_{2}=-{\frac {\Delta H_{\text{fus}}^{\circ }}{R}}\left({\frac {1}{T}}-{\frac {1}{T_{\text{fus}}}}\right)}  

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

x 2 = exp ⁡ [ − 28100   J mol − 1 8.314   J K − 1   mol − 1 ( 1 298   K − 1 442   K ) ] = 0.0248 {\displaystyle x_{2}=\exp {\left[-{\frac {28100~{\text{J mol}}^{-1}}{8.314~{\text{J K}}^{-1}~{\text{mol}}^{-1}}}\left({\frac {1}{298~{\text{K}}}}-{\frac {1}{442~{\text{K}}}}\right)\right]}=0.0248}  

Since the molar mass of water and paracetamol are 18.0153gmol−1 and 151.17gmol−1 and the density of the solution is 1000gL−1, an estimate of the solubility in grams per liter is:

0.0248 × 1000   g L − 1 18.0153   g mol − 1 1 − 0.0248 × 151.17   g mol − 1 = 213.4   g L − 1 {\displaystyle {\frac {0.0248\times {\frac {1000~{\text{g L}}^{-1}}{18.0153~{\text{g mol}}^{-1}}}}{1-0.0248}}\times 151.17~{\text{g mol}}^{-1}=213.4~{\text{g L}}^{-1}}  

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:

μ solid ∘ = μ solution ∘ {\displaystyle \mu _{\text{solid}}^{\circ }=\mu _{\text{solution}}^{\circ }\,}  

or

μ solid ∘ = μ liquid ∘ + R T ln ⁡ X 2 {\displaystyle \mu _{\text{solid}}^{\circ }=\mu _{\text{liquid}}^{\circ }+RT\ln X_{2}\,}  

with R {\displaystyle R\,}   the gas constant and T {\displaystyle T\,}   the temperature.

Rearranging gives:

R T ln ⁡ X 2 = − ( μ liquid ∘ − μ solid ∘ ) {\displaystyle RT\ln X_{2}=-\left(\mu _{\text{liquid}}^{\circ }-\mu _{\text{solid}}^{\circ }\right)\,}  

and since

Δ G fus ∘ = μ liquid ∘ − μ solid ∘ {\displaystyle \Delta G_{\text{fus}}^{\circ }=\mu _{\text{liquid}}^{\circ }-\mu _{\text{solid}}^{\circ }\,}  

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

R T ln ⁡ X 2 = − ( Δ G fus ∘ ) {\displaystyle RT\ln X_{2}=-\left(\Delta G_{\text{fus}}^{\circ }\right)\,}  

Application of the Gibbs–Helmholtz equation:

( ∂ ( Δ G fus ∘ T ) ∂ T ) p = − Δ H fus ∘ T 2 {\displaystyle \left({\frac {\partial \left({\frac {\Delta G_{\text{fus}}^{\circ }}{T}}\right)}{\partial T}}\right)_{p\,}=-{\frac {\Delta H_{\text{fus}}^{\circ }}{T^{2}}}}  

ultimately gives:

( ∂ ( ln ⁡ X 2 ) ∂ T ) = Δ H fus ∘ R T 2 {\displaystyle \left({\frac {\partial \left(\ln X_{2}\right)}{\partial T}}\right)={\frac {\Delta H_{\text{fus}}^{\circ }}{RT^{2}}}}  

or:

∂ ln ⁡ X 2 = Δ H fus ∘ R T 2 × δ T {\displaystyle \partial \ln X_{2}={\frac {\Delta H_{\text{fus}}^{\circ }}{RT^{2}}}\times \delta T}  

and with integration:

∫ X 2 = 1 X 2 = x 2 δ ln ⁡ X 2 = ln ⁡ x 2 = ∫ T fus T Δ H fus ∘ R T 2 × Δ T {\displaystyle \int _{X_{2}=1}^{X_{2}=x_{2}}\delta \ln X_{2}=\ln x_{2}=\int _{T_{\text{fus}}}^{T}{\frac {\Delta H_{\text{fus}}^{\circ }}{RT^{2}}}\times \Delta T}  

the end result is obtained:

ln ⁡ x 2 = − Δ H fus ∘ R ( 1 T − 1 T fus ) {\displaystyle \ln x_{2}=-{\frac {\Delta H_{\text{fus}}^{\circ }}{R}}\left({\frac {1}{T}}-{\frac {1}{T_{\text{fus}}}}\right)}  
  • 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

  1. ^ Atkins & Jones 2008, p. 236.
  2. ^ Ott & Boerio-Goates 2000, pp. 92–93.
  3. ^ 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. S2CID 120473493.
  4. ^ Ibrahim Dincer and Marc A. Rosen. Thermal Energy Storage: Systems and Applications, page 155
  5. ^ 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

  • Atkins, Peter; Jones, Loretta (2008), Chemical Principles: The Quest for Insight (4th ed.), W. H. Freeman and Company, p. 236, ISBN 978-0-7167-7355-9
  • Ott, BJ. Bevan; Boerio-Goates, Juliana (2000), Chemical Thermodynamics: Advanced Applications, Academic Press, ISBN 0-12-530985-6

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