8. Using Hess's Law and experimental data to determine the enthalpy of hydration of anhydrous magnesium sulfate

Method (for experimental details see method 1.3a1)

These experiments illustrate how to indirectly determine an enthalpy change that cannot be determined by direct experiment.

By the calorimetric methods already described from 3. to 6. you can separately determine the enthalpy of solution of magnesium sulfate heptahydrate crystals AND the enthalpy of solution of anhydrous magnesium sulfate.

In this case the hydration of anhydrous magnesium sulfate to give the hydrated version of the salt with its 7 molecules of water of crystallisation.

ΔH^θ for MgSO₄(s) + 7H₂O(l) ===> MgSO₄.7H₂O(s)

By the calorimetric methods already described you can separately determine the enthalpy of solution (solution) of anhydrous magnesium sulfate and hydrated magnesium sulfate crystals using the calculation method already described for experiments 3. to 6.

See example 7. and example 9. for another case of combining experimental enthalpy values to obtain another one not obtainable by experiment.

From this data, using a Hess's Law cycle described below, you can then calculate the enthalpy of hydration of anhydrous magnesium sulfate.

Application of Hess's Law to solving this problem ΔHθ1 A    B ΔHθ2      ΔHθ3 C from Hess's Law:   ΔHθ2  =  ΔHθ1  +  ΔHθ3 or ΔHθ1  =  ΔHθ2  –  ΔHθ3 or ΔHθ3  =  ΔHθ2  –  ΔHθ1

ΔHθ1 = ΔHθhydration[MgSO4(s)]
MgSO4(s) + 7H2O(l) MgSO4.7H2O(s)
ΔHθ2 = ΔHθsolution[MgSO4(s)]		+ aq		ΔHθ3 = ΔHθsolution[MgSO4.7H2O(s)]
MgSO4(aq) + 7H2O(l)
∴ from Hess's Law: ΔHθ2  =  ΔHθ1  +  ΔHθ3       and      ΔHθ1  =  ΔHθ2  –  ΔHθ3 ΔHθhydration[MgSO4(s)] = ΔHθsolution[MgSO4(s)] – ΔHθsolution[MgSO4.7H2O(s)]

The last line is the crucial calculation using 2 experimentally obtained values.

 

DATA and theoretical calculations for experiment examples 7. and 8.

The enthalpy of solution is sometimes just called the enthalpy of solution.

The best enthalpy data I can find from data books or computation from: ΔHθ1  =  ΔHθ2  –  ΔHθ3
salt and enthalpy value in kJmol–1	ΔHθ2 = ΔHθsolution(anhydrous salt)	ΔHθ3 = ΔHθsolution(hydrated salt)	ΔHθ1 = ΔHθhydration(anhydrous salt)
(a) copper(II) sulfate	ΔHθ2 = ΔHθsolution(CuSO4) = –73.3	ΔHθ3 = ΔHθsolution(CuSO4.5H2O) = +4.7	ΔHθ1 = ΔHθhydration(CuSO4) = –73.3 – (+4.7) = –78.0
(b) magnesium sulfate	ΔHθ2 = ΔHθsolution(MgSO4) = –91.2	ΔHθ3 = ΔHθsolution(MgSO4.7H2O) = +12.8	ΔHθ1 = ΔHθhydration(MgSO4) = –91.2 – (+12.8) = –104.0

Theoretical calculations for enthalpy of hydration of anhydrous salt to hydrated salt crystals (examples 7. and 8.)

ΔH^θ₁ = ΔH^θ_hydration(anhydrous salt) cannot be obtained directly by experiment, but can be calculated from experimental data (experiments 7./8.) using a Hess's Law cycle, and can be theoretically calculated from enthalpy of formation data (see below). This is the object of experimental exercises 7. and 8.

ΔH^θ₂ = ΔH^θ_solution(anhydrous salt)  can be obtained by experiment (e.g. like experiments 3. to 6.), and can also be obtained from standard data books!

Note that the values quoted are for infinite dilution, because the enthalpy values are dependent in a small way on the ratio of salt to water - and the specific heat capacity of solutions also depends on concentration!

ΔH^θ₃ = ΔH^θ_solution(hydrated salt) can be obtained by experiment, couldn't find any reliable data in books or internet, but can, theoretically, be calculated from reliable book/internet data for ΔH^θ₁ and ΔH^θ₂.

Theoretical calculation (see methods of enthalpy calculations)

ΔH^θ_reaction,298 = ∑ΔH^θ_f,298(products) – ∑ΔH^θ_f,298(reactants)

 

Hydration of magnesium sulfate calculation using data book information

MgSO₄(s)  +  7H₂O(l)  ===>  MgSO₄.7H₂O(s)

ΔH^θ_f,298(MgSO₄(s)) = –1278, ΔH^θ_f,298(H₂O(l)) = –286, ΔH^θ_f,298(MgSO₄.7H₂O(s)) = –3384

ΔH^θ_reaction,298 = ∑ΔH^θ_f,298(products) – ∑ΔH^θ_f,298(reactants)

ΔH^θ_{hydration,298} = ΔH^θ_f,298(MgSO₄.7H₂O(s)) – {ΔH^θ_f,298(MgSO₄(s)) + 7 x ΔH^θ_f,298(H₂O(l))}

ΔH^θ_{hydration,298}(MgSO₄(s)) = –3384 – {–1278 – 7 x 286} = –104.0 kJ mol^–1

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Experimental methods of determining enthalpy changes

Energetics–Thermochemistry–Thermodynamics Notes INDEX

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