Born-Haber Cycle for MgCl2 (Lattice Energy Steps)
For one mole of MgCl₂(s), the Born–Haber cycle combines magnesium atomization and ionization, chlorine bond breaking and electron gain, and crystal formation. Using the data below and Hess’s law gives a lattice enthalpy of about −2524 kJ mol⁻¹ when defined as formation from gaseous ions. The reverse, crystal separation into gaseous ions, is +2524 kJ mol⁻¹.
It can be easy to lose track of a sign or a coefficient in this calculation. The most common traps are counting chlorine’s bond energy twice, forgetting magnesium’s second ionization energy, or reporting a lattice value without saying which direction it describes. I’ll build the cycle one step at a time, then check the result.
Born–Haber Cycle Overview and Thermochemical Data
A Born–Haber cycle applies Hess’s law to the formation of an ionic solid. It breaks the overall reaction into steps that make gaseous atoms and ions, then join the ions into a crystal. The steps’ enthalpy changes must add to the enthalpy change for the overall reaction.
The balanced standard formation reaction is:
[ \mathrm{Mg(s) + Cl_2(g) \rightarrow MgCl_2(s)} ]
Its standard enthalpy of formation, (\Delta H_f^\circ), is approximately (-642\ \mathrm{kJ\,mol^{-1}}). This means that forming one mole of solid magnesium chloride from its elements in their standard states releases about 642 kJ.
The cycle reaches the same solid by a different route: first turn the elements into gaseous ions, then form the crystal from those ions. Because enthalpy is a state function, the total change is the same along either route. That is Hess’s law in practice.
Data used in the calculation
The values below are approximate, in kilojoules per mole of MgCl₂ formed. The atomization and ionization values refer to one mole of magnesium atoms; the chlorine values are stated for the quantities needed to make one mole of MgCl₂.
| Step | Process | Enthalpy change (kJ mol⁻¹) |
|---|---|---|
| Magnesium atomization | Mg(s) → Mg(g) | +148 |
| First ionization | Mg(g) → Mg⁺(g) + e⁻ | +738 |
| Second ionization | Mg⁺(g) → Mg²⁺(g) + e⁻ | +1451 |
| Chlorine bond dissociation | Cl₂(g) → 2Cl(g) | +243 |
| Electron gain by chlorine | 2Cl(g) + 2e⁻ → 2Cl⁻(g) | −698 |
| Crystal formation from gaseous ions | Mg²⁺(g) + 2Cl⁻(g) → MgCl₂(s) | To calculate |
A mole of MgCl₂ contains one magnesium ion and two chloride ions. That formula sets the required amounts in the cycle: one magnesium atom, one Cl₂ molecule, two electrons gained by chlorine, and both magnesium ionization steps.
The data are rounded teaching values. Published thermochemical values can differ slightly with the data source and temperature, so small differences in a final result do not always signal an error. Keep the values and units consistent throughout one calculation.
MgCl₂ Steps, Stoichiometry, and Enthalpy Requirements
Each step changes the chemical form of the elements, but the atoms and total charge must balance. For magnesium chloride, the endpoint before crystal formation is one gaseous Mg²⁺ ion plus two gaseous Cl⁻ ions. Writing that endpoint first helps show which steps and coefficients the cycle needs.
1. Turn solid magnesium into gaseous atoms
[ \mathrm{Mg(s) \rightarrow Mg(g)} \qquad \Delta H = +148\ \mathrm{kJ\,mol^{-1}} ]
This is magnesium atomization, also called sublimation in this context. It takes energy to separate magnesium from its solid form and make gaseous atoms, so the enthalpy change is positive.
2. Remove two electrons from each magnesium atom
[ \mathrm{Mg(g) \rightarrow Mg^+(g) + e^-} \qquad IE_1 = +738\ \mathrm{kJ\,mol^{-1}} ]
[ \mathrm{Mg^+(g) \rightarrow Mg^{2+}(g) + e^-} \qquad IE_2 = +1451\ \mathrm{kJ\,mol^{-1}} ]
Ionization energy is the energy needed to remove an electron from a gaseous atom or ion. Magnesium must lose two electrons to become Mg²⁺. Include both ionization energies: the first makes Mg⁺, and the second makes Mg²⁺.
Together, the ionization steps require:
[ 738 + 1451 = +2189\ \mathrm{kJ\,mol^{-1}} ]
The second ionization energy is larger because it removes an electron from an already positive ion. It is still a separate required step, not an optional correction.
3. Break chlorine molecules into atoms
[ \mathrm{Cl_2(g) \rightarrow 2Cl(g)} \qquad D(\mathrm{Cl_2}) = +243\ \mathrm{kJ\,mol^{-1}} ]
The bond-dissociation enthalpy shown here is for breaking one mole of Cl₂ molecules into two moles of chlorine atoms. It is not 243 kJ per chlorine atom. MgCl₂ needs two chlorine atoms, so one Cl₂ molecule supplies exactly the amount required.
4. Add two electrons to two chlorine atoms
[ \mathrm{2Cl(g) + 2e^- \rightarrow 2Cl^-(g)} ]
For one chlorine atom, the electron-affinity value used here is about (-349\ \mathrm{kJ\,mol^{-1}}). Since the formula requires two chloride ions:
[ 2 \times (-349) = -698\ \mathrm{kJ\,mol^{-1}} ]
Electron affinity describes the enthalpy change when a gaseous atom gains an electron. With the values used here, chlorine’s electron gain releases energy, so the sign is negative.
5. Form the crystal from gaseous ions
[ \mathrm{Mg^{2+}(g) + 2Cl^-(g) \rightarrow MgCl_2(s)} ]
This final step is the lattice-formation step. Its value is the unknown in the cycle. Forming the solid from separated gaseous ions releases a large amount of energy, so the result is negative when lattice enthalpy means crystal formation.
Sign Conventions, Calculation, and Common Errors
The lattice enthalpy sign depends on the reaction direction named by the term. For the calculation here, lattice enthalpy means the enthalpy change when gaseous ions form the solid. Some courses instead define lattice energy as the energy needed to split the solid into gaseous ions, which has the opposite sign.
The complete Hess’s law equation is:
[ \Delta H_f^\circ = \Delta H_{\mathrm{sub}} + IE_1 + IE_2 + D(\mathrm{Cl_2}) + 2EA(\mathrm{Cl}) + \Delta H_{\mathrm{latt}} ]
Substitute the values:
[ -642 = 148 + 738 + 1451 + 243 – 698 + \Delta H_{\mathrm{latt}} ]
First add the steps before lattice formation:
[ 148 + 738 + 1451 + 243 – 698 = 1882 ]
Then rearrange to isolate the lattice enthalpy:
[ \Delta H_{\mathrm{latt}} = -642 – 1882 = \boxed{-2524\ \mathrm{kJ\,mol^{-1}}} ]
The reverse reaction is crystal dissociation:
[ \mathrm{MgCl_2(s) \rightarrow Mg^{2+}(g) + 2Cl^-(g)} ]
Reversing a reaction reverses the sign, so its enthalpy change is (+2524\ \mathrm{kJ\,mol^{-1}}). Before submitting an answer, state which definition you use.
Quick checks for the working
| Check | Correct result | What it catches |
|---|---|---|
| Formula unit | 1 Mg and 2 Cl | Missing a chloride ion |
| Magnesium charge | Mg²⁺ | Omitting the second ionization energy |
| Chlorine quantity | 1 Cl₂ → 2 Cl | Doubling the 243 kJ value |
| Electron affinity | 2 × (−349) = −698 | Missing the coefficient of two |
| Lattice direction | Formation: negative; separation: positive | Reporting the wrong sign |
| Units | kJ mol⁻¹ of MgCl₂ | Mixing per-mole quantities |
A useful sign check is to read every equation as a physical change. Atomization, bond breaking, and electron removal require energy in this data set, so they are positive. Chlorine gaining electrons releases energy, while gaseous ions joining the solid releases much more; both are negative.
The net equation also provides a balance check. When the five preparatory steps and crystal formation are added, intermediate gaseous species and electrons cancel. What remains is (\mathrm{Mg(s) + Cl_2(g) \rightarrow MgCl_2(s)}), the original formation reaction. If the final equation does not match, review the steps and coefficients.
Interpretation: Lattice Enthalpy and Ionic-Bonding Implications
A lattice enthalpy near (-2524\ \mathrm{kJ\,mol^{-1}}) means that crystal formation from separated gaseous Mg²⁺ and Cl⁻ ions releases a large amount of energy. It is not the enthalpy of formation of MgCl₂ from its elements. The two quantities describe different routes and should not be used as if they were interchangeable.
The magnitude reflects the strong electrostatic attraction between oppositely charged ions in the crystal. The Born–Haber calculation does not claim that the solid consists of isolated pairs of ions; it calculates the energy change for forming the ionic solid from gaseous ions. This distinction matters when interpreting what the number represents.
For a course or exam, follow the stated lattice-energy convention. If none is stated, give the reaction direction alongside the number. For example: “Lattice enthalpy for (\mathrm{Mg^{2+}(g) + 2Cl^-(g) \rightarrow MgCl_2(s)}) is approximately (-2524\ \mathrm{kJ\,mol^{-1}}).” That wording makes the sign unambiguous.
The result depends on the full set of thermochemical values, so a different reliable data table may give a slightly different answer. The method stays the same: balance the overall formation reaction, include every gaseous-ion step, use Hess’s law, and label the lattice direction. Those checks make the calculation easier to verify.
Frequently asked questions
What is the lattice enthalpy of MgCl₂?
For formation from gaseous ions, it is approximately (-2524\ \mathrm{kJ\,mol^{-1}}) using the data in this guide. The reverse dissociation value is (+2524\ \mathrm{kJ\,mol^{-1}}).
Why does the calculation include two ionization energies?
Magnesium becomes Mg²⁺ by losing two electrons. It must undergo both the first and second ionization steps before the cycle reaches the required gaseous ion.
Why is chlorine’s bond-dissociation enthalpy counted once?
One Cl₂ molecule contains the two chlorine atoms needed for one MgCl₂ formula unit. The listed (+243\ \mathrm{kJ\,mol^{-1}}) already covers breaking that molecule into both atoms.
Why is chlorine’s electron affinity multiplied by two?
Two chlorine atoms each gain one electron to form two Cl⁻ ions. Thus, the total is (2 \times (-349) = -698\ \mathrm{kJ\,mol^{-1}}).
Why is lattice formation enthalpy negative?
The reaction joins oppositely charged gaseous ions to form a solid and releases energy. Energy released by the reaction appears as a negative enthalpy change.
Why might a textbook show a positive lattice energy?
It may define lattice energy as the energy needed to separate the solid into gaseous ions. That is the reverse reaction, so its value is positive.
What does (\Delta H_f^\circ) represent here?
It is the enthalpy change for forming one mole of MgCl₂(s) from Mg(s) and Cl₂(g) in their standard states. The approximate value used here is (-642\ \mathrm{kJ\,mol^{-1}}).
What should I do if my answer differs slightly?
Check the data values, coefficients, and signs first. If those are correct, a small difference may come from rounding or a different thermochemical data source.
How can I check that the cycle is balanced?
Add the step equations and cancel intermediate species, such as gaseous atoms, ions, and electrons. The remaining equation should be (\mathrm{Mg(s) + Cl_2(g) \rightarrow MgCl_2(s)}).
What units should I report?
Use (\mathrm{kJ\,mol^{-1}}), and make clear that the mole refers to MgCl₂. State whether the value describes crystal formation or crystal dissociation.
(This article was written by one of our staff writers, Alan Westbrook. Visit our Meet the Team page.)
