Construct a Born-Haber cycle for the formation of the hypothetical compound NaCl , where the sodium ion has a \(2+\) charge (the second ionization energy for sodium is given in Table 7.2 . (a) How large would the lattice energy need to be for the formation of \(\mathrm{NaCl}_{2}\) to be exothermic? (b) If we were to estimate the lattice energy of \(\mathrm{NaCl}_{2}\) to be roughly equal to that of \(\mathrm{MgCl}_{2}(2326 \mathrm{kJ} / \mathrm{mol}\) from Table 8.1\(),\) what value would you obtain for the standard enthalpy of formation, \(\Delta H_{f}^{\circ},\) of \(\mathrm{NaCl}_{2} ?\)

Short Answer

Expert verified
The hypothetical compound NaCl₂ requires a lattice energy less than 5709 kJ/mol to be exothermic in its formation. Using the lattice energy of MgCl₂ (-2326 kJ/mol) as an estimation, the standard enthalpy of formation, ∆Hf°, of NaCl₂ is approximately 3383 kJ/mol.

Step by step solution

01

Write down the Born-Haber cycle for NaCl₂ formation

The Born-Haber cycle for the formation of NaCl₂ is expressed as follows: 1. Sublimation energy of sodium: Na (s) → Na (g) 2. 1st ionization energy of sodium: Na (g) → Na⁺ (g) + e⁻ 3. 2nd ionization energy of sodium: Na⁺ (g) → Na²⁺ (g) + e⁻ 4. Bond dissociation energy of chlorine: Cl₂ (g) → 2 Cl (g) 5. Electron affinity of chlorine: Cl (g) + e⁻ → Cl⁻ (g) 6. Lattice energy: Na²⁺ (g) + 2 Cl⁻ (g) → NaCl₂ (s) Now, we have all elements of the cycle necessary to determine the lattice energy needed for exothermic formation and the standard enthalpy of formation of NaCl₂.
02

Calculate the lattice energy required for NaCl₂ formation to be exothermic

In order for the formation of NaCl₂ to be exothermic, the sum of all the energies in the cycle must be negative. To find out how large the lattice energy should be, we need to calculate the sum of steps 1 through 5. Sublimation energy of sodium: +108 kJ/mol 1st ionization energy of sodium: +496 kJ/mol 2nd ionization energy of sodium: +4,560 kJ/mol Bond dissociation energy of chlorine: +243 kJ/mol Electron affinity of chlorine (twice, since there are 2 chlorines): -2 * 349 = -698 kJ/mol Sum of the energies in steps 1 to 5: +108 + 496 + 4560 + 243 - 698 = 5709 kJ/mol Now, let's denote the unknown lattice energy as LE. We have: LE - 5709 kJ/mol < 0 LE < 5709 kJ/mol So, the lattice energy should be less than 5709 kJ/mol for the NaCl₂ formation to be exothermic.
03

Estimate the standard enthalpy of formation, ∆Hf°, of NaCl₂

We will use the given lattice energy of MgCl₂ as an estimation for NaCl₂: Lattice energy of MgCl₂: -2326 kJ/mol Now, let's calculate the standard enthalpy of formation, ∆Hf°, of NaCl₂ using the estimated lattice energy: ∆Hf° = Sum of the energies in steps 1 to 5 + lattice energy = 5709 kJ/mol - 2326 kJ/mol = 3383 kJ/mol Therefore, the standard enthalpy of formation, ∆Hf°, of NaCl₂ is approximately 3383 kJ/mol.

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