Construct a Born–Haber cycle for the formation of the hypothetical compound NaCl₂, 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 NaCl₂ to be exothermic?

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Identify the steps involved in constructing a Born–Haber cycle for NaCl_2. This includes sublimation of Na, ionization of Na to Na^2+, dissociation of Cl_2, electron affinity of Cl, and lattice energy of NaCl_2.

Write the sublimation energy for Na: Na(s) → Na(g).

Write the first and second ionization energies for Na: Na(g) → Na^+(g) + e^- and Na^+(g) → Na^{2+}(g) + e^-.

Write the dissociation energy for Cl_2: \( \frac{1}{2} \text{Cl}_2(g) \rightarrow \text{Cl}(g) \).

Write the electron affinity for Cl: Cl(g) + e^- → Cl^-(g).

Write the lattice energy for NaCl_2: Na^{2+}(g) + 2Cl^-(g) → NaCl_2(s).

Apply Hess's Law to sum all the energies involved in the cycle to find the lattice energy. The formation of NaCl_2 will be exothermic if the sum of all energies is negative.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Born-Haber Cycle

The Born-Haber cycle is a thermodynamic cycle that relates the lattice energy of an ionic compound to its formation enthalpy. It involves several steps, including the sublimation of the solid, ionization of the metal, and the formation of the ionic lattice from gaseous ions. This cycle allows for the calculation of lattice energy by considering the energy changes associated with each step in the formation of the compound.

Lattice Energy

Lattice energy is the energy released when gaseous ions combine to form an ionic solid. It is a measure of the strength of the ionic bonds in the compound; higher lattice energy indicates stronger interactions between ions. For the formation of NaCl2 to be exothermic, the lattice energy must be sufficiently large to overcome the positive energy contributions from ionization and other processes in the Born-Haber cycle.

Ionization Energy

Ionization energy is the energy required to remove an electron from an atom or ion in the gas phase. For sodium, the first ionization energy is the energy needed to remove one electron, while the second ionization energy is the energy required to remove a second electron after the first has been removed. In the context of NaCl2, understanding the second ionization energy is crucial, as it directly impacts the energy balance in the Born-Haber cycle and the overall feasibility of forming the compound.