Most Probable Second Ionization Enthalpy of Magnesium

To determine the most probable estimated value for the $2^{\text{nd}}$ ionization enthalpy ($\text{IE}_2$) of Magnesium ($\text{Mg}$), we analyze the following concepts:

1. Sign of Ionization Enthalpy: Ionization enthalpy is defined as the amount of energy required to remove an electron from a neutral gaseous atom or a positively charged gaseous ion. Since energy must always be absorbed to overcome the electrostatic attraction between the nucleus and the electron, ionization processes are endothermic. Hence, all successive ionization enthalpies must be positive: $\text{IE} > 0$ This eliminates options (A) $-906\text{ kJ/mol}$ and (B) $-856\text{ kJ/mol}$.

2. Comparison of $\text{IE}_1$ and $\text{IE}_2$: The removal of the first valence electron from a neutral $\text{Mg}$ atom gives a monopositive cation ($\text{Mg}^+$): $\text{Mg}(g) \rightarrow \text{Mg}^+(g) + e^-, \quad \Delta H_1 = \text{IE}_1 = +737\text{ kJ/mol}$ In $\text{Mg}^+$, the ratio of nuclear charge to the number of electrons ($Z/e$) increases, resulting in a higher effective nuclear charge ($Z_{\text{eff}}$) acting on the remaining electrons. As a result, the nucleus attracts the remaining valence electrons more strongly, making it harder to remove the second electron: $\text{Mg}^+(g) \rightarrow \text{Mg}^{2+}(g) + e^-, \quad \Delta H_2 = \text{IE}_2$ Therefore, the $2^{\text{nd}}$ ionization enthalpy must be strictly greater than the $1^{\text{st}}$ ionization enthalpy: $\text{IE}_2 > \text{IE}_1$

3. Evaluation of Options:

   • Option (D) $+590\text{ kJ/mol} < +737\text{ kJ/mol}$, which violates $\text{IE}_2 > \text{IE}_1$.
   • Option (C) $+1450\text{ kJ/mol} > +737\text{ kJ/mol}$, which is consistent with theoretical expectations (the standard experimental value of $\text{IE}_2$ for $\text{Mg}$ is approximately $1451\text{ kJ/mol}$).

Thus, the most probable value for the $2^{\text{nd}}$ ionization enthalpy of $\text{Mg}$ is $+1450\text{ kJ/mol}$.

Correct Option: C