Answer : The value of second ionization energy of Ca is 1010 kJ.
Explanation :
The formation of calcium oxide is,
![Ca(s)+\frac{1}{2}O_2(g)\overset{\Delta H_f}\rightarrow CaO(s)](https://tex.z-dn.net/?f=Ca%28s%29%2B%5Cfrac%7B1%7D%7B2%7DO_2%28g%29%5Coverset%7B%5CDelta%20H_f%7D%5Crightarrow%20CaO%28s%29)
= enthalpy of formation of calcium oxide = -635 kJ
The steps involved in the born-Haber cycle for the formation of
:
(1) Conversion of solid calcium into gaseous calcium atoms.
![Ca(s)\overset{\Delta H_s}\rightarrow Ca(g)](https://tex.z-dn.net/?f=Ca%28s%29%5Coverset%7B%5CDelta%20H_s%7D%5Crightarrow%20Ca%28g%29)
= sublimation energy of calcium = 193 kJ
(2) Conversion of gaseous calcium atoms into gaseous calcium ions.
![Ca(g)\overset{\Delta H_I_1}\rightarrow Ca^{+1}(g)](https://tex.z-dn.net/?f=Ca%28g%29%5Coverset%7B%5CDelta%20H_I_1%7D%5Crightarrow%20Ca%5E%7B%2B1%7D%28g%29)
= first ionization energy of calcium = 590 kJ
(3) Conversion of gaseous calcium ion into gaseous calcium ions.
![Ca^{+1}(g)\overset{\Delta H_I_2}\rightarrow Ca^{+2}(g)](https://tex.z-dn.net/?f=Ca%5E%7B%2B1%7D%28g%29%5Coverset%7B%5CDelta%20H_I_2%7D%5Crightarrow%20Ca%5E%7B%2B2%7D%28g%29)
= second ionization energy of calcium = ?
(4) Conversion of molecular gaseous oxygen into gaseous oxygen atoms.
![O_2(g)\overset{\Delta H_D}\rightarrow OI(g)](https://tex.z-dn.net/?f=O_2%28g%29%5Coverset%7B%5CDelta%20H_D%7D%5Crightarrow%20OI%28g%29)
![\frac{1}{2}O_2(g)\overset{\Delta H_D}\rightarrow O(g)](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B2%7DO_2%28g%29%5Coverset%7B%5CDelta%20H_D%7D%5Crightarrow%20O%28g%29)
= dissociation energy of oxygen = ![\frac{498}{2}=249kJ](https://tex.z-dn.net/?f=%5Cfrac%7B498%7D%7B2%7D%3D249kJ)
(5) Conversion of gaseous oxygen atoms into gaseous oxygen ions.
![O(g)\overset{\Delta H_E_1}\rightarrow O^-(g)](https://tex.z-dn.net/?f=O%28g%29%5Coverset%7B%5CDelta%20H_E_1%7D%5Crightarrow%20O%5E-%28g%29)
= first electron affinity energy of oxygen = -141 kJ
(6) Conversion of gaseous oxygen ion into gaseous oxygen ions.
![O^-(g)\overset{\Delta H_E_2}\rightarrow O^{2-}(g)](https://tex.z-dn.net/?f=O%5E-%28g%29%5Coverset%7B%5CDelta%20H_E_2%7D%5Crightarrow%20O%5E%7B2-%7D%28g%29)
= second electron affinity energy of oxygen = 878 kJ
(7) Conversion of gaseous cations and gaseous anion into solid calcium oxide.
![Ca^{2+}(g)+O^{2-}(g)\overset{\Delta H_L}\rightarrow CaO(s)](https://tex.z-dn.net/?f=Ca%5E%7B2%2B%7D%28g%29%2BO%5E%7B2-%7D%28g%29%5Coverset%7B%5CDelta%20H_L%7D%5Crightarrow%20CaO%28s%29)
= lattice energy of calcium oxide = -3414 kJ
To calculate the overall energy from the born-Haber cycle, the equation used will be:
![\Delta H_f^o=\Delta H_s+\Delta H_I_1+\Delta H_I_2+\Delta H_D+\Delta H_E_1+\Delta H_E_2+\Delta H_L](https://tex.z-dn.net/?f=%5CDelta%20H_f%5Eo%3D%5CDelta%20H_s%2B%5CDelta%20H_I_1%2B%5CDelta%20H_I_2%2B%5CDelta%20H_D%2B%5CDelta%20H_E_1%2B%5CDelta%20H_E_2%2B%5CDelta%20H_L)
Now put all the given values in this equation, we get:
![-635kJ=193kJ+590kJ+\Delta H_I_2+249kJ+(-141kJ)+878kJ+(-3414kJ)](https://tex.z-dn.net/?f=-635kJ%3D193kJ%2B590kJ%2B%5CDelta%20H_I_2%2B249kJ%2B%28-141kJ%29%2B878kJ%2B%28-3414kJ%29)
![\Delta H_I_2=1010kJ](https://tex.z-dn.net/?f=%5CDelta%20H_I_2%3D1010kJ)
Therefore, the value of second ionization energy of Ca is 1010 kJ.