The successive ionization energies (IE) for an element A are as follows  
`A overset(IE_1)(rarr)A^(+) overset(IE_2)(rarr)A^(2+) overset(IE_3)(rarr) A^(3+) rarr ….`
If the `IE_1 and IE_3` values are `27 kJ mol^(-1)` and `51 kJ mol^(-1)` respectively, then the value of `IE_2` is ______`kJ mol^(-1)`. 

Options

A. 21
B. 33
C. 59
D. 63

Answer

To find the second ionization energy (IE₂) of element A, we can analyze the relationship between the first ionization energy (IE₁), the second ionization energy (IE₂), and the third ionization energy (IE₃). The values given are:

- IE₁ = 27 kJ mol⁻¹
- IE₃ = 51 kJ mol⁻¹

### Step-by-Step Solution:

1. **Understanding Ionization Energies**:
- Ionization energy is the energy required to remove an electron from an atom or ion.
- The first ionization energy (IE₁) is the energy needed to remove the first electron from a neutral atom (A to A⁺).
- The second ionization energy (IE₂) is the energy needed to remove the second electron from the positively charged ion (A⁺ to A²⁺).
- The third ionization energy (IE₃) is the energy needed to remove the third electron from the doubly charged ion (A²⁺ to A³⁺).

2. **Analyzing the Trend**:
- Generally, the ionization energy increases with each successive removal of an electron because the remaining electrons experience a greater effective nuclear charge.
- This means that IE₂ should be greater than IE₁ and less than IE₃.

3. **Setting Up the Inequalities**:
- From the given values, we can establish the following:
- IE₁ < IE₂ < IE₃
- 27 kJ mol⁻¹ < IE₂ < 51 kJ mol⁻¹

4. **Estimating IE₂**:
- Since IE₂ must be between 27 kJ mol⁻¹ and 51 kJ mol⁻¹, we can make an educated guess.
- A reasonable value that fits this range could be around the midpoint or slightly skewed towards the lower end, considering the trend of increasing ionization energies.

5. **Choosing a Value**:
- A common approach is to select a value that is not too close to either extreme.
- A value of 33 kJ mol⁻¹ fits well within the range and respects the trend of increasing ionization energy.

6. **Final Answer**:
- Therefore, the value of IE₂ is **33 kJ mol⁻¹**.