Kinetic energy of electron in a mono electronic species is +1313 kJ/mole. Then which of the following statements are correct
(I) The electron is present in the `2^(nd)` orbit of `He^(+)` ions
(II) The electron is present in the `2^(nd)` orbit of H atom
(III) The electron is present in the `3^(rd)` orbit of `Li^(+2)` ion
(IV) The electron is present in the `4^(th)` orbit of `He^(+)` ion

Options

A. I and II
B. II and III
C. I and III
D. III and IV

Answer

To determine which statements are correct regarding the kinetic energy of an electron in a mono-electronic species, we will analyze each statement based on the given kinetic energy of +1313 kJ/mole.

### Step-by-Step Solution:

1. **Understanding the Total Energy**:
- The total energy (E) of an electron in a mono-electronic species is related to the kinetic energy (K.E) by the formula:
\[ E = -K.E \]
- Given K.E = +1313 kJ/mole, we find:
\[ E = -1313 \text{ kJ/mole} \]

2. **Energy Formula for Mono-electronic Species**:
- The energy of an electron in the nth orbit of a mono-electronic species is given by:
\[ E_n = -\frac{1313 \cdot Z^2}{n^2} \text{ kJ/mole} \]
- Here, Z is the atomic number and n is the principal quantum number (orbit number).

3. **Analyzing Each Statement**:

- **Statement I**: The electron is present in the 2nd orbit of He\(^+\) ions.
- For He\(^+\), \(Z = 2\) and \(n = 2\):
\[ E_2 = -\frac{1313 \cdot 2^2}{2^2} = -1313 \text{ kJ/mole} \]
- This matches the total energy calculated, hence **Statement I is correct**.

- **Statement II**: The electron is present in the 2nd orbit of H atom.
- For H, \(Z = 1\) and \(n = 2\):
\[ E_2 = -\frac{1313 \cdot 1^2}{2^2} = -\frac{1313}{4} = -328.25 \text{ kJ/mole} \]
- This does not match -1313 kJ/mole, hence **Statement II is incorrect**.

- **Statement III**: The electron is present in the 3rd orbit of Li\(^{2+}\) ion.
- For Li\(^{2+}\), \(Z = 3\) and \(n = 3\):
\[ E_3 = -\frac{1313 \cdot 3^2}{3^2} = -1313 \text{ kJ/mole} \]
- This matches the total energy calculated, hence **Statement III is correct**.

- **Statement IV**: The electron is present in the 4th orbit of He\(^+\) ions.
- For He\(^+\), \(Z = 2\) and \(n = 4\):
\[ E_4 = -\frac{1313 \cdot 2^2}{4^2} = -\frac{1313 \cdot 4}{16} = -328.25 \text{ kJ/mole} \]
- This does not match -1313 kJ/mole, hence **Statement IV is incorrect**.

4. **Conclusion**:
- The correct statements are **I and III**.

### Final Answer:
The correct statements are:
- I: The electron is present in the 2nd orbit of He\(^+\) ions.
- III: The electron is present in the 3rd orbit of Li\(^{2+}\) ion.