Supposing the energy of fourth shell for hydrogen atom is - 50 a.u. (arbitrary unit). What would be its ionization potential -

Options

A. 50
B. 800
C. `15.4`
D. `20.8`

Answer

To find the ionization potential of the hydrogen atom given that the energy of the fourth shell is -50 a.u., we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Energy Formula**:
The energy of an electron in the nth orbit of a hydrogen atom is given by the formula:
\[ E_n = -\frac{I \cdot Z^2}{n^2} \]
where \(E_n\) is the energy of the nth orbit, \(I\) is the ionization energy, \(Z\) is the atomic number (which is 1 for hydrogen), and \(n\) is the principal quantum number.

2. **Identify Given Values**:
- The energy of the fourth shell (\(E_4\)) is given as -50 a.u.
- For hydrogen, \(Z = 1\) and \(n = 4\).

3. **Substitute Values into the Formula**:
Substitute the known values into the energy formula:
\[ -50 = -\frac{I \cdot 1^2}{4^2} \]
This simplifies to:
\[ -50 = -\frac{I}{16} \]

4. **Eliminate the Negative Sign**:
Remove the negative sign from both sides:
\[ 50 = \frac{I}{16} \]

5. **Solve for Ionization Energy (I)**:
Multiply both sides by 16 to solve for \(I\):
\[ I = 50 \times 16 \]
\[ I = 800 \text{ a.u.} \]

6. **Conclusion**:
The ionization potential of the hydrogen atom is 800 a.u.

### Final Answer:
The ionization potential of the hydrogen atom is **800 a.u.**

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