The energy of second Bohr orbit of the hydrogen atom is `- 328 k J mol^-1`, hence the energy of fourth Bohr orbit would be.

Options

A. (a) `-41 kJ mol^(-1)`
B. (b) `-1312 kJ mol^(-1)`
C. (c) `-164 kJ mol^(-1)`
D. (d) `-82 kJ mol^(-1)`

Explanation

To find the energy of the fourth Bohr orbit of the hydrogen atom, we can follow these steps:

### Step 1: Understand the Energy Formula
According to Bohr's model, the energy of the nth Bohr orbit of a hydrogen atom is given by the formula:
\[ E_n = \frac{E_1 \cdot Z}{n^2} \]
where:
- \( E_n \) is the energy of the nth orbit,
- \( E_1 \) is the energy of the first orbit,
- \( Z \) is the atomic number (for hydrogen, \( Z = 1 \)),
- \( n \) is the principal quantum number (orbit number).

### Step 2: Relate the Given Energy to the First Orbit
We know from the question that the energy of the second Bohr orbit \( E_2 \) is given as:
\[ E_2 = -328 \, \text{kJ/mol} \]

Using the formula for the second orbit (\( n = 2 \)):
\[ E_2 = \frac{E_1 \cdot Z}{2^2} = \frac{E_1 \cdot 1}{4} = \frac{E_1}{4} \]

### Step 3: Solve for \( E_1 \)
From the above equation, we can express \( E_1 \) in terms of \( E_2 \):
\[ E_2 = \frac{E_1}{4} \]
\[ -328 \, \text{kJ/mol} = \frac{E_1}{4} \]

To find \( E_1 \):
\[ E_1 = 4 \cdot (-328 \, \text{kJ/mol}) \]
\[ E_1 = -1312 \, \text{kJ/mol} \]

### Step 4: Calculate the Energy of the Fourth Orbit
Now, we can find the energy of the fourth Bohr orbit (\( E_4 \)) using the formula:
\[ E_4 = \frac{E_1 \cdot Z}{4^2} = \frac{E_1 \cdot 1}{16} = \frac{E_1}{16} \]

Substituting the value of \( E_1 \):
\[ E_4 = \frac{-1312 \, \text{kJ/mol}}{16} \]
\[ E_4 = -82 \, \text{kJ/mol} \]

### Conclusion
The energy of the fourth Bohr orbit of the hydrogen atom is:
\[ E_4 = -82 \, \text{kJ/mol} \]

### Final Answer
The correct option is:
**Option D: -82 kJ/mol**

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