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. `-41kJ" "mol^(-1)`
B. `-1312" kJ "mol^(-1)`
C. `-164" kJ "mol^(-1)`
D. `-82" kJ "mol^(-1)`

Answer

To find the energy of the fourth Bohr orbit of the hydrogen atom given that the energy of the second Bohr orbit is -328 kJ/mol, we can follow these steps:

### Step 1: Understand the formula for the energy of the nth Bohr orbit
The energy of the nth Bohr orbit (En) is given by the formula:
\[ E_n = -\frac{R_H}{n^2} \]
where \( R_H \) is the Rydberg constant for hydrogen.

### Step 2: Use the given energy of the second orbit
We know that for the second orbit (n=2):
\[ E_2 = -\frac{R_H}{2^2} = -\frac{R_H}{4} \]
We are given that:
\[ E_2 = -328 \text{ kJ/mol} \]
This implies:
\[ -\frac{R_H}{4} = -328 \text{ kJ/mol} \]

### Step 3: Solve for R_H
From the equation above, we can find \( R_H \):
\[ \frac{R_H}{4} = 328 \text{ kJ/mol} \]
Multiplying both sides by 4 gives:
\[ R_H = 328 \times 4 = 1312 \text{ kJ/mol} \]

### Step 4: Calculate the energy of the fourth orbit
Now we can find the energy of the fourth orbit (n=4):
\[ E_4 = -\frac{R_H}{4^2} = -\frac{R_H}{16} \]
Substituting the value of \( R_H \):
\[ E_4 = -\frac{1312}{16} \text{ kJ/mol} \]

### Step 5: Perform the calculation
Calculating \( E_4 \):
\[ E_4 = -82 \text{ kJ/mol} \]

### Final Answer
Thus, the energy of the fourth Bohr orbit is:
\[ E_4 = -82 \text{ kJ/mol} \]

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