The energy of second bohr orbit of the hydrogen atom is `-328 kJ mol^(-1)` hence the energy of third bohr orbit would be

Options

A. `-41 kJ mol^(-1)`
B. `-1312 Kj mol^(-1)`
C. `-164 kJ mol^(-1)`
D. `-145 kj mol^(-1)`

Answer

To find the energy of the third Bohr orbit of the hydrogen atom, we can use the relationship between the energies of different orbits. The energy of the nth orbit in a hydrogen atom is given by the formula:

\[ E_n = -\frac{2.18 \times 10^{-21}}{n^2} \text{ kJ/mol} \]

Given that the energy of the second Bohr orbit (n=2) is \(-328 \text{ kJ/mol}\), we can set up the following relationship:

### Step 1: Write the formula for the energy of the nth orbit
The energy of the nth orbit is given by:
\[ E_n = -\frac{2.18 \times 10^{-21}}{n^2} \text{ kJ/mol} \]

### Step 2: Calculate the energy of the second orbit (n=2)
Using the formula:
\[ E_2 = -\frac{2.18 \times 10^{-21}}{2^2} = -\frac{2.18 \times 10^{-21}}{4} = -0.545 \times 10^{-21} \text{ kJ/mol} \]
However, we know from the problem statement that \(E_2 = -328 \text{ kJ/mol}\).

### Step 3: Set up the ratio of energies for orbits n=2 and n=3
The energies of the orbits are inversely proportional to the square of the principal quantum number (n):
\[ \frac{E_2}{E_3} = \frac{n_3^2}{n_2^2} \]
Where \(n_2 = 2\) and \(n_3 = 3\):
\[ \frac{-328}{E_3} = \frac{3^2}{2^2} = \frac{9}{4} \]

### Step 4: Rearrange to find \(E_3\)
Rearranging the equation gives:
\[ E_3 = -328 \times \frac{4}{9} \]

### Step 5: Calculate \(E_3\)
Now, calculate \(E_3\):
\[ E_3 = -328 \times \frac{4}{9} = -145.777 \text{ kJ/mol} \]

### Step 6: Round the answer
Rounding the answer gives:
\[ E_3 \approx -145 \text{ kJ/mol} \]

### Final Answer
Thus, the energy of the third Bohr orbit is approximately:
\[ E_3 \approx -145 \text{ kJ/mol} \]