The potential energy of an electron in the second Bohr's orbit of the` He^(+)` ion having total energy -3.4 eV is:-

Options

A. `-54.4 eV`
B. `-27.2 eV `
C. `-108.8 e V`
D. `-13.6 e V`

Explanation

To find the potential energy of an electron in the second Bohr's orbit of the He\(^+\) ion, given that the total energy is -3.4 eV, we can follow these steps:

### Step 1: Understand the relationship between potential energy and total energy
In the Bohr model, the potential energy (PE) of an electron is related to its total energy (TE) by the formula:
\[ PE = 2 \times TE \times Z^2 \]
where \(Z\) is the atomic number of the ion.

### Step 2: Identify the atomic number of He\(^+\)
For the helium ion (He\(^+\)), the atomic number \(Z\) is 2.

### Step 3: Substitute the values into the formula
Given that the total energy \(TE\) is -3.4 eV, we substitute \(TE\) and \(Z\) into the formula:
\[ PE = 2 \times (-3.4 \, \text{eV}) \times (2^2) \]

### Step 4: Calculate \(Z^2\)
Calculate \(Z^2\):
\[ Z^2 = 2^2 = 4 \]

### Step 5: Substitute \(Z^2\) back into the equation
Now substitute \(Z^2\) back into the potential energy equation:
\[ PE = 2 \times (-3.4 \, \text{eV}) \times 4 \]

### Step 6: Perform the multiplication
Calculate the multiplication:
\[ PE = 2 \times (-3.4) \times 4 = -27.2 \, \text{eV} \]

### Step 7: Final result
Thus, the potential energy of the electron in the second Bohr's orbit of the He\(^+\) ion is:
\[ PE = -27.2 \, \text{eV} \]

### Conclusion
The potential energy of the electron in the second Bohr's orbit of the He\(^+\) ion is -27.2 eV.

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