The energy of the second Bohr orbit of the hydrogen atom is -3.41 eV. The energy of the third Bohr orbit of the He ion will be

Options

A. `-30.69` eV
B. `-13.64` eV
C. `-7.67` eV
D. `-6.06` eV

Answer

To find the energy of the third Bohr orbit of the He ion (He+), we can use the formula for the energy levels of a hydrogen-like atom:

\[ E_n = -\frac{Z^2 \cdot 13.6 \, \text{eV}}{n^2} \]

where:
- \( E_n \) is the energy of the nth orbit,
- \( Z \) is the atomic number,
- \( n \) is the principal quantum number.

### Step 1: Identify the values
For the He ion (He+):
- The atomic number \( Z = 2 \) (since helium has 2 protons).
- We want to find the energy for the third orbit, so \( n = 3 \).

### Step 2: Substitute the values into the formula
Now we can substitute these values into the energy formula:

\[ E_3 = -\frac{2^2 \cdot 13.6 \, \text{eV}}{3^2} \]

### Step 3: Calculate \( Z^2 \)
Calculate \( Z^2 \):
\[ Z^2 = 2^2 = 4 \]

### Step 4: Calculate \( n^2 \)
Calculate \( n^2 \):
\[ n^2 = 3^2 = 9 \]

### Step 5: Substitute and simplify
Now substitute \( Z^2 \) and \( n^2 \) into the equation:

\[ E_3 = -\frac{4 \cdot 13.6 \, \text{eV}}{9} \]

### Step 6: Calculate the energy
Now calculate the energy:

\[ E_3 = -\frac{54.4 \, \text{eV}}{9} \]
\[ E_3 = -6.04 \, \text{eV} \]

### Final Answer
The energy of the third Bohr orbit of the He ion is approximately **-6.04 eV**.

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