calculate the energy assoclated with the first orbit of `He^(+)` . What is the radius of this orbit?

Answer

To calculate the energy associated with the first orbit of \( \text{He}^+ \) and the radius of this orbit, we can follow these steps:

### Step 1: Calculate the Energy
The formula for the energy of an electron in the nth orbit of a hydrogen-like atom is given by:

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

Where:
- \( E \) is the energy in electron volts (eV),
- \( Z \) is the atomic number,
- \( n \) is the principal quantum number (orbit number).

For \( \text{He}^+ \):
- \( Z = 2 \) (since helium has an atomic number of 2),
- \( n = 1 \) (first orbit).

Substituting these values into the formula:

\[ E = -\frac{2^2 \cdot 13.6}{1^2} = -\frac{4 \cdot 13.6}{1} = -54.4 \, \text{eV} \]

To convert this energy into joules, we use the conversion factor \( 1 \, \text{eV} = 1.6 \times 10^{-19} \, \text{J} \):

\[ E = -54.4 \, \text{eV} \times 1.6 \times 10^{-19} \, \text{J/eV} = -8.704 \times 10^{-18} \, \text{J} \]

### Step 2: Calculate the Radius
The formula for the radius of the nth orbit in a hydrogen-like atom is given by:

\[ r_n = \frac{0.0529 \, \text{nm}}{Z} \cdot n^2 \]

Where:
- \( r_n \) is the radius in nanometers,
- \( Z \) is the atomic number,
- \( n \) is the principal quantum number.

For \( \text{He}^+ \):
- \( Z = 2 \),
- \( n = 1 \).

Substituting these values into the formula:

\[ r_1 = \frac{0.0529 \, \text{nm}}{2} \cdot 1^2 = \frac{0.0529}{2} = 0.02645 \, \text{nm} \]

### Final Answers
- The energy associated with the first orbit of \( \text{He}^+ \) is approximately \( -8.704 \times 10^{-18} \, \text{J} \).
- The radius of this orbit is \( 0.02645 \, \text{nm} \).