Question

The energy associated with first orbit is He⁺ is

• A. 0J
• B. -8.72×10^-18 J
• C. -4.58×10^-18 J
• D. -0.545×10^-18 J

Answer 1

The Correct Option is B

The energy associated with the first orbit of a hydrogen-like ion like \( \text{He}^+ \) can be calculated using the formula:

\[ E_n = - \dfrac{Z^2 \cdot R_H}{n^2} \]

Where:

• \( E_n \) is the energy of the electron in the \( n \)-th orbit.
• \( Z \) is the atomic number (for \( \text{He}^+ \), \( Z = 2 \)).
• \( R_H \) is the Rydberg constant in energy (\( R_H = 2.18 \times 10^{-18} \, \text{J} \)).
• \( n \) is the principal quantum number (for the first orbit, \( n = 1 \)).

Substituting values for \( \text{He}^+ \):

\[ E_1 = - \dfrac{2^2 \cdot 2.18 \times 10^{-18}}{1^2} \]

\[ E_1 = - 8.72 \times 10^{-18} \, \text{J} \]

Thus, the energy associated with the first orbit of \( \text{He}^+ \) is \( -8.72 \times 10^{-18} \, \text{J} \), so the correct answer is (B).

Answer 2

For hydrogen-like ions (He\(^+\) is similar to hydrogen, but with 2 protons), the energy associated with the first orbit is given by the formula: \[ E = - \frac{13.6 \times Z^2}{n^2} \, \text{eV} \] Where:
\( Z \) is the atomic number (for He\(^+\), \( Z = 2 \)),
\( n \) is the principal quantum number (for the first orbit, \( n = 1 \)).

Substituting the values: \[ E = - \frac{13.6 \times 2^2}{1^2} = - 54.4 \, \text{eV} = - 8.72 \times 10^{-18} \, \text{J} \]