Question

The longest wavelength of light absorbed by a hydrogen-like atom is 2.48 nm. The nuclear charge (Z) of the atom is ........
(Round off to nearest integer) (Rydberg constant \( R_{\infty} = 109700 \, \text{cm}^{-1} \))

Hint:

To calculate the nuclear charge, use the Rydberg formula and solve for \( Z \). The longest wavelength corresponds to the transition between the n=1 and n=2 states.

Answer 1

Correct Answer: 7

The wavelength \( \lambda = 2.48 \, \text{nm} = 2.48 \times 10^{-7} \, \text{cm} \). The Rydberg formula for the wavelength of light absorbed/emitted by a hydrogen-like atom is given by:

\[ \dfrac{1}{\lambda} = R_{\infty}Z^2\left(\dfrac{1}{n_1^2}-\dfrac{1}{n_2^2}\right) \]

 

For the longest wavelength, the transition is between consecutive levels \( n_1 = 1 \) and \( n_2 = 2 \):

\[ \dfrac{1}{\lambda} = R_{\infty}Z^2\left(1-\dfrac{1}{4}\right) = \dfrac{3R_{\infty}Z^2}{4} \]

 

Substituting the known values:

\[ \dfrac{1}{2.48 \times 10^{-7}} = \dfrac{3 \times 109700 \times Z^2}{4} \]

 

Simplifying gives:

\[ Z^2 = \dfrac{4}{3 \times 109700 \times 2.48 \times 10^{-7}} \]

 

Calculate \( Z^2 \):

\[ Z^2 = \dfrac{4}{3 \times 109700 \times 2.48 \times 10^{-7}} \approx 52.34 \]

 

Computing \( Z \):

\[ Z = \sqrt{52.34} \approx 7.23 \]

 

Rounding to the nearest integer: \( Z = 7 \)