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

Step 1: Using the Rydberg formula.
The wavelength of light absorbed corresponds to the transition in the hydrogen-like atom. The Rydberg formula for the wavelength is given by: \[ \frac{1}{\lambda} = R_{\infty} \left( Z^2 \left( \frac{1}{1^2} - \frac{1}{2^2} \right) \right) \]

Step 2: Plugging in the values.
Given \( \lambda = 2.48 \, \text{nm} \), substitute the values into the equation and solve for \( Z \). After solving, we get \( Z = 5 \).

Step 3: Conclusion.
The nuclear charge \( Z \) is \( \boxed{5} \).