Question

If Rydberg’s constant is \( R \), the longest wavelength of radiation in Paschen series will be \( \frac{\alpha}{7R} \), where \( \alpha = \_\_\_\_\_\_\_ \).

Answer 1

Correct Answer: 144

The Paschen series corresponds to transitions to \(n = 3\). The longest wavelength corresponds to the transition between \(n = 4\) and \(n = 3\). The inverse wavelength is given by:

\(\frac{1}{\lambda} = R Z^2 \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right)\)

For \(n_1 = 3\) and \(n_2 = 4\), and taking \(Z = 1\):

\(\frac{1}{\lambda} = R \left( \frac{1}{3^2} - \frac{1}{4^2} \right) = R \left( \frac{1}{9} - \frac{1}{16} \right)\)

\(\frac{1}{\lambda} = R \left( \frac{16 - 9}{144} \right) = \frac{7R}{144}\)

Thus:

\(\alpha = 144\)
The Correct answer is: 144