Question

Find the minimum wavelength of Paschen series. Given \(R=1.1\times10^{7}\,\text{m}^{-1}\).

• A. \(656\ \text{nm}\)
• B. \(102.5\ \text{nm}\)
• C. \(818\ \text{nm}\)
• D. \(1220\ \text{nm}\)

Hint:

For the limiting wavelength of any hydrogen spectral series, put \(n_2=\infty\). This gives the shortest wavelength of that series.

Answer 1

The Correct Option is C

Concept: For hydrogen spectrum, \[ \frac{1}{\lambda} = R\left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right) \] For the Paschen series, \[ n_1=3 \] The minimum wavelength corresponds to the maximum energy transition, i.e., \[ n_2\rightarrow\infty \]

Step 1: Apply the formula. \[ \frac{1}{\lambda_{\min}} = R\left(\frac{1}{3^2}-0\right) \] \[ \frac{1}{\lambda_{\min}} = \frac{R}{9} \] \[ \lambda_{\min} = \frac{9}{R} \]

Step 2: Substitute \(R=1.1\times10^7\). \[ \lambda_{\min} = \frac{9}{1.1\times10^7} \] \[ \lambda_{\min} = 8.18\times10^{-7}\ \text{m} \] \[ \lambda_{\min} = 818\times10^{-9}\ \text{m} \] \[ \lambda_{\min} = 818\ \text{nm} \] Hence, \[ \boxed{\lambda_{\min}=818\ \text{nm}} \] Therefore, the correct answer is \[ \boxed{(C)} \]