Question

$n = 4$ $\_\_\_\_$

$n = 3$ $\_\_\_\_$ paschen

$n = 2$ $\_\_\_\_$ Balmer

$n = 1$ $\_\_\_\_$ Lyman

Smallest wavelength of lyman

$${\frac{1}{\lambda} = R\left( \frac{1}{12} - \frac{1}{\infty^{2}} \right) }{R = \frac{1}{\lambda} = \frac{1}{91}{\text{ }nm}^{- 1} }$$$\lambda_{\text{max~}}$ for balmer series

$${n_{1} = 2 \rightarrow n_{2} = 3 }{\frac{1}{\lambda_{B}} = R\left( \frac{1}{4} - \frac{1}{9} \right) }{\frac{1}{\lambda_{B}} = \frac{1}{91}\left( \frac{5}{36} \right) }{\lambda_{B} = \left( \frac{91 \times 36}{5} \right) = 655.2\text{ }nm }$$$\lambda_{\text{max~}}$ paschen

$${n_{1} = 3 \rightarrow n_{2} = 4 }{\frac{1}{\lambda_{p}} = \frac{1}{91}\left( \frac{1}{3^{2}} - \frac{1}{4^{2}} \right) = \frac{1}{91} \times \frac{7}{144} }{\lambda_{p} = \left( \frac{91 \times 144}{7} \right) = 1872\text{ }nm }{\Delta\lambda = \lambda_{P} - \lambda_{B} = 1872 - 655.2 }{\Delta\lambda = 1216.8 }{\Delta\lambda \approx 1217}$$