Determine the separation of the first line of the Balmer series (in $\text{Å}$) in a spectrum of ordinary hydrogen and tritium (mass number 3). Take Rydberg's constants = 10967800 m^-1.

First line of Balmer series of H-atom is

$\frac{1}{{\lambda }_{\mathrm{H}}}=R\left(\frac{1}{4}-\frac{1}{9}\right)\times \frac{m_{\mathrm{H}}}{m_e+m_{\mathrm{H}}}$

First line of Balmer series for Z-atom is

$\frac{1}{{\lambda }_{\mathrm{T}}}=R\left(\frac{1}{4}-\frac{1}{9}\right)\times \frac{m_{\mathrm{T}}}{m_e+m_{\mathrm{H}}}$

$\Rightarrow {\lambda }_{\mathrm{H}}-{\lambda }_{\mathrm{T}}=\frac{36\left(m_e+m_{\mathrm{H}}\right)}{5R{m}_{\mathrm{H}}}-\frac{36\left(m_e+m_{\mathrm{T}}\right)}{5R{m}_{\mathrm{T}}}$

$\begin{array}{c}=\frac{36}{5R}\left[\frac{\left(m_e+m_{\mathrm{H}}\right)m_{\mathrm{T}}-\left(m_e+m_{\mathrm{T}}\right)m_{\mathrm{H}}}{m_{\mathrm{H}}{m}_{\mathrm{T}}}\right]\\ =\frac{36}{5R}\left[\frac{m_e\left(m_{\mathrm{T}}-m_{\mathrm{H}}\right)}{m_{\mathrm{H}}{m}_{\mathrm{T}}}\right]\\ =\frac{36}{5\times 10967800}\times \frac{9.109\times {10}^{-31}\times 2\times 1.67\times {10}^{-27}}{3\times \left(1.67\times {10}^{-27}\right)}=2.387\text{Å}\end{array}$