Question

The first four spectral lines in the Lyman series of a $H-atom$ are $\lambda =1218\stackrel{o}{A},974.3\stackrel{o}{A}$ and $951\stackrel{o}{A}$. If instead of Hydrogen, we consider Deuterium, calculate the shift in the wavelength of these lines.

Answer 1

$\text{Find the relation of wavelength with reduced mass}.$

Let reduced masses of hydrogen $={\mu }_H$

Reduced masses of deuterium $={\mu }_D$

As wave number is given by

$\frac{1}{\lambda }=R[\frac{1}{n_f^2}-\frac{1}{n_i^2}]$

Here $n_i$ and $n_f$ are fixed,

$\lambda \propto \frac{1}{R}$ or $\frac{{\lambda }_D}{{\lambda }_H}=\frac{R_H}{R_D}$ …$\left(i\right)$

$R_H=\frac{m_e{e}^4}{8{\epsilon }_0^{2}c{h}^3}=\frac{{\mu }_H{e}^4}{8{\epsilon }_0^{2}c{h}^3}$

$R_D=\frac{m_e{e}^4}{8{\epsilon }_0^{2}c{h}^3}=\frac{{\mu }_D{e}^4}{8{\epsilon }_0^{2}c{h}^3}$

$\therefore \frac{R_H}{R_D}=\frac{{\mu }_H}{{\mu }_D}$ …$\left(ii\right)$

From equation $\left(i\right)$ and $\left(ii\right)$:

$\frac{{\lambda }_D}{{\lambda }_H}=\frac{{\mu }_H}{{\mu }_D}$ …$\left(iii\right)$

$\text{Find the ratio of reduced mass of hydrogen and deuterium}$

Reduced mass for hydrogen,

${\mu }_H=\frac{m_e}{1+m_e/M}\simeq m_e(1-\frac{m_e}{M})$

Reduced mass for deuterium,

${\mu }_D=\frac{2M\cdot m_e}{2M(1+\frac{m_e}{2M})}\simeq m_e(1-\frac{m_e}{2M})$

Where $M$ is mass of proton

$\frac{{\mu }_H}{{\mu }_D}=\frac{m_e(1-\frac{m_e}{M})}{m_e(1-\frac{m_e}{2M})}$

$=(1-\frac{m_e}{M}){(1-\frac{m_e}{2M})}^{-1}$

$\Rightarrow \frac{{\mu }_H}{{\mu }_D}=(1-\frac{m_e}{M})(1+\frac{m_e}{2M})$

or $\frac{{\mu }_H}{{\mu }_D}\simeq (1-\frac{1}{2\times 1840})\simeq 0.99973$ …$\left(iv\right)$

$(\because M=1840m_e)$

$\text{Find the relation of wavelength of hydrogen and deuterium}.$

From $\left(iii\right)$ and $\left(iv\right)$

$\frac{{\lambda }_D}{{\lambda }_H}=0.99973,{\lambda }_D=0.99973{\lambda }_H$

Using ${\lambda }_H=1218\stackrel{o}{A},1028\stackrel{o}{A},974.3\stackrel{o}{A}$ and $951.4\stackrel{o}{A}$, we get

${\lambda }_D=1217.7\stackrel{o}{A},1027.7\stackrel{o}{A},974.04\stackrel{o}{A},951.1\stackrel{o}{A}$

$\text{Find the shift in wavelength}.$

Shift in wavelength $({\lambda }_H-{\lambda }_D)\approx 0.3\stackrel{o}{A}.$

$\text{Final Answer}:0.3\stackrel{o}{A}.$