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(B) For a hydrogen-like ion,the energy of an orbit is given by $E_n = -13.6 \frac{Z^2}{n^2} \; eV$.For $Li^{++}$,the atomic number $Z = 3$.The energy

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$114$

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(B) For a hydrogen-like ion,the energy of an orbit is given by $E_n = -13.6 \frac{Z^2}{n^2} \; eV$.For $Li^{++}$,the atomic number $Z = 3$.The energy of the first orbit $(n_1 = 1)$ is $E_1 = -13.6 	imes \frac{3^2}{1^2} = -13.6 	imes 9 = -122.4 \; eV$.The energy of the third orbit $(n_2 = 3)$ is $E_3 = -13.6 	imes \frac{3^2}{3^2} = -13.6 \; eV$.The energy required to excite the electron is $\Delta E = E_3 - E_1 = -13.6 - (-122.4) = 108.8 \; eV$.The wavelength $\lambda$ of the photon is given by $\lambda = \frac{hc}{\Delta E}$.Given $hc = 1242 \; eV \cdot nm = 12420 \; eV \cdot \mathring{A}$.$\lambda = \frac{12420 \; eV \cdot \mathring{A}}{108.8 \; eV} \approx 114.15 \; \mathring{A}$.Since $1 \; \mathring{A} = 10^{-10} \; m$,we have $\lambda \approx 114 	imes 10^{-10} \; m$.Thus,the value of $x$ is $114$.

$A$ beam of monochromatic light is used to excite the electron in $Li^{++}$ from the first orbit to the third orbit. The wavelength of monochromatic light is found to be $x \times 10^{-10} \; m$. The value of $x$ is $\dots$. [Given $hc = 1242 \; eV \cdot nm$]

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