Question

The hydrogen spectrum consists of several spectral lines in Lyman series (L$_1$, L$_2$, L$_3$ ...; L$_1$ has lowest energy among Lyman series). Similarly, it consists of several spectral lines in Balmer series (B$_1$, B$_2$, B$_3$ ...; B$_1$ has lowest energy among Balmer lines). The energy of L$_1$ is $x$ times the energy of B$_1$. The value of $x$ is ___ $\times 10^{-1}$. (Nearest integer)

Hint:

Always convert the final ratio into the exact form asked in the question before rounding.

Answer 1

Correct Answer: 54

Step 1: Identify the electronic transitions. 
L$_1$ corresponds to transition from $n = 2$ to $n = 1$ (Lyman series). 
B$_1$ corresponds to transition from $n = 3$ to $n = 2$ (Balmer series). 
Step 2: Write the energy expression for hydrogen atom. 
\[ E = 13.6 \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right) \text{ eV} \] 
Step 3: Calculate energy of L$_1$. 
\[ E_{L_1} = 13.6 \left( 1 - \frac{1}{4} \right) \] \[ E_{L_1} = 13.6 \times \frac{3}{4} = 10.2 \text{ eV} \] 
Step 4: Calculate energy of B$_1$. 
\[ E_{B_1} = 13.6 \left( \frac{1}{4} - \frac{1}{9} \right) \] \[ E_{B_1} = 13.6 \times \frac{5}{36} \approx 1.89 \text{ eV} \] 
Step 5: Calculate the ratio. 
\[ x = \frac{E_{L_1}}{E_{B_1}} = \frac{10.2}{1.89} \approx 5.4 \] \[ x = 54 \times 10^{-1} \] 
Step 6: Final conclusion. 
Nearest integer value of $x \times 10^{-1}$ is 54