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

To determine the relationship between the energy of Lyman series lines (L$_1$) and Balmer series lines (B$_1$) in the hydrogen spectrum, we utilize the formula for the energy of a photon emitted due to an electron transition: 
E = hν = 13.6 eV (1/n$_1^2$ - 1/n$_2^2$), where $n_1$ and $n_2$ are principal quantum numbers with $n_2 > n_1$.
The transition in Lyman series is from $n_2$ to $n_1 = 1$, and for L$_1$, $n_2 = 2$. For Balmer series, the transition is from $n_2$ to $n_1 = 2$, and for B$_1$, $n_2 = 3$.
Calculating the energy difference for L$_1$:
E$_{L_1}$ = 13.6 eV (1/1² - 1/2²) = 13.6 eV (1 - 1/4) = 13.6 eV × 3/4 = 10.2 eV.
Calculating the energy difference for B$_1$:
E$_{B_1}$ = 13.6 eV (1/2² - 1/3²) = 13.6 eV (1/4 - 1/9).
Simplifying:
E$_{B_1}$ = 13.6 eV × (5/36) = 1.89 eV.
Now, find $x$:
$x = \frac{E_{L_1}}{E_{B_1}} = \frac{10.2}{1.89} ≈ 5.40$.
The required answer is $x \space \times \space 10^{-1} = 54$ (nearest integer).