Question

Line corresponding to lyman series are $\text{L}_1, \text{L}_2, \text{L}_3, \text{L}_4, \ldots$, among these $\text{L}_1$ line corresponds to lowest energy. Similarly lines corresponding to balmer series are $\text{B}_1, \text{B}_2, \text{B}_3, \text{B}_4, \ldots$, among these $\text{B}_1$ line corresponds to lowest energy $\Delta E_L = \text{Energy of } 1^{\text{st}} \text{ line of lyman series}$, $\Delta E_B = \text{Energy of } 1^{\text{st}} \text{ line of balmer series}$. If $\Delta E_L = x \cdot \Delta E_B$. Calculate $(x \times 10^{-1})$

• A. 54
• B. 27
• C. 18
• D. 36

Hint:

The energy of the first line (lowest energy) in the Lyman series ($n=2 \rightarrow 1$) is $3/4$ Rydberg units. The first line in the Balmer series ($n=3 \rightarrow 2$) is $5/36$ Rydberg units.

Answer 1

The Correct Option is A

To find \( x \) in the given relation \(\Delta E_L = x \cdot \Delta E_B\), we need to consider the Lyman and Balmer series in hydrogen atoms.

The Lyman series involves electronic transitions from higher energy levels to the first energy level (\( n_1 = 1 \)). The formula for energy in the Lyman series is given by:

\(\Delta E_L = 13.6 \left(\dfrac{1}{1^2} - \dfrac{1}{n^2}\right)\), where \( n = 2, 3, 4, \ldots \).

The first line (L₁) corresponds to \( n = 2 \), so:

\(\Delta E_L = 13.6 \left(\dfrac{1}{1^2} - \dfrac{1}{2^2}\right) = 13.6 \left(\dfrac{3}{4}\right) = 10.2 \, \text{eV}\)

The Balmer series involves transitions from higher energy levels to the second energy level (\( n_1 = 2 \)). The formula for energy in the Balmer series is:

\(\Delta E_B = 13.6 \left(\dfrac{1}{2^2} - \dfrac{1}{n^2}\right)\), where \( n = 3, 4, 5, \ldots \).

The first line (B₁) corresponds to \( n = 3 \), so:

\(\Delta E_B = 13.6 \left(\dfrac{1}{4} - \dfrac{1}{9}\right) = 13.6 \left(\dfrac{5}{36}\right) \approx 1.89 \, \text{eV}\)

Now, we calculate \( x \) using the relation \(\Delta E_L = x \cdot \Delta E_B\):

\(10.2 = x \cdot 1.89\)

\(x = \dfrac{10.2}{1.89} \approx 5.4\)

Finally, we need to calculate \( x \times 10^{-1} \):

\(x \times 10^{-1} = 5.4 \times 10 = 54\)

Thus, the correct answer is 54.