Question

Hydrogen atoms are excited on n=4 state. In the spectrum of emitted radiation, number of lines in the unltraviolet and visible regions are respectively-

• A. 2 and 3
• B. 3 and 2
• C. 1 and 3
• D. 3 and 1

Answer 1

Answer: (B)

3 and 2

Answer 2

When hydrogen atoms are excited to the $n=4$ state, electrons can de-excite to any lower energy level ($n=3, n=2, n=1$). The emitted radiation corresponds to these transitions. The total number of possible spectral lines is given by $\frac{n(n-1)}{2}$. For $n=4$, this is $\frac{4(3)}{2} = 6$ lines. These lines correspond to all possible transitions $n_i \to n_f$ where $4 \ge n_i > n_f \ge 1$.

The spectral lines are classified into series based on the final energy level ($n_f$):

• Lyman Series ($n_f=1$): Transitions to the ground state. These lines fall in the Ultraviolet (UV) region.
• Balmer Series ($n_f=2$): Transitions to the first excited state. These lines fall in the Visible region.
• Paschen Series ($n_f=3$): Transitions to the second excited state. These lines fall in the Infrared (IR) region.

Given that the initial excitation is to $n=4$, the possible transitions are:

1. $4 \to 3$: Belongs to Paschen series ($n_f=3$), which is in the IR region.
2. $4 \to 2$: Belongs to Balmer series ($n_f=2$), which is in the Visible region.
3. $4 \to 1$: Belongs to Lyman series ($n_f=1$), which is in the UV region.
4. $3 \to 2$: Belongs to Balmer series ($n_f=2$), which is in the Visible region.
5. $3 \to 1$: Belongs to Lyman series ($n_f=1$), which is in the UV region.
6. $2 \to 1$: Belongs to Lyman series ($n_f=1$), which is in the UV region.

Counting the lines in each region:

• Ultraviolet (UV) region: Lines ending at $n_f=1$ (Lyman series): $4 \to 1$, $3 \to 1$, $2 \to 1$. There are 3 lines in the UV region.
• Visible region: Lines ending at $n_f=2$ (Balmer series): $4 \to 2$, $3 \to 2$. There are 2 lines in the Visible region.
• Infrared (IR) region: Lines ending at $n_f=3$ (Paschen series): $4 \to 3$. There is 1 line in the IR region.

The question asks for the number of lines in the ultraviolet and visible regions, respectively. Therefore, the answer is 3 and 2.