Question 73: Atoms & Nuclei

Question

In the hydrogen atom, the electron makes a transition from the higher orbit (i) to a lower orbit $(f)$. The ratio of the radius of the orbits in given by $r_i: r_f=16: 4$. The wavelength of photon emitted due to this transition is $\_\_\_\_$ nm.
(Given Rydberg constant $=1.0973 	imes 10^7 / \mathrm{m}$ )

Accepted Answer

Step 1: Find the quantum numbers of the orbits
In the Bohr model of the hydrogen atom, the radius of the $n^{	ext{th}}$ orbit is given by:
$$r_n \propto n^2$$
So, the ratio of radii is:
$$\frac{r_i}{r_f} = \frac{n_i^2}{n_f^2} = \frac{16}{4}$$
Taking the square root on both sides:
$$\frac{n_i}{n_f} = \frac{4}{2} = \frac{2}{1}$$
Therefore:
$n_i = 4$ (higher orbit)
$n_f = 2$ (lower orbit)
Step 2: Apply the Rydberg formula
The wavelength of the emitted photon is given by the Rydberg formula:
$$\frac{1}{\lambda} = R \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} ight)$$
Substituting the values:
$$\frac{1}{\lambda} = 1.0973 	imes 10^7 \left( \frac{1}{2^2} - \frac{1}{4^2} ight)$$
$$\frac{1}{\lambda} = 1.0973 	imes 10^7 \left( \frac{1}{4} - \frac{1}{16} ight)$$
$$\frac{1}{\lambda} = 1.0973 	imes 10^7 \left( \frac{4-1}{16} ight) = 1.0973 	imes 10^7 	imes \frac{3}{16}$$
Step 3: Calculate the wavelength
$$\frac{1}{\lambda} = \frac{3 	imes 1.0973 	imes 10^7}{16} = \frac{3.2919 	imes 10^7}{16}$$
$$\frac{1}{\lambda} = 2.0574 	imes 10^6 	ext{ m}^{-1}$$
Therefore:
$$\lambda = \frac{1}{2.0574 	imes 10^6} 	ext{ m} \approx 4.86 	imes 10^{-7} 	ext{ m}$$
Converting to nanometres:
$$\lambda \approx 486 	ext{ nm}$$
Answer
Option C: 486 nm
Note: This transition ($n=4 	o n=2$) belongs to the Balmer series and corresponds to the $H_\beta$ line of the hydrogen spectrum, which lies in the visible region.

Answer

121

Answer

242

Answer

486

Answer

974

Atoms & Nuclei

In the hydrogen atom, the electron makes a transition from the higher orbit (i) to a lower orbit (f). The ratio of the radius of the orbits in given by r_i: r_f=16: 4. The wavelength of photon emitted due to this transition is \_\_\_\_ nm. (Given Rydberg constant =1.0973 \times 10^7 / \mathrm{m} )

Choose the correct answer: