Name: Moseley’s Law
Uploaded: 2019-07-19
Duration: 7 min 40 s
Description: Moseley’s law is an empirical law concerning the characteristic x-rays that are emitted by atoms. Let's watch this video to learn about Moseley's law.

Question

In a hydrogen atom, an electron undergoes transition from the second excited state to the first excited state and then to the ground state. Find out the ratio of the wavelengths of the emitted radiations in the two cases.

Accepted Answer

According to the question, the electron is initially in the third orbit of quantum number $n = 3$ $(∵ it is in 2^{nd} excited state)$ . This electron now makes a transition to the second orbit of quantum number $n = 2$ $(∵ it is in 1^{s t} excited state)$ and then finally, it jumps to the ground state whose quantum number is $n = 1$ .

We know that the wavelength of the emitted radiation of light when an electron of an element of atomic number $Z$ makes a transition from one orbit $(n_{i})$ to another orbit $(n_{f})$ is give as:

$\frac{1}{λ} = R \times (\frac{1}{{(n_{f})}^{2}} - \frac{1}{{(n_{i})}^{2}}) \times Z^{2} (R is the Rydberg's constant)$

Case 1- Transition of electron from second excited state $(n_{i} = 3)$ to the first excited state $(n_{f} = 2)$ :

$\frac{1}{λ_{1}} = R \times (\frac{1}{{(n_{f})}^{2}} - \frac{1}{{(n_{i})}^{2}}) \times Z^{2} (For hydrogen, Z = 1) \Rightarrow \frac{1}{λ_{1}} = R \times (\frac{1}{4} - \frac{1}{9}) \Rightarrow λ_{1} = \frac{1}{R} \times \frac{5}{36} . . . (1)$

Case 2- Transition of electron from first excited state $(n_{i} = 2)$ to the ground state $(n_{f} = 1)$ :

$\frac{1}{λ_{2}} = R \times (\frac{1}{{(n_{f})}^{2}} - \frac{1}{{(n_{i})}^{2}}) \times Z^{2} (For hydrogen, Z = 1) \Rightarrow \frac{1}{λ_{2}} = R \times (\frac{1}{1} - \frac{1}{4}) \Rightarrow λ_{2} = \frac{1}{R} \times \frac{3}{4} . . . (2)$

The ratio of wavelengths of the emitted radiations in both the cases can be calculated by dividing the equation $(1)$ by the equation $(2)$ .

$∴ \frac{λ_{1}}{λ_{2}} = (\frac{1}{R} \times \frac{5}{36}) : (\frac{1}{R} \times \frac{3}{4}) \Rightarrow λ_{1} : λ_{2} = \frac{5}{36} \times \frac{4}{3} \Rightarrow λ_{1} : λ_{2} = 5 : 27$

In a hydrogen atom, an electron undergoes transition from the second excited state to the first excited state and then to the ground state. Find out the ratio of the wavelengths of the emitted radiations in the two cases.

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