SATHEE: atoms Question 25

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Question: Q. 3. (i) State Bohr’s quantization condition for defining stationary orbits. How does de-Broglie hypothesis explain the stationary orbits?

(ii) Find the relation between the three wavelengths $λ_{1}, λ_{2}$ and $λ_{3}$ from the energy level diagram shown below.

R [Delhi I, II, III 2016]

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Solution:

Ans. (i) $L = \frac{n h}{2 π}$ i.e., angular momentum of orbiting electron is quantised.]

According to de-Broglie hypothesis

$1 / 2$

Linear momentum

$p = \frac{h}{λ}$

And for circular orbit, $L = r_{n} p$ where ’ $r_{n}$ ’ is the radius of $n^{th}$ orbit

$= \frac{r_{n} h}{λ}$

Also

$L = \frac{n h}{2 π}$

$∴ \frac{r_{n}}{λ} = \frac{n h}{2 π}$

$\Rightarrow$ $2 π r = n λ$ $1 / 2$

$∴$ Circumference of permitted orbits are integral multiples of the wave-length $λ$ .

(ii)

$Multiple \tag$

Adding (i) & (ii)

$\begin{matrix} (iv) & E_{C} - E_{A} = \frac{h c}{λ_{1}} + \frac{h c}{λ_{2}} \end{matrix}$

Using equation (iii) and (iv)

$\begin{aligned} \frac{h c}{λ_{3}} & = \frac{h c}{λ_{1}} + \frac{h c}{λ_{2}} \Rightarrow \frac{1}{λ_{3}} & = \frac{1}{λ_{1}} + \frac{1}{λ_{2}} \end{aligned}$

[CBSE Marking Scheme 2016]

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