Question

The electron of a hydrogen atom is in its \( n \)th Bohr orbit having de Broglie wavelength of 13.4 Å. The value of \( n \) is ............ (rounded up to the nearest integer). \[ \text{[Radius of \( n \)th Bohr orbit is } r_n = 0.53n^2 \, \text{Å}, \, \pi = 3.14\text{]} \]

Hint:

Use the de Broglie wavelength formula to calculate the quantum number \( n \) of an electron in a hydrogen atom.

Answer 1

Correct Answer: 4

1. Relate de Broglie Wavelength and Bohr Radius

According to Bohr's quantization condition (postulate), the angular momentum of an electron in a stationary orbit is quantized:

$$mvr_n = n\frac{h}{2\pi}$$

where $m$ is the electron mass, $v$ is its velocity, $r_n$ is the orbit radius, $h$ is Planck's constant, and $n$ is the principal quantum number.

According to the de Broglie hypothesis, the wavelength ($\lambda$) associated with the electron is:

$$\lambda = \frac{h}{mv}$$

Rearranging this gives:

$$mv = \frac{h}{\lambda}$$

Substitute the de Broglie relation for $mv$ into the Bohr quantization condition:

$$\left(\frac{h}{\lambda}\right) r_n = n\frac{h}{2\pi}$$

Simplifying the equation by canceling $h$ on both sides, we get the fundamental relationship between the orbit radius ($r_n$) and the de Broglie wavelength ($\lambda$):

$$\frac{r_n}{\lambda} = \frac{n}{2\pi}$$

$$2\pi r_n = n\lambda$$

2. Substitute Given Values and Formulas

We are given the following values and formula:

de Broglie wavelength: $\lambda = 13.4 \text{ Å}$

Radius of $n^{\text{th}}$ Bohr orbit: $r_n = 0.53n^2 \text{ Å}$

Value of $\pi = 3.14$

Substitute the expressions for $r_n$ and $\lambda$ into the derived equation $2\pi r_n = n\lambda$:

 

$$2(3.14) (0.53n^2 \text{ Å}) = n (13.4 \text{ Å})$$

3. Solve for $n$

First, simplify the constants on the left side:

$$(2 \times 3.14 \times 0.53) n^2 = 13.4 n$$

$$3.3392 n^2 = 13.4 n$$

Since $n$ is the principal quantum number, it cannot be zero. We can divide both sides by $n$:

$$3.3392 n = 13.4$$

$$n = \frac{13.4}{3.3392}$$

$$n \approx 4.0128$$

4. Round to the Nearest Integer

The question asks for the value of $n$ rounded up to the nearest integer.

$$n \approx 4.0128$$

Rounding $4.0128$ to the nearest integer gives:

$$n = 4$$

The value of $n$ is 4.