the number of de broglie wavelength contained in the second bohr orbit of hydrogen atom is

Options

A. 1
B. 2
C. 3
D. 4

Explanation

To find the number of de Broglie wavelengths contained in the second Bohr orbit of a hydrogen atom, we can follow these steps:

### Step 1: Understand the de Broglie wavelength
The de Broglie wavelength (\( \lambda \)) of a particle is given by the formula:
\[ \lambda = \frac{h}{p} \]
where \( h \) is Planck's constant and \( p \) is the momentum of the particle.

### Step 2: Bohr's model of the hydrogen atom
In Bohr's model, the electron moves in circular orbits around the nucleus. The radius of the \( n \)-th orbit is given by:
\[ r_n = n^2 \cdot r_1 \]
where \( r_1 \) is the radius of the first orbit (approximately \( 0.529 \, \text{Å} \)) and \( n \) is the principal quantum number.

### Step 3: Calculate the circumference of the second orbit
For the second Bohr orbit (\( n = 2 \)):
\[ r_2 = 2^2 \cdot r_1 = 4 \cdot r_1 \]
The circumference \( C \) of the second orbit is:
\[ C = 2 \pi r_2 = 2 \pi (4 r_1) = 8 \pi r_1 \]

### Step 4: Relate the circumference to the de Broglie wavelength
In the Bohr model, the circumference of the orbit must be an integer multiple of the de Broglie wavelength:
\[ C = n \cdot \lambda \]
For the second orbit, we can express this as:
\[ 8 \pi r_1 = n \cdot \lambda \]

### Step 5: Determine the number of wavelengths
Since we are looking for the number of de Broglie wavelengths in the second orbit, we can rearrange the equation:
\[ n = \frac{C}{\lambda} \]
From the previous steps, we know that:
\[ C = 8 \pi r_1 \]
Thus, we can conclude that:
\[ n = \frac{8 \pi r_1}{\lambda} \]

### Step 6: Calculate the number of wavelengths
In the case of the second Bohr orbit, it can be shown that:
\[ n = 2 \]
This means there are 2 de Broglie wavelengths contained in the second Bohr orbit of the hydrogen atom.

### Final Answer
The number of de Broglie wavelengths contained in the second Bohr orbit of a hydrogen atom is **2**.

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