Question

The de-Broglie wavelength of an electron in 4th orbit is (where \( r \) = radius of the 1st orbit)

• A. \( 2\pi r \)
• B. \( 4\pi r \)
• C. \( 8\pi r \)
• D. \( 16\pi r \)

Hint:

The general formula for the de-Broglie wavelength in the \( n \)-th orbit is \( \lambda = 2\pi r n \). Since the orbit radius grows quadratically with \( n \) and the wavelength is \( \frac{2\pi r_n}{n} \), the wavelength increases linearly with \( n \) as \( \lambda_n = n \cdot \lambda_1 \). Since \( \lambda_1 = 2\pi r \), for \( n = 4 \) the wavelength is simply \( 4 \times 2\pi r = 8\pi r \).

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
Bohr's quantization condition for atomic orbits states that the angular momentum of an electron in a stable orbit must be an integral multiple of \(\hbar\).
Louis de Broglie later provided a physical interpretation of this: the electron's orbit must contain an integral number of its wavelengths.
This means that for an electron to exist in the \(n\)-th stable orbit, the circumference of that orbit must be exactly equal to \(n\) times the de Broglie wavelength \(\lambda\).
Combining this with the knowledge of how the radius of Bohr orbits scales with the principal quantum number \(n\), we can determine the wavelength.
Step 2: Key Formula or Approach:
1. Orbit Circumference condition: \(2\pi r_n = n\lambda\).
2. Radius scaling in Bohr's model: \(r_n = n^2 r_1\).
3. Given: \(r_1 = r\).
Step 3: Detailed Explanation:
For the 4th orbit, the principal quantum number is \(n = 4\).
The radius of the 4th orbit \(r_4\) in terms of the 1st orbit radius \(r\) is:
\[ r_4 = n^2 r = 4^2 r = 16r \]
According to de Broglie's hypothesis for stationary orbits:
\[ 2\pi r_n = n\lambda \]
Substituting \(n = 4\) and \(r_n = r_4 = 16r\):
\[ 2\pi (16r) = 4\lambda \]
\[ 32\pi r = 4\lambda \]
Dividing both sides by 4 to solve for \(\lambda\):
\[ \lambda = \frac{32\pi r}{4} = 8\pi r \]
This matches option (C).
Step 4: Final Answer:
The de-Broglie wavelength of the electron in the 4th orbit is \(8\pi r\), derived from the standing wave condition on the circumference of the expanded Bohr orbit.