The wavelength of the electron in the first orbit of the Hydrogen atom is x. The wave length of the electron in the third orbit and the circumference of the third orbit of the Hydrogen atom are respectively

Options

A. 3x, 9x
B. 9x, 27x
C. x, 3x
D. x/3,x

Answer

To solve the problem, we need to find the wavelength of the electron in the third orbit of the hydrogen atom and the circumference of the third orbit. We know that the wavelength of the electron in the first orbit is given as \( x \).

### Step 1: Understand the relationship between wavelength and orbit
According to de Broglie's hypothesis, the wavelength \( \lambda \) of an electron can be expressed as:
\[ \lambda = \frac{h}{mv} \]
where \( h \) is Planck's constant, \( m \) is the mass of the electron, and \( v \) is its velocity.

### Step 2: Use the quantization condition for circular orbits
For an electron in a circular orbit, the angular momentum is quantized:
\[ mvr = \frac{nh}{2\pi} \]
where \( n \) is the principal quantum number (1 for the first orbit, 3 for the third orbit).

### Step 3: Calculate the circumference of the first orbit
For the first orbit (\( n = 1 \)):
\[ mvr = \frac{1h}{2\pi} \]
From this, we can express the circumference \( C \) of the first orbit as:
\[ C_1 = 2\pi r_1 = \frac{h}{mv} \]
This means:
\[ C_1 = \lambda_1 = x \]

### Step 4: Calculate the circumference of the third orbit
For the third orbit (\( n = 3 \)):
\[ mvr = \frac{3h}{2\pi} \]
Thus, the circumference \( C_3 \) of the third orbit is:
\[ C_3 = 2\pi r_3 = \frac{h}{mv} \cdot 3 = 3 \cdot \lambda_1 = 3x \]

### Step 5: Calculate the wavelength in the third orbit
Using the relationship from de Broglie's equation:
\[ \lambda_3 = \frac{h}{mv} \cdot \frac{1}{3} = \frac{\lambda_1}{3} = \frac{x}{3} \]

### Final Results
Thus, the wavelength of the electron in the third orbit is \( \frac{x}{3} \) and the circumference of the third orbit is \( 3x \).

### Summary of the Results
- Wavelength of the electron in the third orbit: \( \frac{x}{3} \)
- Circumference of the third orbit: \( 3x \)