If the radius of first Bohr's of hydrogen is x , then de - Broglie wavelength of electron in its 3rd orbit is

Options

A. `2pir`
B. `6pix`
C. 9x
D. `x/3`

Answer

To find the de Broglie wavelength of the electron in the third orbit of hydrogen, given that the radius of the first Bohr orbit is \( x \), we can follow these steps:

### Step 1: Understand the Radius of the nth Orbit
The radius of the nth orbit in the Bohr model is given by the formula:
\[ R_n = n^2 \cdot R_1 \]
where \( R_1 \) is the radius of the first orbit.

### Step 2: Substitute the Given Values
We know that the radius of the first orbit \( R_1 = x \). Therefore, for the third orbit (\( n = 3 \)):
\[ R_3 = 3^2 \cdot R_1 = 9 \cdot R_1 = 9x \]

### Step 3: Use the de Broglie Wavelength Formula
The de Broglie wavelength (\( \lambda \)) of an electron in an orbit is given by:
\[ \lambda = \frac{h}{m v} \]
where \( h \) is Planck's constant, \( m \) is the mass of the electron, and \( v \) is the velocity of the electron in that orbit.

### Step 4: Relate Velocity to the Radius
From Bohr's model, the velocity of the electron in the nth orbit can be expressed as:
\[ v_n = \frac{n h}{2 \pi m R_n} \]
For the third orbit (\( n = 3 \)):
\[ v_3 = \frac{3h}{2 \pi m R_3} \]

### Step 5: Substitute \( R_3 \) into the Velocity Equation
Substituting \( R_3 = 9x \) into the velocity equation:
\[ v_3 = \frac{3h}{2 \pi m (9x)} = \frac{3h}{18 \pi m x} = \frac{h}{6 \pi m x} \]

### Step 6: Substitute Velocity into the de Broglie Wavelength Formula
Now substituting \( v_3 \) back into the de Broglie wavelength formula:
\[ \lambda = \frac{h}{m v_3} = \frac{h}{m \left(\frac{h}{6 \pi m x}\right)} = \frac{h \cdot 6 \pi m x}{h} \]
The \( h \) cancels out:
\[ \lambda = 6 \pi x \]

### Final Answer
Thus, the de Broglie wavelength of the electron in its third orbit is:
\[ \lambda = 6 \pi x \]