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

Options

A. `2pir`
B. `4pir`
C. `8pir`
D. `16pir`

Explanation

To find the de-Broglie wavelength of an electron in the 4th orbit, we can follow these steps:

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

### Step 2: Express momentum in terms of mass and velocity
The momentum \(p\) can be expressed as:
\[ p = mv \]
where \(m\) is the mass of the electron and \(v\) is its velocity.

### Step 3: Relate momentum to angular momentum
According to Bohr's model, the angular momentum \(L\) of an electron in the nth orbit is quantized and given by:
\[ L = n \frac{h}{2\pi} \]
For a circular orbit, angular momentum can also be expressed as:
\[ L = mvr \]
where \(r\) is the radius of the orbit.

### Step 4: Set the two expressions for angular momentum equal
Setting the two expressions for angular momentum equal gives:
\[ mvr = n \frac{h}{2\pi} \]
From this, we can solve for \(p\):
\[ p = mv = \frac{nh}{2\pi r} \]

### Step 5: Substitute momentum into the de-Broglie wavelength formula
Substituting \(p\) into the de-Broglie wavelength formula gives:
\[ \lambda = \frac{h}{p} = \frac{h}{\frac{nh}{2\pi r}} = \frac{2\pi r}{n} \]

### Step 6: Determine the radius of the 4th orbit
According to Bohr's model, the radius of the nth orbit is given by:
\[ r_n = n^2 r_1 \]
where \(r_1\) is the radius of the first orbit. For the 4th orbit:
\[ r_4 = 4^2 r_1 = 16 r_1 \]

### Step 7: Substitute \(r_4\) into the wavelength formula
Now substituting \(n = 4\) and \(r = r_4\) into the wavelength formula:
\[ \lambda = \frac{2\pi r_4}{4} = \frac{2\pi (16 r_1)}{4} = 8\pi r_1 \]

### Final Result
Thus, the de-Broglie wavelength of an electron in the 4th orbit is:
\[ \lambda = 8\pi r_1 \]

---