According to de Broglie's explanation of Bohr's second postulate of quantisation, the standing particle wave on a circular orbit for n = 4 is given by

Options

A. `2pir_n=lambda`
B. `(2pi)/lambda=2r_n`
C. `2pir_n=4lambda`
D. `lambda/(2lambda)=4r_n`

Answer

To find the standing particle wave on a circular orbit for n = 4 according to de Broglie's explanation of Bohr's second postulate of quantization, we can follow these steps:

### Step-by-Step Solution

1. **Understanding Bohr's Second Postulate**:
According to Bohr's second postulate, the angular momentum (L) of an electron in a circular orbit is quantized and given by:
\[ L = mvr = n\frac{h}{2\pi} \]
where:
- \(m\) is the mass of the electron,
- \(v\) is the velocity of the electron,
- \(r\) is the radius of the orbit,
- \(n\) is the principal quantum number,
- \(h\) is Planck's constant.

2. **Substituting for Angular Momentum**:
We can express the angular momentum in terms of the radius and velocity:
\[ mvr = n\frac{h}{2\pi} \]
For the nth orbit, we denote the radius as \(r_n\):
\[ mv r_n = n\frac{h}{2\pi} \]

3. **Using de Broglie Wavelength**:
The de Broglie wavelength (\(\lambda\)) of a particle is given by:
\[ \lambda = \frac{h}{p} \]
where \(p\) is the momentum. Since \(p = mv\), we can rewrite this as:
\[ \lambda = \frac{h}{mv} \]

4. **Relating Wavelength to Orbit**:
According to the standing wave condition for the electron in a circular orbit, the circumference of the orbit must be an integer multiple of the wavelength:
\[ 2\pi r_n = n\lambda \]
Substituting the expression for \(\lambda\):
\[ 2\pi r_n = n \left(\frac{h}{mv}\right) \]

5. **Combining the Equations**:
Now, we can combine the two equations:
\[ 2\pi r_n = n \left(\frac{h}{mv}\right) \]
From the angular momentum equation, we can express \(mv\) in terms of \(n\):
\[ mv = \frac{nh}{2\pi r_n} \]
Substituting this into the wavelength equation gives:
\[ 2\pi r_n = n \left(\frac{h}{\frac{nh}{2\pi r_n}}\right) \]
Simplifying this, we find:
\[ 2\pi r_n = 2\pi r_n \]
which confirms the relationship.

6. **Substituting n = 4**:
Now, substituting \(n = 4\):
\[ 2\pi r_4 = 4\lambda \]
Thus, we can express the standing wave condition for \(n = 4\):
\[ r_4 = \frac{4\lambda}{2\pi} = \frac{2\lambda}{\pi} \]

### Final Result:
The standing particle wave on a circular orbit for \(n = 4\) is given by:
\[ r_4 = \frac{2\lambda}{\pi} \]