The de Broglie wavelength of an electron in the 3rd Bohr orbit is

Options

A. `2pia_0`
B. `4pia_0`
C. `6pia_0`
D. `8pia_0`

Answer

To find the de Broglie wavelength of an electron in the 3rd Bohr orbit, we can follow these steps:

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

### Step 2: Relate Momentum to Mass and Velocity
Since momentum \( p \) is defined as:
\[ p = mv \]
where \( m \) is the mass of the electron and \( v \) is its velocity, we can rewrite the de Broglie wavelength as:
\[ \lambda = \frac{h}{mv} \]

### Step 3: Use Bohr's Model for Angular Momentum
According to Bohr's model, the angular momentum of an electron in the nth orbit is quantized and given by:
\[ mvr = \frac{nh}{2\pi} \]
where \( n \) is the principal quantum number and \( r \) is the radius of the orbit.

### Step 4: Express Velocity in Terms of n and r
From the angular momentum equation, we can express \( mv \) as:
\[ mv = \frac{nh}{2\pi r} \]

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

### Step 6: Use the Radius of the Bohr Orbit
The radius of the nth Bohr orbit is given by:
\[ r_n = n^2 \cdot a_0 \]
where \( a_0 \) is the Bohr radius.

### Step 7: Substitute the Radius into the Wavelength Formula
Substituting \( r_n \) into the wavelength formula gives:
\[ \lambda = \frac{2\pi (n^2 a_0)}{n} = 2\pi n a_0 \]

### Step 8: Calculate for the 3rd Orbit
For the 3rd Bohr orbit (\( n = 3 \)):
\[ \lambda = 2\pi (3) a_0 = 6\pi a_0 \]

### Final Answer
Thus, the de Broglie wavelength of an electron in the 3rd Bohr orbit is:
\[ \lambda = 6\pi a_0 \]

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