The de-Broglie wavelength associated with electrons revolving round the nucleus in a hydrogen atom in the ground state will be

Options

A. 3.3 Å
B. 1.3 Å
C. 6.6 Å
D. 20 Å

Explanation

To find the de-Broglie wavelength associated with electrons revolving around the nucleus in a hydrogen atom in the ground state, 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:
\[ \lambda = \frac{h}{mv} \]
where:
- \( h \) is the Planck's constant (\(6.626 \times 10^{-34} \, \text{Js}\)),
- \( m \) is the mass of the electron (\(9.11 \times 10^{-31} \, \text{kg}\)),
- \( v \) is the velocity of the electron.

### Step 2: Use Bohr's model for the hydrogen atom
According to Bohr's model, the angular momentum of an electron in an orbit is quantized and given by:
\[ mvr = n \frac{h}{2\pi} \]
For the ground state of hydrogen, \( n = 1 \). Thus, we have:
\[ mvr = \frac{h}{2\pi} \]

### Step 3: Rearrange the equation to find velocity
From the angular momentum equation, we can express the velocity \( v \) as:
\[ v = \frac{h}{2\pi mr} \]

### Step 4: Determine the radius of the electron's orbit
The radius of the electron's orbit in the ground state (n=1) for hydrogen is given by:
\[ r = 0.529 \, \text{Å} = 0.529 \times 10^{-10} \, \text{m} \]

### Step 5: Substitute the radius into the velocity equation
Now substituting the value of \( r \) into the velocity equation:
\[ v = \frac{h}{2\pi m (0.529 \times 10^{-10})} \]

### Step 6: Substitute \( v \) into the de-Broglie wavelength formula
Now we can substitute \( v \) back into the de-Broglie wavelength formula:
\[ \lambda = \frac{h}{m \left(\frac{h}{2\pi m (0.529 \times 10^{-10})}\right)} \]

### Step 7: Simplify the expression
This simplifies to:
\[ \lambda = 2\pi (0.529 \times 10^{-10}) = 3.3 \times 10^{-10} \, \text{m} = 3.3 \, \text{Å} \]

### Final Answer
Thus, the de-Broglie wavelength associated with the electron in the ground state of a hydrogen atom is:
\[ \lambda = 3.3 \, \text{Å} \]