The acceleration of electron in the first orbit of hydrogen atom is

Options

A. `(4 pi^(2) m)/(h^(3))`
B. `(h^(2))/(4 pi^(2)mr)`
C. `(h)/(4 pi^(2)m^(2)r^(3))`
D. `(m^(2)h^(2))/(4 pi^(2)r^(3))`

Explanation

To find the acceleration of the electron in the first orbit of a hydrogen atom, we can follow these steps:

### Step 1: Understand the Bohr Model
According to the Bohr model of the atom, the angular momentum of an electron in orbit is quantized and is given by the formula:
\[ L = mvr = n\frac{h}{2\pi} \]
where:
- \( L \) is the angular momentum,
- \( m \) 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 (for the first orbit, \( n = 1 \)),
- \( h \) is Planck's constant.

### Step 2: Apply the Formula for the First Orbit
For the first orbit (\( n = 1 \)):
\[ mvr = \frac{h}{2\pi} \]
From this, we can express the velocity \( v \) as:
\[ v = \frac{h}{2\pi m r} \]

### Step 3: Use Centripetal Force Concept
The electron is in circular motion, which requires a centripetal force. The centripetal force \( F_c \) is given by:
\[ F_c = \frac{mv^2}{r} \]
This centripetal force is provided by the electrostatic force between the electron and the nucleus.

### Step 4: Relate Centripetal Force to Acceleration
The centripetal acceleration \( a \) can be expressed as:
\[ a = \frac{v^2}{r} \]

### Step 5: Substitute for Velocity
Now, substituting the expression for \( v \) into the acceleration formula:
\[ a = \frac{v^2}{r} = \frac{\left(\frac{h}{2\pi m r}\right)^2}{r} \]

### Step 6: Simplify the Expression
Now, simplifying the expression:
\[ a = \frac{h^2}{(2\pi m)^2 r^2} \cdot \frac{1}{r} = \frac{h^2}{(2\pi m)^2 r^3} \]

### Final Result
Thus, the acceleration of the electron in the first orbit of the hydrogen atom is:
\[ a = \frac{h^2}{4\pi^2 m^2 r^3} \]