The kinetic energy of an electron in the second Bohr orbit of a hydrogen atom is [`a_0` is Bohr radius] :

Options

A. `(h^(2))/(4 pi^2 ma_0^2)`
B. `(h^2)/(16 pi^2 ma_0^2)`
C. `(4 h^2)/(32 pi^2 ma_0^2)`
D. `(h^2)/(64 pi^2 ma_0^2)`

Explanation

To find the kinetic energy of an electron in the second Bohr orbit of a hydrogen atom, we can follow these steps:

### Step 1: Understand the Bohr Model
The Bohr model describes the electron in a hydrogen atom as moving in circular orbits around the nucleus. The kinetic energy (KE) of an electron in a circular orbit can be expressed in terms of its mass (m), velocity (v), and radius (R) of the orbit.

### Step 2: Write the Expression for Kinetic Energy
The kinetic energy of an electron is given by the formula:
\[ KE = \frac{1}{2} mv^2 \]

### Step 3: Relate Velocity to the Bohr Model
According to the Bohr model, the centripetal force required for circular motion is provided by the electrostatic force between the positively charged nucleus and the negatively charged electron. This can be expressed as:
\[ \frac{mv^2}{R} = \frac{ke^2}{R^2} \]
where \( k \) is Coulomb's constant, and \( e \) is the charge of the electron.

From this, we can rearrange to find \( v^2 \):
\[ mv^2 = \frac{ke^2}{R} \]

### Step 4: Substitute into the Kinetic Energy Formula
Substituting \( mv^2 \) into the kinetic energy formula gives:
\[ KE = \frac{1}{2} \left(\frac{ke^2}{R}\right) \]

### Step 5: Use Bohr's Radius for the Second Orbit
For the nth orbit, the radius \( R_n \) is given by:
\[ R_n = n^2 a_0 \]
where \( a_0 \) is the Bohr radius. For the second orbit (n=2):
\[ R_2 = 4a_0 \]

### Step 6: Substitute \( R_2 \) into the Kinetic Energy Expression
Now substituting \( R_2 \) into the kinetic energy formula:
\[ KE = \frac{1}{2} \left(\frac{ke^2}{4a_0}\right) \]

### Step 7: Use the Expression for \( k \)
The value of \( k \) can be expressed in terms of fundamental constants:
\[ k = \frac{1}{4\pi \epsilon_0} \]
Substituting this into the kinetic energy expression gives:
\[ KE = \frac{1}{2} \left(\frac{1}{4\pi \epsilon_0} \cdot \frac{e^2}{4a_0}\right) \]

### Step 8: Simplify the Expression
After simplification, we can express the kinetic energy in terms of \( a_0 \):
\[ KE = \frac{e^2}{32\pi \epsilon_0 a_0} \]

### Step 9: Final Expression for Kinetic Energy
For hydrogen, we can relate \( e^2 \) and other constants to find the final expression:
\[ KE = \frac{n^2 h^2}{8 \pi^2 m a_0^2} \]
For \( n = 2 \):
\[ KE = \frac{4h^2}{8 \pi^2 m a_0^2} = \frac{h^2}{32 \pi^2 m a_0^2} \]

### Final Answer
Thus, the kinetic energy of an electron in the second Bohr orbit of a hydrogen atom is:
\[ KE = \frac{h^2}{32 \pi^2 m a_0^2} \]

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