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))/(4pi^(2)ma_(0)^(2))`
B. `(h^(2))/(16pi^(2)ma_(0)^(2))`
C. `(h^(2))/(32pi^(2)ma_(0)^(2))`
D. `(h^(2))/(64pi^(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 Formula for Kinetic Energy
The kinetic energy (KE) of an electron is given by the formula:
\[ KE = \frac{1}{2} mv^2 \]
where \( m \) is the mass of the electron and \( v \) is its velocity.

### Step 2: Use Bohr's Theory
According to Bohr's theory, the kinetic energy can also be expressed in terms of the principal quantum number \( n \):
\[ KE = \frac{n^2 h^2}{8 \pi^2 m r^2} \]
where \( h \) is Planck's constant, \( m \) is the mass of the electron, and \( r \) is the radius of the orbit.

### Step 3: Determine the Radius for the Second Orbit
The radius of the \( n \)-th orbit in a hydrogen atom is given by:
\[ r = n_0 \frac{n^2}{Z} \]
For hydrogen, \( Z = 1 \), so:
\[ r = n_0 n^2 \]
where \( n_0 \) (Bohr radius) is approximately \( 0.529 \, \text{Å} \).

For the second orbit (\( n = 2 \)):
\[ r = n_0 \cdot 2^2 = n_0 \cdot 4 = 4 n_0 \]

### Step 4: Calculate \( r^2 \)
Now, we need to find \( r^2 \):
\[ r^2 = (4 n_0)^2 = 16 n_0^2 \]

### Step 5: Substitute \( r^2 \) into the Kinetic Energy Formula
Substituting \( r^2 \) into the kinetic energy formula:
\[ KE = \frac{n^2 h^2}{8 \pi^2 m (16 n_0^2)} \]
\[ KE = \frac{n^2 h^2}{128 \pi^2 m n_0^2} \]

### Step 6: Substitute \( n = 2 \)
For the second orbit, \( n = 2 \):
\[ KE = \frac{2^2 h^2}{128 \pi^2 m n_0^2} \]
\[ KE = \frac{4 h^2}{128 \pi^2 m n_0^2} \]
\[ KE = \frac{h^2}{32 \pi^2 m n_0^2} \]

### Final Answer
Thus, the kinetic energy of an electron in the second Bohr orbit of a hydrogen atom is:
\[ KE = \frac{h^2}{32 \pi^2 m n_0^2} \]