An electron makes a transition from orbit `n = 4` to the orbit `n = 2` of a hydrogen atom. The wave number of the emitted radiations `(R =` Rydberg's constant) will be

Options

A. `(16)/(3R)`
B. `(2R)/(16)`
C. `(3R)/(16)`
D. `(4R)/(16)`

Explanation

To find the wave number of the emitted radiation when an electron transitions from the orbit \( n = 4 \) to \( n = 2 \) in a hydrogen atom, we can use the Rydberg formula for hydrogen. The wave number \( \bar{\nu} \) (which is the reciprocal of the wavelength \( \lambda \)) is given by:

\[ \bar{\nu} = R \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \]

where:
- \( R \) is the Rydberg constant,
- \( n_1 \) is the lower energy level (final state),
- \( n_2 \) is the higher energy level (initial state).

### Step 1: Identify the values of \( n_1 \) and \( n_2 \)
In this case:
- The electron transitions from \( n = 4 \) (initial state) to \( n = 2 \) (final state).
- Thus, \( n_1 = 2 \) and \( n_2 = 4 \).

### Step 2: Substitute the values into the Rydberg formula
Now, we can substitute these values into the Rydberg formula:

\[ \bar{\nu} = R \left( \frac{1}{2^2} - \frac{1}{4^2} \right) \]

### Step 3: Calculate \( \frac{1}{2^2} \) and \( \frac{1}{4^2} \)
Calculating the squares:

\[ \frac{1}{2^2} = \frac{1}{4} \]
\[ \frac{1}{4^2} = \frac{1}{16} \]

### Step 4: Substitute the values back into the equation
Now we can substitute these values back into the equation:

\[ \bar{\nu} = R \left( \frac{1}{4} - \frac{1}{16} \right) \]

### Step 5: Find a common denominator and simplify
To subtract these fractions, we need a common denominator. The common denominator of 4 and 16 is 16:

\[ \frac{1}{4} = \frac{4}{16} \]

So,

\[ \bar{\nu} = R \left( \frac{4}{16} - \frac{1}{16} \right) = R \left( \frac{3}{16} \right) \]

### Step 6: Final expression for wave number
Thus, the wave number of the emitted radiation is:

\[ \bar{\nu} = \frac{3R}{16} \]

### Summary
The wave number of the emitted radiation when an electron transitions from orbit \( n = 4 \) to orbit \( n = 2 \) in a hydrogen atom is \( \frac{3R}{16} \).

---