Calculate the wave number of radiations having a frequency of `4 xx 10^(-11) kHz.`

Explanation

To calculate the wave number of radiations having a frequency of \(4 \times 10^{-11}\) kHz, we will follow these steps:

### Step 1: Convert the frequency from kilohertz to hertz
The frequency is given in kilohertz (kHz), and we need to convert it to hertz (Hz) because the standard unit for frequency in the wave number formula is hertz.

\[ \text{Frequency in Hz} = 4 \times 10^{-11} \, \text{kHz} \times 10^3 \, \text{Hz/kHz} = 4 \times 10^{-8} \, \text{Hz} \]

### Step 2: Use the formula for wave number
The wave number (\( \bar{\nu} \)) is defined as the reciprocal of the wavelength (\( \lambda \)), which can be calculated using the formula:

\[ \bar{\nu} = \frac{1}{\lambda} = \frac{f}{c} \]

where:
- \( f \) is the frequency in hertz,
- \( c \) is the speed of light in vacuum, approximately \( 3 \times 10^8 \, \text{m/s} \).

### Step 3: Substitute the values into the formula
Now we can substitute the frequency and the speed of light into the wave number formula:

\[ \bar{\nu} = \frac{4 \times 10^{-8} \, \text{Hz}}{3 \times 10^8 \, \text{m/s}} \]

### Step 4: Calculate the wave number
Now, we perform the calculation:

\[ \bar{\nu} = \frac{4 \times 10^{-8}}{3 \times 10^8} = \frac{4}{3} \times 10^{-16} \, \text{m}^{-1} \]

Calculating the numerical value:

\[ \bar{\nu} \approx 1.333 \times 10^{-16} \, \text{m}^{-1} \]

### Final Answer
Thus, the wave number of the radiation is approximately:

\[ \bar{\nu} \approx 1.333 \times 10^{-16} \, \text{m}^{-1} \]