Calculate the wave number of radiations having a frequency of `4 xx 10^(14) Hz`

Explanation

To calculate the wave number of radiation with a frequency of \(4 \times 10^{14} \, \text{Hz}\), we can follow these steps:

### Step 1: Understand the relationship between wave number, frequency, and the speed of light
The wave number (\(\bar{\mu}\)) is defined as the reciprocal of the wavelength (\(\lambda\)):
\[ \bar{\mu} = \frac{1}{\lambda} \]
Additionally, the frequency (\(f\)) and wavelength are related through the speed of light (\(c\)):
\[ c = f \cdot \lambda \]

### Step 2: Rearrange the equation to find the wave number
From the equation \(c = f \cdot \lambda\), we can express wavelength in terms of frequency:
\[ \lambda = \frac{c}{f} \]
Substituting this into the wave number equation gives:
\[ \bar{\mu} = \frac{1}{\lambda} = \frac{f}{c} \]

### Step 3: Substitute the known values
We know:
- Frequency, \(f = 4 \times 10^{14} \, \text{Hz}\)
- Speed of light, \(c = 3 \times 10^{8} \, \text{m/s}\)

Now we can substitute these values into the wave number equation:
\[ \bar{\mu} = \frac{4 \times 10^{14} \, \text{Hz}}{3 \times 10^{8} \, \text{m/s}} \]

### Step 4: Calculate the wave number
Performing the calculation:
\[ \bar{\mu} = \frac{4}{3} \times 10^{14 - 8} = \frac{4}{3} \times 10^{6} \, \text{m}^{-1} \]
Calculating \(\frac{4}{3}\):
\[ \bar{\mu} \approx 1.33 \times 10^{6} \, \text{m}^{-1} \]

### Step 5: Convert to centimeter inverse
Since \(1 \, \text{m} = 100 \, \text{cm}\), we convert the wave number to centimeter inverse:
\[ \bar{\mu} = 1.33 \times 10^{6} \, \text{m}^{-1} \times \frac{1 \, \text{cm}}{0.01 \, \text{m}} = 1.33 \times 10^{8} \, \text{cm}^{-1} \]

### Final Answer
Thus, the wave number of the radiation is:
\[ \bar{\mu} \approx 1.33 \times 10^{6} \, \text{m}^{-1} \text{ or } 1.33 \times 10^{8} \, \text{cm}^{-1} \]