The wavelength of light having wave number 4000 cm is

Options

A. `10 mu`
B. `1.5 mu`
C. `2.0 mu`
D. `2.5 mu`

Answer

To find the wavelength of light given its wave number, we can follow these steps:

### Step 1: Understand the relationship between wave number and wavelength
The wave number (denoted as \( \bar{\nu} \)) is defined as the number of waves per unit distance and is given by the formula:
\[ \bar{\nu} = \frac{1}{\lambda} \]
where \( \lambda \) is the wavelength.

### Step 2: Rearrange the formula to find wavelength
From the wave number formula, we can rearrange it to solve for wavelength:
\[ \lambda = \frac{1}{\bar{\nu}} \]

### Step 3: Substitute the given wave number
We are given a wave number of \( 4000 \, \text{cm}^{-1} \). Substituting this value into the formula gives:
\[ \lambda = \frac{1}{4000 \, \text{cm}^{-1}} \]

### Step 4: Calculate the wavelength
Now, we perform the calculation:
\[ \lambda = \frac{1}{4000} = 0.00025 \, \text{cm} \]
This can also be expressed in scientific notation:
\[ \lambda = 2.5 \times 10^{-4} \, \text{cm} \]

### Step 5: Convert the wavelength from centimeters to meters
To convert the wavelength from centimeters to meters, we use the conversion factor \( 1 \, \text{cm} = 10^{-2} \, \text{m} \):
\[ \lambda = 2.5 \times 10^{-4} \, \text{cm} \times \frac{1 \, \text{m}}{10^{2} \, \text{cm}} = 2.5 \times 10^{-6} \, \text{m} \]

### Step 6: Express the wavelength in micrometers
Since \( 1 \, \mu\text{m} = 10^{-6} \, \text{m} \), we can express the wavelength in micrometers:
\[ \lambda = 2.5 \, \mu\text{m} \]

### Conclusion
Thus, the wavelength of light having a wave number of \( 4000 \, \text{cm}^{-1} \) is:
\[ \lambda = 2.5 \, \mu\text{m} \]