The value of Planck's constant is `6.63 xx 10^-34 Js`. The velocity of light is `3 xx 10^8 m//sec`. Which value is closest to the wavelength of quantum of light with frequency of `8 xx 10^15 sec^-1` ?

Options

A. `5 xx 10^-18 m`
B. `4 xx 10^-8 m`
C. `3 xx 10^7 m`
D. `2 xx 10^-25 m`

Explanation

To find the wavelength of a quantum of light with a given frequency, we can use the formula that relates the speed of light (c), wavelength (λ), and frequency (ν):

\[ c = \lambda \cdot \nu \]

Where:
- \( c \) is the speed of light (3 x 10^8 m/s),
- \( \lambda \) is the wavelength,
- \( \nu \) is the frequency (8 x 10^15 s^-1).

### Step-by-step Solution:

1. **Identify the given values**:
- Speed of light, \( c = 3 \times 10^8 \, \text{m/s} \)
- Frequency, \( \nu = 8 \times 10^{15} \, \text{s}^{-1} \)

2. **Rearrange the formula to solve for wavelength (λ)**:
\[ \lambda = \frac{c}{\nu} \]

3. **Substitute the known values into the equation**:
\[ \lambda = \frac{3 \times 10^8 \, \text{m/s}}{8 \times 10^{15} \, \text{s}^{-1}} \]

4. **Perform the division**:
- First, divide the coefficients:
\[ \frac{3}{8} = 0.375 \]
- Next, divide the powers of ten:
\[ 10^8 / 10^{15} = 10^{8-15} = 10^{-7} \]
- Combine the results:
\[ \lambda = 0.375 \times 10^{-7} \, \text{m} = 3.75 \times 10^{-8} \, \text{m} \]

5. **Convert to nanometers (if needed)**:
- Since \( 1 \, \text{m} = 10^9 \, \text{nm} \):
\[ \lambda = 3.75 \times 10^{-8} \, \text{m} \times 10^9 \, \text{nm/m} = 37.5 \, \text{nm} \]

6. **Determine the closest value from the options provided**:
- Based on the calculated wavelength, the closest value to \( 37.5 \, \text{nm} \) should be selected from the options given.

### Final Answer:
The closest value to the wavelength of the quantum of light with a frequency of \( 8 \times 10^{15} \, \text{s}^{-1} \) is approximately \( 3.75 \times 10^{-8} \, \text{m} \) or \( 37.5 \, \text{nm} \).