The value of Planck's constant is `6.63 xx 10^(-34)Js`. The velocity of light is `3.0 xx 10^(8)ms^(-1)`. Which value is closest to the wavelength in nanometers of a quantum of light with frequency `8xx10^(15)s^(-1)`?

Options

A. `2xx10^(-15)`
B. `5xx10^(-18)`
C. `4xx10^(1)`
D. `3xx10^(7)`

Explanation

To find the wavelength of a quantum of light with a given frequency, we can use the relationship between wavelength (λ), frequency (ν), and the speed of light (c). The formula is:

\[ \lambda = \frac{c}{\nu} \]

Where:
- \( \lambda \) is the wavelength in meters,
- \( c \) is the speed of light in meters per second (m/s),
- \( \nu \) is the frequency in hertz (s\(^{-1}\)).

### Step 1: Identify the given values
- Planck's constant: \( 6.63 \times 10^{-34} \, \text{Js} \) (not needed for this calculation)
- Speed of light, \( c = 3.0 \times 10^{8} \, \text{m/s} \)
- Frequency, \( \nu = 8 \times 10^{15} \, \text{s}^{-1} \)

### Step 2: Substitute the values into the formula
Using the formula \( \lambda = \frac{c}{\nu} \):

\[ \lambda = \frac{3.0 \times 10^{8} \, \text{m/s}}{8 \times 10^{15} \, \text{s}^{-1}} \]

### Step 3: Perform the calculation
Calculating the above expression:

\[ \lambda = \frac{3.0}{8} \times \frac{10^{8}}{10^{15}} \]

\[ \lambda = 0.375 \times 10^{-7} \, \text{m} \]

### Step 4: Convert to nanometers
Since \( 1 \, \text{nm} = 10^{-9} \, \text{m} \), we convert meters to nanometers:

\[ \lambda = 0.375 \times 10^{-7} \, \text{m} = 3.75 \times 10^{-8} \, \text{m} \]

Now converting to nanometers:

\[ \lambda = 3.75 \times 10^{-8} \, \text{m} \times \frac{10^{9} \, \text{nm}}{1 \, \text{m}} = 37.5 \, \text{nm} \]

### Step 5: Conclusion
The wavelength of the quantum of light with a frequency of \( 8 \times 10^{15} \, \text{s}^{-1} \) is approximately \( 37.5 \, \text{nm} \).