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. `4xx10^(1)`
B. `3xx10^(7)`
C. `2xx10^(-25)`
D. `5xx10^(-18)`

Explanation

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

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

From this formula, we can rearrange it to solve for wavelength (λ):

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

### Step 1: Identify the given values
- Planck's constant (h) is not needed for this calculation.
- The speed of light (c) = \(3.0 \times 10^8 \, \text{m/s}\)
- The frequency (ν) = \(8.0 \times 10^{15} \, \text{s}^{-1}\)

### Step 2: Substitute the values into the formula
Now, we can substitute the known values into the rearranged formula:

\[ \lambda = \frac{3.0 \times 10^8 \, \text{m/s}}{8.0 \times 10^{15} \, \text{s}^{-1}} \]

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

1. Divide the coefficients:
\[ \frac{3.0}{8.0} = 0.375 \]

2. Subtract the exponents (using the property of exponents):
\[ 10^{8 - 15} = 10^{-7} \]

Combining these results gives:

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

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

\[ \lambda = 0.375 \times 10^{-7} \, \text{m} \times \frac{10^9 \, \text{nm}}{1 \, \text{m}} \]

This simplifies to:

\[ \lambda = 0.375 \times 10^{2} \, \text{nm} \]

\[ \lambda = 37.5 \, \text{nm} \]

### Step 5: Round to the nearest value
The closest value to \(37.5 \, \text{nm}\) is \(40 \, \text{nm}\).

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

**40 nm**