What is the energy in joule of photon of light whose wave length is 2 x` 10^5 `nm

Options

A. 2.5 x `10^(-19)`
B. 9.9 x `10^(-22)`
C. 5 x `10^(-20)`
D. 6.5 x `10^(-22)`

Explanation

To find the energy of a photon of light with a given wavelength, we can use the formula:

\[ E = \frac{hc}{\lambda} \]

where:
- \( E \) is the energy of the photon,
- \( h \) is Planck's constant (\( 6.626 \times 10^{-34} \, \text{J s} \)),
- \( c \) is the speed of light (\( 3 \times 10^{8} \, \text{m/s} \)),
- \( \lambda \) is the wavelength in meters.

### Step-by-step Solution:

**Step 1: Convert the wavelength from nanometers to meters.**

Given:
\[ \lambda = 2 \times 10^{5} \, \text{nm} \]

To convert nanometers to meters, we use the conversion factor \( 1 \, \text{nm} = 10^{-9} \, \text{m} \):
\[ \lambda = 2 \times 10^{5} \, \text{nm} \times 10^{-9} \, \text{m/nm} = 2 \times 10^{-4} \, \text{m} \]

**Step 2: Substitute the values into the energy formula.**

Now we substitute \( h \), \( c \), and \( \lambda \) into the energy formula:
\[ E = \frac{(6.626 \times 10^{-34} \, \text{J s}) \times (3 \times 10^{8} \, \text{m/s})}{2 \times 10^{-4} \, \text{m}} \]

**Step 3: Calculate the energy.**

First, calculate the numerator:
\[ 6.626 \times 10^{-34} \times 3 \times 10^{8} = 1.9878 \times 10^{-25} \, \text{J m} \]

Now, divide by the wavelength:
\[ E = \frac{1.9878 \times 10^{-25}}{2 \times 10^{-4}} \]

Calculating this gives:
\[ E = 9.939 \times 10^{-22} \, \text{J} \]

**Step 4: Round the answer.**

Rounding to three significant figures, we get:
\[ E \approx 9.94 \times 10^{-22} \, \text{J} \]

### Final Answer:
The energy of the photon is approximately \( 9.94 \times 10^{-22} \, \text{J} \).

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