Question

a photon has a wavelength of 665 nm. calculate the energy of the photon in joules. be sure your answer has the correct number of significant digits. note: reference the fundamental constants table for additional information.

Explanation:

Step1: Recall the formula for photon energy

The energy \( E \) of a photon is given by the formula \( E = \frac{hc}{\lambda} \), where \( h \) is Planck's constant (\( h = 6.626 \times 10^{-34} \, \text{J·s} \)), \( c \) is the speed of light (\( c = 3.00 \times 10^{8} \, \text{m/s} \)), and \( \lambda \) is the wavelength of the photon.

First, convert the wavelength from nanometers to meters. Since \( 1 \, \text{nm} = 10^{-9} \, \text{m} \), the wavelength \( \lambda = 665 \, \text{nm} = 665 \times 10^{-9} \, \text{m} = 6.65 \times 10^{-7} \, \text{m} \).

Step2: Substitute the values into the formula

Substitute \( h = 6.626 \times 10^{-34} \, \text{J·s} \), \( c = 3.00 \times 10^{8} \, \text{m/s} \), and \( \lambda = 6.65 \times 10^{-7} \, \text{m} \) into the formula:

\[
E = \frac{(6.626 \times 10^{-34} \, \text{J·s})(3.00 \times 10^{8} \, \text{m/s})}{6.65 \times 10^{-7} \, \text{m}}
\]

First, calculate the numerator: \( (6.626 \times 10^{-34})(3.00 \times 10^{8}) = 1.9878 \times 10^{-25} \, \text{J·m} \)

Then, divide by the denominator: \( E = \frac{1.9878 \times 10^{-25} \, \text{J·m}}{6.65 \times 10^{-7} \, \text{m}} \approx 2.99 \times 10^{-19} \, \text{J} \)

Answer:

\( 2.99 \times 10^{-19} \)