The wavelength of a particular color of yellow light is 584 nm. The energy of this wavelength of light is kJ/photon. (109 nm = 1 m)
The correct answer and explanation is :
To calculate the energy of a photon with a wavelength of 584 nm (nanometers), we can use the formula:
[ E = \frac{h \cdot c}{\lambda} ]
Where:
- ( E ) is the energy of the photon.
- ( h ) is Planck’s constant, approximately ( 6.626 \times 10^{-34} \, \text{J} \cdot \text{s} ).
- ( c ) is the speed of light, approximately ( 3.00 \times 10^{8} \, \text{m/s} ).
- ( \lambda ) is the wavelength of the light in meters.
Step 1: Convert the wavelength to meters.
Given that ( 1 \, \text{nm} = 10^{-9} \, \text{m} ), a wavelength of 584 nm is:
[ \lambda = 584 \, \text{nm} \times 10^{-9} \, \frac{\text{m}}{\text{nm}} = 5.84 \times 10^{-7} \, \text{m} ]
Step 2: Calculate the energy in joules.
Substitute the known values into the energy formula:
[ E = \frac{(6.626 \times 10^{-34} \, \text{J} \cdot \text{s}) \times (3.00 \times 10^{8} \, \text{m/s})}{5.84 \times 10^{-7} \, \text{m}} ]
Performing the calculation:
[ E \approx 3.41 \times 10^{-19} \, \text{J} ]
Step 3: Convert the energy to kilojoules.
Since ( 1 \, \text{kJ} = 10^{3} \, \text{J} ):
[ E = 3.41 \times 10^{-19} \, \text{J} \times \frac{1 \, \text{kJ}}{10^{3} \, \text{J}} = 3.41 \times 10^{-22} \, \text{kJ} ]
Conclusion:
The energy of a photon with a wavelength of 584 nm is approximately ( 3.41 \times 10^{-22} \, \text{kJ} ).
Explanation:
The energy of a photon is inversely proportional to its wavelength; that is, shorter wavelengths correspond to higher energy photons. This relationship is described by Planck’s equation, which links the energy of a photon to its frequency (and thus its wavelength) through Planck’s constant and the speed of light. Understanding this relationship is fundamental in fields like spectroscopy and quantum mechanics, where the energy of photons determines their interaction with matter.