Calculate the energy in kJ/mol of light with a wavelength of 360 nm.
The Correct Answer and Explanation is:
To calculate the energy in kilojoules per mole (kJ/mol) of light with a wavelength of 360 nm, we can use the following formula:E=hcλE = \frac{hc}{\lambda}E=λhc
Where:
- EEE is the energy of a single photon in joules
- h=6.626×10−34 J\cdotpsh = 6.626 \times 10^{-34} \, \text{J·s}h=6.626×10−34J\cdotps (Planck’s constant)
- c=3.00×108 m/sc = 3.00 \times 10^8 \, \text{m/s}c=3.00×108m/s (speed of light)
- λ=360 nm=360×10−9 m\lambda = 360 \, \text{nm} = 360 \times 10^{-9} \, \text{m}λ=360nm=360×10−9m
Step 1: Energy of one photon
E=(6.626×10−34)(3.00×108)360×10−9=5.52×10−19 JE = \frac{(6.626 \times 10^{-34}) (3.00 \times 10^8)}{360 \times 10^{-9}} = 5.52 \times 10^{-19} \, \text{J}E=360×10−9(6.626×10−34)(3.00×108)=5.52×10−19J
Step 2: Convert to kJ/mol
One mole of photons contains Avogadro’s number of photons:6.022×1023 photons/mol6.022 \times 10^{23} \, \text{photons/mol}6.022×1023photons/molEnergy per mole=(5.52×10−19 J)×(6.022×1023)=3.32×105 J/mol\text{Energy per mole} = (5.52 \times 10^{-19} \, \text{J}) \times (6.022 \times 10^{23}) = 3.32 \times 10^5 \, \text{J/mol}Energy per mole=(5.52×10−19J)×(6.022×1023)=3.32×105J/mol
Convert joules to kilojoules:3.32×105 J/mol=332 kJ/mol3.32 \times 10^5 \, \text{J/mol} = 332 \, \text{kJ/mol}3.32×105J/mol=332kJ/mol
Final Answer: 332 kJ/mol
Explanation
Light behaves both like a wave and a particle. Each particle of light, known as a photon, carries a discrete amount of energy that depends on the wavelength. Shorter wavelengths correspond to higher energy photons, while longer wavelengths have lower energy.
The energy of a photon is calculated using the equation E=hcλE = \frac{hc}{\lambda}E=λhc, where hhh is Planck’s constant, ccc is the speed of light, and λ\lambdaλ is the wavelength. Because the wavelength in this question is given in nanometers, it must be converted to meters for consistency with the SI units used in the constants. Since 1 nanometer is equal to 1×10−91 \times 10^{-9}1×10−9 meters, 360 nm becomes 3.60×10−73.60 \times 10^{-7}3.60×10−7 meters.
Using the formula, we find that a single photon of 360 nm light has an energy of approximately 5.52×10−195.52 \times 10^{-19}5.52×10−19 joules. While this value seems small, it is for just one photon. In chemistry, we are usually interested in the energy per mole, so we multiply the single-photon energy by Avogadro’s number, 6.022×10236.022 \times 10^{23}6.022×1023, to get the total energy in joules per mole. This gives about 3.32×1053.32 \times 10^53.32×105 joules per mole.
Finally, we convert joules to kilojoules by dividing by 1000. This yields a final result of 332 kilojoules per mole, which is a reasonable value for ultraviolet light. This energy is high enough to break chemical bonds, which is why UV light can cause damage to living tissues.
