If light has a wavelength of 585 nm, what is the energy of this light expressed with units of kJ/mol? A beam of radiation has an energy of 3.11×10^2 kJ/mol. What is the wavelength of this light
The Correct Answer and Explanation is:
To answer both parts of this question, we’ll use two key equations:
1. Energy of a photon:
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 is Planck’s constant,
- c=3.00×108 m/sc = 3.00 \times 10^8 \, \text{m/s}c=3.00×108m/s is the speed of light,
- λ\lambdaλ is the wavelength in meters.
To find energy per mole of photons (kJ/mol), we multiply the result by Avogadro’s number:NA=6.022×1023 mol−1N_A = 6.022 \times 10^{23} \, \text{mol}^{-1}NA=6.022×1023mol−1
Part 1: Energy of light with wavelength 585 nm
Convert 585 nm to meters:585 nm=585×10−9 m585 \, \text{nm} = 585 \times 10^{-9} \, \text{m}585nm=585×10−9m
Now plug into the equation:E=6.626×10−34×3.00×108585×10−9=3.398×10−19 JE = \frac{6.626 \times 10^{-34} \times 3.00 \times 10^8}{585 \times 10^{-9}} = 3.398 \times 10^{-19} \, \text{J}E=585×10−96.626×10−34×3.00×108=3.398×10−19J
Energy per mole:Emol=3.398×10−19×6.022×1023=204.5 kJ/molE_{\text{mol}} = 3.398 \times 10^{-19} \times 6.022 \times 10^{23} = 204.5 \, \text{kJ/mol}Emol=3.398×10−19×6.022×1023=204.5kJ/mol
Answer: 204.5 kJ/mol
Part 2: Wavelength of light with energy 311 kJ/mol
Convert kJ/mol to J per photon:311×1036.022×1023=5.164×10−19 J\frac{311 \times 10^3}{6.022 \times 10^{23}} = 5.164 \times 10^{-19} \, \text{J}6.022×1023311×103=5.164×10−19J
Use the photon energy formula solved for λ\lambdaλ:λ=hcE=6.626×10−34×3.00×1085.164×10−19=3.85×10−7 m\lambda = \frac{hc}{E} = \frac{6.626 \times 10^{-34} \times 3.00 \times 10^8}{5.164 \times 10^{-19}} = 3.85 \times 10^{-7} \, \text{m}λ=Ehc=5.164×10−196.626×10−34×3.00×108=3.85×10−7m
Convert to nanometers:3.85×10−7 m=385 nm3.85 \times 10^{-7} \, \text{m} = 385 \, \text{nm}3.85×10−7m=385nm
Answer: 385 nm
Explanation
The energy of light is inversely related to its wavelength. This means that shorter wavelengths carry more energy per photon, while longer wavelengths carry less. To compute the energy of light from its wavelength, we use the equation E=hc/λE = hc/\lambdaE=hc/λ, where hhh is Planck’s constant and ccc is the speed of light. These two constants allow us to convert a light wave’s length into the energy associated with a single photon.
In the first part, we converted the given wavelength of 585 nm into meters and calculated the energy of one photon. Since chemists typically work with moles, we scaled this energy up by multiplying by Avogadro’s number, giving us the energy per mole of photons, which came out to approximately 204.5 kJ/mol.
In the second part, we worked the process in reverse. Given the energy in kJ/mol, we first converted this to joules per photon by dividing by Avogadro’s number. With that energy per photon, we then rearranged the same formula to solve for wavelength. The result was 385 nm, which lies in the visible light spectrum, close to violet.
These relationships between energy, wavelength, and frequency help explain various phenomena in chemistry and physics, from how molecules absorb light to how solar panels generate electricity. Understanding these conversions is fundamental in spectroscopy and photochemistry.
