1. Convert Energy to Joules per Molecule
* We're given 210 kJ/mol.
* Avogadro's number (6.022 x 10²³ molecules/mol) is the conversion factor for molecules to moles.
* Calculate energy per molecule:
(210 kJ/mol) * (1000 J/kJ) / (6.022 x 10²³ molecules/mol) = 3.49 x 10⁻¹⁹ J/molecule
2. Use the Planck-Einstein Relation
* The Planck-Einstein relation connects energy (E) and frequency (ν) of light: E = hν, where h is Planck's constant (6.626 x 10⁻³⁴ J⋅s).
* Rearrange to solve for frequency: ν = E/h
* Calculate frequency: ν = (3.49 x 10⁻¹⁹ J) / (6.626 x 10⁻³⁴ J⋅s) = 5.27 x 10¹⁴ Hz
3. Calculate Wavelength
* The relationship between frequency (ν) and wavelength (λ) is given by the equation: c = νλ, where c is the speed of light (3.00 x 10⁸ m/s).
* Rearrange to solve for wavelength: λ = c/ν
* Calculate wavelength: λ = (3.00 x 10⁸ m/s) / (5.27 x 10¹⁴ Hz) = 5.69 x 10⁻⁷ m
4. Convert to Nanometers
* 1 meter (m) = 10⁹ nanometers (nm)
* Convert wavelength to nanometers: 5.69 x 10⁻⁷ m * (10⁹ nm/m) = 569 nm
Answer: The longest wavelength of light that can break an oxygen bond in hydrogen peroxide using 210 kJ per mole is approximately 569 nm. This corresponds to the green region of the visible light spectrum.
