What is the energy (in kJ) of one mole of photons of ultraviolet light with a wavelength of 85 nm? (LO 5.3) (a) 1.4 * 10-6 kJ (b) 1.4 * 103 kJ (c) 2.4 * 1014 kJ (d) 2.4 * 10-15 kJ

Verified step by step guidance

1. Convert the wavelength from nanometers to meters by using the conversion factor: 1 nm = 1 x 10^{-9} m.
2. Use the speed of light equation, c = \lambda \nu, to find the frequency (\nu) of the light, where c is the speed of light (3.00 x 10^8 m/s) and \lambda is the wavelength in meters.
3. Apply Planck's equation, E = h\nu, to calculate the energy of a single photon, where h is Planck's constant (6.626 x 10^{-34} J·s).
4. Convert the energy from joules to kilojoules by using the conversion factor: 1 kJ = 1000 J.
5. Multiply the energy of a single photon by Avogadro's number (6.022 x 10^{23} mol^{-1}) to find the energy of one mole of photons.

Key Concepts

Photon Energy

The energy of a photon is directly related to its frequency and inversely related to its wavelength. This relationship is described by the equation E = hν, where E is energy, h is Planck's constant (6.626 x 10^-34 J·s), and ν is the frequency of the light. Since frequency and wavelength are inversely related (ν = c/λ), where c is the speed of light, this allows us to calculate the energy of photons based on their wavelength.

Wavelength and Frequency

Wavelength (λ) is the distance between successive peaks of a wave, while frequency (ν) is the number of waves that pass a point in one second. The speed of light (c) is constant at approximately 3.00 x 10^8 m/s, and the relationship between these quantities is given by the equation c = λν. For ultraviolet light with a wavelength of 85 nm, converting this to meters is essential for accurate calculations.

Unit Conversion

In chemistry, it is often necessary to convert units to ensure consistency and accuracy in calculations. In this case, energy is typically expressed in kilojoules (kJ), while photon energy calculations yield results in joules (J). To convert joules to kilojoules, one must divide by 1,000. Understanding how to perform these conversions is crucial for arriving at the correct answer in energy-related problems.