(b) Calculate the energy of a photon of radiation whose wavelength is 413 nm.

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, where c is the speed of light (3.00 x 10^8 m/s), \lambda is the wavelength in meters, and \nu is the frequency in s^{-1}. Rearrange the equation to solve for frequency: \nu = \frac{c}{\lambda}.
3. Substitute the wavelength in meters into the equation to calculate the frequency.
4. Use Planck's equation to find the energy of the photon: E = h\nu, where E is the energy in joules, h is Planck's constant (6.626 x 10^{-34} J·s), and \nu is the frequency calculated in the previous step.
5. Substitute the frequency and Planck's constant into the equation to calculate the energy of the photon.

Key Concepts

Photon Energy

The energy of a photon is directly related to its frequency and inversely related to its wavelength. It can be calculated using the equation E = hν, where E is energy, h is Planck's constant (6.626 x 10^-34 J·s), and ν (nu) is the frequency of the radiation. The frequency can be derived from the wavelength using the speed of light equation, c = λν.

Wavelength and Frequency Relationship

Wavelength (λ) and frequency (ν) are inversely related through the speed of light (c), which is approximately 3.00 x 10^8 m/s. The relationship is expressed as c = λν, meaning that as the wavelength increases, the frequency decreases, and vice versa. This relationship is crucial for converting wavelength measurements into frequency for energy calculations.

Planck's Constant

Planck's constant is a fundamental constant in quantum mechanics that relates the energy of a photon to its frequency. It has a value of approximately 6.626 x 10^-34 J·s. This constant is essential for calculating photon energy and highlights the quantized nature of electromagnetic radiation, where energy is emitted or absorbed in discrete packets called quanta.