Calculate the energy of a photon of electromagnetic radiation at each of the wavelengths indicated in Problem 39. a. 632.8 nm (wavelength of red light from helium–neon laser) b. 503 nm (wavelength of maximum solar radiation) c. 0.052 nm (wavelength contained in medical X-rays)

Verified step by step guidance

1. 1. The energy of a photon can be calculated using the formula: E = h * c / λ, where E is the energy, h is Planck's constant (6.626 x 10^-34 J*s), c is the speed of light (3.00 x 10^8 m/s), and λ is the wavelength. Note that the wavelength must be in meters for the units to cancel out correctly.
2. 2. Convert the given wavelengths from nanometers to meters. Remember that 1 nm = 1 x 10^-9 m. So, for example, 632.8 nm would be 632.8 x 10^-9 m.
3. 3. Substitute the values of h, c, and λ into the formula for each of the given wavelengths. For example, for a wavelength of 632.8 nm, the calculation would be E = (6.626 x 10^-34 J*s) * (3.00 x 10^8 m/s) / (632.8 x 10^-9 m).
4. 4. Repeat the calculation for the other given wavelengths (503 nm and 0.052 nm), remembering to convert the wavelengths to meters before substituting into the formula.
5. 5. The results of these calculations will give the energy of a photon at each of the given wavelengths. Remember that the energy will be in joules (J) because of the units of Planck's constant (J*s) and the speed of light (m/s).

Key Concepts

Photon Energy

The energy of a photon is directly proportional to its frequency and inversely proportional 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 (nu) is the frequency of the electromagnetic radiation. Since frequency and wavelength are related by the speed of light (c = λν), this can also be expressed as E = hc/λ, allowing for energy calculations based on wavelength.

Electromagnetic Spectrum

The electromagnetic spectrum encompasses all types of electromagnetic radiation, ranging from radio waves to gamma rays. Each type of radiation is characterized by its wavelength and frequency. In the context of the question, visible light (like red and green light) and X-rays are part of this spectrum, with X-rays having much shorter wavelengths and higher energy compared to visible light.

Units of Measurement

In calculations involving photon energy, it is crucial to use consistent units. Wavelength is often given in nanometers (nm), where 1 nm = 10^-9 meters. Energy is typically expressed in joules (J). When calculating energy from wavelength, ensure to convert units appropriately, such as converting nm to meters, to maintain accuracy in the results.