Question

What type of relationship (direct or inverse) exists between wavelength, frequency, and photon energy? What does a photon energy unit of a joule equal?

Short answer

The relationship between wavelength, frequency, and photon energy is described by the equation \(E = h \cdot f\), where \(E\) is photon energy, \(h\) is Planck's constant, and \(f\) is frequency. The relationship between photon energy and frequency is direct, while the relationship between photon energy and wavelength is inverse. A photon energy unit of a joule (J) represents the energy of producing one wave cycle per second (1 Hz) of a photon in the electromagnetic spectrum, but it is more common to encounter photon energies in units like electron volts (eV) instead of joules, with the conversion factor \(1 \, eV \approx 1.602 \times 10^{-19} \, J\).

Step-by-step solution

Define wavelength, frequency, and photon energy.

Wavelength (\(λ\)) is the distance between two consecutive points in a wave, such as the distance between two peaks. Frequency (\(f\)) is the number of wave cycles that pass a given point per unit of time, measured in hertz (Hz). Photon energy (\(E\)) refers to the energy carried by a single photon, which is directly proportional to its frequency and inversely proportional to its wavelength.

Describe the relationship between wavelength, frequency, and photon energy.

The relationship between these three variables is given by the equation: \[E = h \cdot f\] Here, \(E\) is the photon energy, \(h\) is the Planck's constant (\(6.626 \times 10^{-34} \, Js\)), and \(f\) is the frequency of the photon. Since the speed of light (\(c\)) is constant (\(3.00 \times 10^8 \, m/s\)), and the speed of light is equal to the product of wavelength and frequency (\(c = λ \cdot f\)), we can rewrite frequency as: \[f = \frac{c}{λ}\] Now, substituting this expression into the photon energy equation, we get: \[E = h \cdot \frac{c}{λ}\]

Identify the type of relationship between wavelength, frequency, and photon energy.

From the equation \(E = h \cdot f\), we can see that the relationship between photon energy and frequency is direct, meaning that as the frequency increases, the photon energy increases, and vice versa. From the equation \(E = h \cdot \frac{c}{λ}\), we can observe that the relationship between photon energy and wavelength is inverse. This means that as the wavelength increases, the photon energy decreases, and vice versa.

Explain what a photon energy unit of a joule equals.

A photon energy unit of a joule (J) is the energy required to produce one wave cycle per second (1 Hz) of a photon in the electromagnetic spectrum. In the context of photon energy, this unit is typically quite large, as photon energies are often found to be in extremely small values. Therefore, it is more common to encounter photon energies expressed in units like electronvolts (eV) rather than joules. To convert photon energy from joules to electron volts, we use the conversion factor \(1 \, eV \approx 1.602 \times 10^{-19} \, J\).