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

An inverse relationship exists between wavelength and frequency, and between wavelength and photon energy, while a direct relationship exists between frequency and photon energy. Photon energy is measured in joules (J), with smaller wavelengths corresponding to higher energy photons and vice versa.

Step-by-step solution

1. Key Concepts and Equations

First, let's recall the key concepts and equations relevant to the problem: - Wavelength: It is the distance between two equivalent points in a continuous wave and is represented by the symbol 'λ', measured in meters (m). - Frequency: It is the number of oscillations of a wave per unit time, represented by the symbol 'ν' (nu), and its unit is Hertz (Hz). - Photon Energy: It is the energy carried by a single photon, represented by the symbol 'E', and its unit is Joules (J). The relationship between wavelength, frequency, and the speed of light (c) is given by the equation: \(c = λν\) Where \(c\) is the speed of light in a vacuum, approximately equal to \(3.0 \times 10^8\) meters per second (m/s). The energy of a photon is given by Planck's equation: \(E = hν\) Where \(h\) is Planck's constant, approximately equal to \(6.63 \times 10^{-34}\) Joule-seconds (J.s).

2. Relationship between Wavelength and Frequency

Using the equation \(c = λν\), we can determine the relationship between wavelength and frequency by isolating one variable: If we solve for frequency (ν), we get: \(ν = \frac{c}{λ}\) From this equation, we can see that frequency is inversely proportional to wavelength. As the wavelength increases, the frequency decreases, and vice versa.

3. Relationship between Frequency and Photon Energy

Now, let's examine the relationship between frequency and photon energy using Planck's equation: \(E = hν\) This equation shows that photon energy is directly proportional to frequency. As the frequency increases, the photon energy also increases, and vice versa.

4. Relationship between Wavelength and Photon Energy

We have found the relationship between wavelength and frequency (inverse) and between frequency and photon energy (direct). To find the relationship between wavelength and photon energy, we can combine the two equations \(c = λν\) and \(E = hν\): \(E = hν\) \(E = h\frac{c}{λ}\) Here, we can observe that the photon energy is inversely proportional to the wavelength. As the wavelength increases, the photon energy decreases, and vice versa.

5. The Photon Energy Unit - Joule

The unit of photon energy is the joule (J). A single joule is equivalent to \(1\) kg.m²/s² which is also the unit of work done by a force of one Newton when it moves an object by one meter in the direction of the force. This unit is used to measure the energy of a photon, and in our context, it is important to remember that a smaller wavelength corresponds to a higher energy photon in joules, and vice versa. To conclude, there is an inverse relationship between wavelength and frequency, between wavelength and photon energy, and a direct relationship between frequency and photon energy. Photon energy is measured in joules.