Problem 55 - Chemistry and Chemical Reactivity

Question

An advertising sign gives off red light and green light. (a) Which light has higher-energy photons? (b) One of the colors has a wavelength of \(680 \mathrm{nm}\) and the other has a wavelength of \(500 \mathrm{nm}\). Which color has which wavelength? (c) Which light has the higher frequency?

Step-by-Step Solution

Answer

(a) Green light has higher-energy photons. (b) Green: 500 nm, Red: 680 nm. (c) Green light has the higher frequency.

Step 1: Understanding Photon Energy

The energy of a photon is inversely related to its wavelength. The shorter the wavelength, the higher the energy. Therefore, we need to identify which color has a shorter wavelength.

Step 2: Color Identification

In the visible spectrum, red light has a longer wavelength compared to green light. Thus, green light typically has wavelengths around 500 nm, while red light has wavelengths around 680 nm. Therefore, red light is at 680 nm and green light is at 500 nm.

Step 3: Determining Higher-Energy Photons

Since photon energy is inversely related to wavelength (as given by the formula: \( E = \frac{hc}{\lambda} \), where \( h \) is Planck's constant and \( c \) is the speed of light), green light with a wavelength of 500 nm has higher-energy photons than red light with 680 nm wavelength.

Step 4: Frequency and Wavelength Relationship

The frequency of light is inversely related to its wavelength according to the formula \( f = \frac{c}{\lambda} \). Therefore, a shorter wavelength results in a higher frequency. Since green light has a shorter wavelength (500 nm), it also has a higher frequency.

Key Concepts

Wavelength

Wavelength is a fundamental property of waves, including light. It is the distance between consecutive peaks (or troughs) in a wave.

• Measured usually in nanometers (nm) or meters (m).
• Denoted by the Greek letter lambda, \( \lambda \).
• Different wavelengths correspond to different colors in the visible spectrum.

Light with shorter wavelengths has higher energy, while longer wavelengths have lower energy. For example, in the visible light spectrum, red light has a longer wavelength compared to green light. This means green light — having a shorter wavelength — carries more energy than red light. Wavelength is inversely proportional to both the energy and frequency as described by the formula:\[ E = \frac{hc}{\lambda} \] where \( E \) is the energy, \( h \) is Planck's constant, and \( c \) is the speed of light.

Frequency

Frequency refers to how many wave cycles pass a particular point per second. It is measured in Hertz (Hz), where 1 Hz equals one cycle per second.

• Denoted by the letter \( f \).
• Inversely related to wavelength: shorter wavelengths correspond to higher frequencies.

For light, the frequency determines the color within the visible spectrum. Using the equation:\[ f = \frac{c}{\lambda} \] where \( c \) is the speed of light and \( \lambda \) is the wavelength, we understand that green light, due to its shorter wavelength as compared to red light, has a higher frequency. This means the waves of green light pass a point more frequently within a given time span.

Visible Spectrum

The visible spectrum is the portion of the electromagnetic spectrum that can be seen by the human eye. It ranges approximately from 380 nm (violet) to 750 nm (red).

• Contains all the colors perceived by the human eye, such as violet, blue, green, yellow, orange, and red.
• Each color corresponds to a specific range of wavelengths.

Within the visible spectrum, colors with shorter wavelengths, like violet and blue, appear closer to the beginning, whereas colors like red have longer wavelengths and appear towards the end. In the context of the given exercise, green light (wavelength around 500 nm) and red light (wavelength of 680 nm) are part of this spectrum. Green light, with its shorter wavelength, is more energetic than red, which is why it appears at the higher-energy end of the visible spectrum.

Planck's Constant

Planck's constant is a fundamental value used to describe the sizes of quanta in quantum mechanics. Denoted by \( h \), its value is approximately \( 6.626 \times 10^{-34} \text{Js} \).

• Essential in calculating the energy of photons.
• Appears in the equation \( E = hf \), where \( E \) is energy and \( f \) is frequency.

Furthermore, it also appears in the equation \( E = \frac{hc}{\lambda} \), linking energy to the wavelength of light. Using this constant, one can determine the energy carried by a photon of a specific wavelength or frequency.Planck's constant plays a crucial role in understanding interactions at the quantum level, and it underlines the principle that energy increments are quantized. It's not just abstract—it's an everyday practical vital link to phenomena such as calculating the energy of light in the exercise's context.