Problem 53 - Chemistry and Chemical Reactivity

Question

An advertising sign gives off red light and green light. (a) Which light has the 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; (b) Red: 680 nm, Green: 500 nm; (c) Green light.

Step 1: Understanding Photon Energy

The energy of a photon is given by the equation \(E = \frac{hc}{\lambda}\), where \(E\) is the energy, \(h\) is Planck's constant, \(c\) is the speed of light, and \(\lambda\) is the wavelength. Photons with shorter wavelengths have higher energy because energy is inversely proportional to wavelength.

Step 2: Identifying Energy of Red and Green Light

Red light has a longer wavelength compared to green light. Consequently, according to the equation \(E = \frac{hc}{\lambda}\), the green light has higher-energy photons than red light because it has a shorter wavelength.

Step 3: Matching Wavelengths to Colors

Typically, red light has wavelengths ranging from roughly 620 nm to 750 nm, while green light has wavelengths from 495 nm to 570 nm. Therefore, the 680 nm wavelength corresponds to red light, and the 500 nm wavelength corresponds to green light.

Step 4: Calculating Frequency

The frequency \(f\) of a wave is given by the equation \(f = \frac{c}{\lambda}\). A shorter wavelength results in a higher frequency. Therefore, green light with a wavelength of 500 nm has a higher frequency than red light with a wavelength of 680 nm.

Key Concepts

Wavelength and Frequency

To dip our toes into the world of light, it's crucial to understand the relationship between wavelength and frequency. Light behaves both as a wave and a particle. When we talk about its wave nature, the key characteristics to consider are its wavelength and frequency.

Wavelength (\( \lambda \)) is the distance between two consecutive peaks of a wave. Different colors of light have different wavelengths. For example, red light has a longer wavelength compared to green light. This difference is what allows us to see these colors as distinct from one another.

Frequency (\( f \)), on the other hand, is the number of waves passing a point in one second. It's measured in Hertz (Hz). There's an inverse relationship between the wavelength and frequency, given by the equation \( f = \frac{c}{\lambda} \). Here,\( c \) refers to the speed of light, which is a constant value of approximately \(3 \, \times \, 10^8 \, \text{m/s} \).

So, as wavelength increases, frequency decreases and vice-versa. Thus, green light, having a shorter wavelength than red light, also has a higher frequency.

Red and Green Light

The colors red and green aren't just distinct to our eyes, they differ in terms of their light properties as well. These colors exist on different segments of the visible spectrum. Understanding their wavelengths is crucial for identifying their energy levels.

Red light typically has a longer wavelength, usually ranging from 620 nm to 750 nm. In our exercise, the red light has a wavelength of 680 nm. Green light falls within a wavelength spectrum of 495 nm to 570 nm. Here, it measures at 500 nm.

• The longer the wavelength of light, the less energy its photons possess.
• In contrast, the shorter the wavelength, the more energy the photons have.

The implications of these differences become apparent when considering photon energy, with shorter wavelengths - like green light - being associated with higher energy photons compared to longer wavelengths, such as red light.

Planck's Equation

One of the key equations to understand the energy of light is Planck's Equation. This equation establishes a direct relationship between the energy of a photon and its wavelength. The energy \(E\) of a photon is given by:\[ E = \frac{hc}{\lambda} \]Here:

• \(h\) is Planck's constant (\(6.626 \, \times \, 10^{-34} \, \text{Js}\)),
• \(c\) is the speed of light (\(3 \, \times \, 10^8 \, \text{m/s}\)),
• \(\lambda\) is the wavelength of the photon.

According to this formula, as the wavelength \(\lambda\) decreases, the energy \(E\) increases. This is why photons of green light, which have a shorter wavelength, possess more energy than photons of red light.

Using Planck's Equation helps to explain and calculate the differing energy levels of various colors of light, forming the basis for understanding photon energy in physics and chemistry.