Problem 1 - Chemistry Matter and Change

Question

Objects get their colors from reflecting only certain wavelengths when hit with white light. Light reflected from a green leaf is found to have a wavelength of \(4.90 \times 10^{-7} \mathrm{m} .\) What is the frequency of the light?

Step-by-Step Solution

Answer

The frequency of the light is approximately \(6.12 \times 10^{14}\) Hz.

Step 1: Understand the Relationship

The frequency of light \(f\) and the wavelength \(\lambda\) are related through the speed of light \(c\) using the formula \(c = f \cdot \lambda\). The speed of light \(c\) is approximately \(3.00 \times 10^8\) meters per second.

Step 2: Rearrange the Formula

We need to find the frequency \(f\). By rearranging the formula \(c = f \cdot \lambda\), we obtain \(f = \frac{c}{\lambda}\).

Step 3: Substitute Given Values

Substitute the known values into the rearranged formula. \(\lambda = 4.90 \times 10^{-7} \text{ m}\) and \(c = 3.00 \times 10^8 \text{ m/s}\). Thus, the formula becomes \(f = \frac{3.00 \times 10^8}{4.90 \times 10^{-7}}\).

Step 4: Calculate the Frequency

Perform the division to find the frequency: \(f = \frac{3.00 \times 10^8}{4.90 \times 10^{-7}} \approx 6.12 \times 10^{14} \text{ Hz}\).

Key Concepts

Speed of Light

The speed of light is a fundamental constant in physics. It represents how fast light travels through a vacuum, which is quite fast! Light moves at approximately \(3.00 \times 10^8\) meters per second. This is an essential figure because it links the two properties of light: wavelength and frequency.

When scientists talk about the speed of light, it helps in understanding how light behaves when it interacts with different materials. The speed of light is not just important for the calculations; it is also a backbone of many theories in physics, particularly in Einstein's theory of relativity.

• Light travels quickly, taking only about 8.3 minutes to reach Earth from the Sun.
• It is used in calculations involving energy, wavelength, and frequency.
• Often abbreviated as \(c\), from the Latin word "celeritas" meaning speed.

Understanding the speed of light provides insight into how light waves move and interact with the objects they encounter.

Calculating Frequency

Calculating frequency involves understanding its relationship with wavelength and the speed of light. Frequency, denoted as \(f\), tells us how many wave cycles occur per second. It is measured in Hertz (Hz), which means cycles per second.

To find the frequency of light when you know its wavelength, you rearrange the formula \(c = f \cdot \lambda\), where \(\lambda\) is the wavelength. Rearranging gives us the formula: \(f = \frac{c}{\lambda}\). This step is key in calculations involving light.

• Start with a known wavelength and the constant speed of light.
• Apply the formula by dividing the speed of light by the wavelength.
• This provides the frequency, offering insights into the energy had by the reflected light.

Practicing this calculation helps you understand how light of different wavelengths behaves, such as the green light reflected from a leaf.

Wavelength of Light

The wavelength of light is the distance between consecutive peaks of a wave. It is typically expressed in meters (m), and different wavelengths correspond to different colors. For instance, the green color we see from a leaf has a specific wavelength in the visible light spectrum.

The wavelength of visible light ranges approximately from 400 nm to 700 nm (nanometers). A green leaf might reflect a wavelength around \(4.90 \times 10^{-7}\) meters, which falls within the visible spectrum. This wavelength corresponds to a color we perceive as green.

• Visible light ranges across many colors, each with its wavelength.
• Wavelength determines the energy and color of the light.
• Shorter wavelengths have higher frequencies and typically more energy.

Knowing the wavelength can help predict the color of the light, just like how we can identify that a particular wavelength reflects as the green seen in leaves.