Question

Photosynthesis uses 660 -nm light to convert \(\mathrm{CO}_{2}\) and \(\mathrm{H}_{2} \mathrm{O}\) into glucose and \(\mathrm{O}_{2}\) . Calculate the frequency of this light

Short answer

The frequency of the 660 nm light used in photosynthesis is approximately \( 4.55 \times 10^{14} Hz \).

Step-by-step solution

Convert the wavelength to meters

Given that the wavelength is 660 nm, we need to convert it to meters before we can solve for the frequency. We know that 1 meter equals \( 10^9 \) nanometers, so we can convert as follows: \( 660\, nm \times \frac{1\, m}{10^9\, nm} = 660 \times 10^{-9}\, m = 6.6 \times 10^{-7} m \)

Find the frequency

Now that we know the wavelength (λ) in meters, we can use the formula \( c = νλ \) to solve for the frequency (ν). We are given the speed of light (c) as \( 3.0 \times 10^8 m/s \). Divide both sides of the equation by λ to solve for the frequency: \( ν = \frac{c}{λ} \) Plugging in the values we have: \( ν = \frac{3.0 \times 10^8\, m/s}{6.6 \times 10^{-7}\, m} \)

Calculate the frequency

Now, we just need to perform the division to find the frequency: \( ν ≈ \frac{3.0 \times 10^8}{6.6 \times 10^{-7}} = 4.55 \times 10^{14}\, Hz \) Therefore, the frequency of the 660 nm light used in photosynthesis is approximately \( 4.55 \times 10^{14} Hz \).

Key concepts

Wavelength to Frequency Conversion

To understand wavelength to frequency conversion, let's dive into some basics. Light exhibits wave properties, which means it has a certain wavelength (λ) and frequency (ν). The relationship between these two is crucial in understanding how light behaves. This relationship is governed by the formula:\[ c = ν ⋅ λ \] where

• \(c\) represents the speed of light (approximately \(3.0 \times 10^8\) meters per second),
• \(ν\) is the frequency, and
• \(λ\) is the wavelength.

To find the frequency when the wavelength is known, you rearrange the formula to solve for frequency:\[ ν = \frac{c}{λ} \] This formula shows that the frequency is inversely proportional to the wavelength. This means as the wavelength increases, the frequency decreases and vice versa. Converting between these two using the speed of light is essential in various scientific applications, including understanding how photosynthesis uses light to synthesize food in plants.

Speed of Light

The speed of light, denoted as \(c\), is a fundamental constant in physics. It is the speed at which light waves propagate through a vacuum. The value is approximately \(3.0 \times 10^8\) meters per second, which is incredibly fast. This constant forms the backbone of many calculations and is essential when working with electromagnetic waves like light.

By knowing the speed of light, you can calculate how light waves will behave under different circumstances. For instance, when light waves interact with matter, such as during photosynthesis, understanding the speed of light helps us understand how the energy is transferred from light to chemical bonds.

This understanding is pivotal in studying various natural phenomena and technologies including optics, lasers, and telecommunications. Because light travels so fast, it provides rapid communication and efficiency in processes that rely on it.

Nanometers to Meters Conversion

Converting units is a fundamental skill in science. When dealing with light, wavelengths are often given in nanometers (nm). However, scientific calculations frequently require meters. That's why it's important to know how to convert nanometers to meters.

One meter is equal to one billion nanometers (\(10^9\) nm). Therefore, to convert nanometers to meters, you divide the number of nanometers by \(10^9\). This is especially helpful in scientific realms where precision matters, such as physics and chemistry.

For example, if you have a wavelength of 660 nm, the conversion would be:\[ 660\, nm \times \frac{1\, m}{10^9\, nm} = 660 \times 10^{-9}\, m = 6.6 \times 10^{-7} m \] The correct conversion ensures that calculations for properties like frequency are accurate. Each correct measurement feeds into larger scientific understanding, like calculating the energy photons carry or assessing how they interact in natural processes such as photosynthesis.