Problem 8 - Chemistry

Question

(a) What is the frequency of light having a wavelength of \(456 \mathrm{nm} ?\) (b) What is the wavelength (in \(\mathrm{nm}\) ) of radiation having a frequency of \(2.45 \times 10^{9} \mathrm{~Hz}\) ? (This is the type of radiation used in microwave ovens.)

Step-by-Step Solution

Answer

The frequency of light having a wavelength of \(456 nm\) is approximately \(6.58 \times 10^{14} Hz\). The wavelength of radiation having a frequency of \(2.45 \times 10^{9} Hz\) is approximately \(122.5 nm\).

Step 1 - Finding the Frequency

Using the relation \(c = \lambda \nu\), we want to solve for \(\nu\). So, we get \(\nu = \frac{c}{\lambda}\). We know that \(\lambda = 456 nm = 456 \times 10^{-9} m\) and \(c \approx 3.0 \times 10^8 m/s\). Substituting these values, we get: \(\nu = \frac{3.0 \times 10^{8}}{456 \times 10^{-9}} Hz\).

Step 2 - Finding the Wavelength

In this part, we want to find the wavelength \(\lambda\) using the same formula. Thus, we obtain \(\lambda = \frac{c}{\nu}\). Given that \(c \approx 3.0 \times 10^8 m/s\) and \(\nu = 2.45 \times 10^9 Hz\), substituting these values in the equation gives us: \(\lambda = \frac{3.0 \times 10^{8}}{2.45 \times 10^{9}} m\). As the wavelength is asked in \(nm\), we convert the result to nanometers by multiplying it by \(10^9\).

Step 3 - Calculating and Providing the Final Answers

By doing the actual calculations in Steps 1 and 2, we obtain the numerical values for the frequency and the wavelength, which represent the final answers to the problems.

Key Concepts

Frequency Calculation

Electromagnetic radiation, such as light, can be characterized by its frequency, which tells us how often the wave oscillates per second. To find the frequency (\(u\)), we can use the formula \[u = \frac{c}{\lambda}\]where \(c\) is the speed of light (approximately \(3.0 \times 10^8 \text{ m/s}\)) and \(\lambda\) is the wavelength.
In practical terms, if you know the wavelength of a light wave, you can determine its frequency by dividing the speed of light by the wavelength:

• Ensure the wavelength is in meters before calculation by converting nanometers to meters (\(1 \text{ nm} = 10^{-9} \text{ m}\)).
• Plug the speed of light and the converted wavelength into the formula.

For example, if the wavelength is \(456 \text{ nm}\), convert it to meters, giving \(456 \times 10^{-9} \text{ m}\), and use the formula to calculate the frequency:
\[u = \frac{3.0 \times 10^8}{456 \times 10^{-9}} \text{ Hz}\]This calculation gives you the frequency of the light wave.

Wavelength Calculation

Just like frequency, the wavelength of electromagnetic radiation can also be calculated if the frequency is known. The formula used is the same:\[\lambda = \frac{c}{u}\]where again \(c\) is the speed of light, and \(u\) is the frequency of the wave.
When given a frequency, simply divide the speed of light by the frequency to find the wavelength.

• First, verify that the frequency is in hertz (Hz).
• Input both the speed of light and the frequency into the formula.
• Remember to convert the final wavelength from meters to nanometers if necessary, by multiplying by \(10^9\).

For example, for a frequency of \(2.45 \times 10^{9} \text{ Hz}\) (like in a microwave oven), the calculated wavelength will be:\[\lambda = \frac{3.0 \times 10^8}{2.45 \times 10^9} \text{ m}\]Then, convert the result to nanometers:
\[ \lambda = \left(\frac{3.0 \times 10^8}{2.45 \times 10^9}\right) \times 10^9 \text{ nm}\]

Speed of Light

The speed of light (\(c\)) is a fundamental constant utilized in many calculations in physics, particularly those involving electromagnetic waves. It tells us how quickly light travels in a vacuum and is approximately equal to \(3.0 \times 10^8 \text{ meters per second (m/s)}\).
This constant is critical for calculations like those involving frequency and wavelength since it links these two dependent properties of a wave. When solving problems involving light, such as frequency and wavelength computations, use this value for accurate results.

• It serves as the basis for converting between the speed, frequency, and wavelength of light.
• Understanding the speed of light helps clarify how fast electromagnetic waves propagate and how the wavelength and frequency are interrelated.
• The speed of light allows us to determine how changes in one property (e.g., wavelength) will inversely affect another (e.g., frequency).