Question

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

Short answer

For part (a), the frequency is approximately \( 6.58 \times 10^{14} Hz \).For part (b), the wavelength is approximately \( 122.45 nm \).

Step-by-step solution

Converting \( \lambda \) from nm to m (for part a)

You start by converting wavelength from nanometers to meters using the conversion factor \( 1m = 10^9 nm \). So, for a wavelength of \(456 nm\), in metres would be \( 456 \times 10^{-9} m \).

Finding the frequency (for part a)

Substitute the given values into the equation from Step 1. So, rearranging the equation to solve for \( f \), we get \( f = \frac{v}{\lambda} \). Substituting \(v = 3.0 \times 10^8 m/s\) and \(\lambda = 456 \times 10^{-9} m\), you solve to find the value of \(f\).

Calculating wavelength from given frequency (for part b)

Similarly, for the second part, substitute the given values for the frequency and speed of light into \( v = \lambda f \) and solve for \( \lambda \) to find the wavelength. Make sure to convert the frequency from Hz to 1/s (remembering that 1 Hz = 1 1/s) before substitution.

Conversion from m to nm (for part b)

As the final step, you need to convert the wavelength from meters back to nanometers using the conversion \( 1m = 10^9 nm \). This gives the answer in the desired unit.

Key concepts

Wavelength to Frequency Conversion

Understanding the relationship between wavelength and frequency is fundamental in chemistry, particularly in spectroscopy and quantum mechanics. The formula that connects these two quantities is given by the equation:
\[ f = \frac{v}{\lambda} \]
where:\

\
• \( f \) is the frequency in hertz (Hz),\
• \( v \) is the speed of light in meters per second (m/s),\
• \( \lambda \) is the wavelength in meters (m).\

In this context, 'speed of light' (\( v \)) is usually a constant value of approximately \(3.0 \times 10^8 m/s\) in a vacuum. When you know the wavelength of light, you can rearrange this formula to solve for frequency, as seen in the exercise's solution.To make this concept clearer, let's take the example provided where wavelength is given in nanometers (nm). Since frequency requires wavelength to be in meters, conversion is the first step, which is simply dividing the value in nanometers by \(10^9\). Then, using the equation, we divide the speed of light by this converted wavelength to find the frequency. This process is particularly useful in identifying the type of radiation based on its wavelength, from visible light to microwaves and radio waves.
Understanding how to manipulate this equation is crucial for students and helps in a wide range of applications like calculating the energy of photons or analyzing spectral lines.

Speed of Light Equation

The speed of light equation is a cornerstone of physics and underpins much of chemistry, especially in the study of electromagnetic radiation. It states that:
\[ c = \lambda f \]
where:\

\
• \( c \) represents the speed of light,\
• \( \lambda \) is the wavelength,\
• \( f \) is the frequency.\

The numerical value of the speed of light in a vacuum is approximately \(3.0 \times 10^8 m/s\), and it serves as a link between the frequency and the wavelength of electromagnetic waves.In the context of the exercise provided, the speed of light equation allows us to find the wavelength once the frequency is known, as demonstrated in the second part of the problem. Here, students must remember that the value of the speed of light is a constant and that manipulating the equation by isolating the desired variable is the key to finding the answer.
This equation not only assists in simple calculations but is also integral to understanding more complex concepts such as the behavior of light when it interacts with matter or even the theory of relativity.

Unit Conversion in Chemistry

Unit conversion might seem a straightforward task, yet it's a crucial skill that ensures accuracy in chemical calculations. Chemistry often requires the conversion between different units of measurement due to the diverse scales on which chemical processes occur. In our exercise, the conversion between nanometers (nm) and meters (m) is paramount when dealing with wavelengths.
As a rule of thumb, \(1\) meter equals \(10^9\) nanometers. This means that to convert from nanometers to meters, you divide the number of nanometers by \(10^9\), and to convert from meters to nanometers, you multiply the number of meters by \(10^9\).

Important Conversion Factors

\
• 1 meter (m) = \(10^9\) nanometers (nm),\
• 1 kilogram (kg) = \(10^3\) grams (g),\
• 1 mole = Avogadro's number \(6.022 \times 10^{23}\) of particles.\

These conversions are not just simple arithmetic; they often involve significant figures and can affect the precision of the results. Thus, students should practice unit conversions diligently to become competent in solving chemistry problems accurately and effectively.
An understanding of how to correctly convert units, along with a clear sense of when it is necessary, will certainly enhance a student's ability to work through an array of problems in chemistry, physics, and beyond. Accuracy in unit conversion can often make the difference between a correct solution and an erroneous one, especially in scientific calculations where precision is paramount.