Problem 39 - Chemistry: Structure and Properties

Question

Calculate the frequency of each wavelength of electromagnetic radiation: a. 632.8 nm (wavelength of red light from helium neon laser) b. 503 nm (wavelength of maximum solar radiation) c. 0.052 nm (wavelength contained in medical X rays)

Step-by-Step Solution

Answer

a. Frequency for 632.8 nm is \(4.74 \times 10^{14} \text{ Hz}\), b. Frequency for 503 nm is \(5.96 \times 10^{14} \text{ Hz}\), c. Frequency for 0.052 nm is \(5.77 \times 10^{18} \text{ Hz}\).

Step 1: Convert Wavelengths to Meters

To calculate the frequency, we first need to convert the wavelength from nanometers to meters, because the speed of light (\(c\)) is given in meters per second. We use the conversion \(1 \text{ nm} = 1 \times 10^{-9} \text{ m}\). Multiply the given wavelengths by \(10^{-9}\) to convert them to meters.

Step 2: Formula for Frequency

The frequency \(f\) can be calculated using the formula \(f = \frac{c}{\lambda}\), where \(c = 3 \times 10^{8} \text{ m/s}\) is the speed of light in a vacuum, and \(\lambda\) is the wavelength in meters.

Step 3: Calculate the Frequency for 632.8 nm Wavelength

First, convert 632.8 nm to meters: \((632.8 \text{ nm}) \times (10^{-9} \text{ m/nm}) = 6.328 \times 10^{-7} \text{ m}\). Then, use the frequency formula to find the frequency: \(f = \frac{3 \times 10^{8}}{6.328 \times 10^{-7}}\).

Step 4: Calculate the Frequency for 503 nm Wavelength

First, convert 503 nm to meters: \((503 \text{ nm}) \times (10^{-9} \text{ m/nm}) = 5.03 \times 10^{-7} \text{ m}\). Then, calculate the frequency using the formula: \(f = \frac{3 \times 10^{8}}{5.03 \times 10^{-7}}\).

Step 5: Calculate the Frequency for 0.052 nm Wavelength

First, convert 0.052 nm to meters: \((0.052 \text{ nm}) \times (10^{-9} \text{ m/nm}) = 5.2 \times 10^{-11} \text{ m}\). Next, calculate the frequency: \(f = \frac{3 \times 10^{8}}{5.2 \times 10^{-11}}\).

Step 6: Final Calculations

Perform the division to obtain the frequencies for all three wavelengths.

Key Concepts

Speed of Light

The speed of light, often denoted by the symbol 'c', is one of the most fundamental constants in physics. It is the speed at which all electromagnetic radiation travels in a vacuum, and it has a value of approximately 299,792,458 meters per second (m/s). In many calculations, however, we round this figure to a more manageable number, using the approximation of exactly 3 x 10^8 m/s to simplify computations. This value is crucial when working with electromagnetic phenomena, including the calculation of frequency for different wavelengths of light, as seen in our problem.

Understanding that light travels at this constant speed allows us to determine how frequently a wave oscillates as it moves through space—a concept tightly linked to the wavelength-to-frequency conversion. Because of its consistency, the speed of light is the keystone in our equations and understanding electromagnetic radiation.

Wavelength to Frequency Conversion

To convert the wavelength of electromagnetic radiation to its frequency, we can apply a basic relationship utilizing the speed of light. The formula for frequency (f) is expressed as \( f = \frac{c}{\lambda} \), where 'c' is the speed of light and '\(\lambda\)' is the wavelength in meters. This relationship shows that the frequency is inversely proportional to the wavelength; as the wavelength increases, the frequency decreases.

When applied to the exercise problem, this mathematical relationship provides a straightforward method to calculate electromagnetic radiation frequency. By converting the given wavelength into meters, we can plug our values directly into this formula, leading to precise frequency calculations. This fundamental conversion is essential for various applications in physics, astronomy, and other sciences where the characteristics of light and other waves are crucial.

Wavelength and Frequency Relationship

The relationship between the wavelength and frequency of electromagnetic radiation is inversely proportional, which means if one increases, the other decreases. This concept is encapsulated in the formula mentioned previously, \( f = \frac{c}{\lambda} \).

The wavelength ('\(\lambda\)') is the distance between successive peaks of a wave. It is often measured in meters, but for very small distances like light waves, it may be given in nanometers (nm). Conversely, frequency ('f') refers to the number of wave cycles that occur in one second, measured in hertz (Hz). A higher frequency means more cycles per second, and this corresponds to a shorter wavelength given that the speed of light remains constant.

In the textbook exercise, we can see this relationship applied. For instance, red light from a helium-neon laser with a wavelength of 632.8 nm has a different frequency than medical X-rays with a much smaller wavelength of 0.052 nm. This inverse relationship is foundational to understanding the behavior of waves across the electromagnetic spectrum.

Units of Measurement Conversion

In the world of science, precise units of measurement are vital, and often, we need to convert these units to conduct calculations correctly. Our exercise required converting nanometers (nm) to meters (m) because the value for the speed of light is given in meters per second (m/s).since there are 1 x 10^9 nanometers in a meter, we convert by multiplying the wavelength in nanometers by 1 x 10^-9. This step is an essential preliminary procedure to ensure that we use the correct units for the wavelength when applying the formula for frequency.

Students might encounter this kind of unit conversion in a wide array of physics and chemistry problems. It is not only about being able to translate one unit to another but understanding that coherent units are a prerequisite for accurate calculations. Misaligned units can lead to incorrect results, so proper conversion is a fundamental skill in the sciences that should be mastered for effective problem-solving.