Question

A particular form of electromagnetic radiation has a frequency of \(9.87 \times 10^{15} \mathrm{~Hz}\). (a) What is its wavelength in nanometers? In meters? (b) To what region of the electromagnetic spectrum would you assign it? (c) What is the energy (in joules) of one quantum of this radiation?

Short answer

(a) 30.4 nm; \(3.04 \times 10^{-8}\) m. (b) Ultraviolet region. (c) \(6.54 \times 10^{-18}\) J.

Step-by-step solution

Calculate the Wavelength in Meters

We start by using the speed of light equation, where the speed of light, \( c \), is \( 3.00 \times 10^8 \) m/s. The wavelength \( \lambda \) is given by the relationship \( c = \lambda u \). Rearranging for wavelength, we have \( \lambda = \frac{c}{u} \). Substituting the given frequency (\( u = 9.87 \times 10^{15} \) Hz), we find:\[ \lambda = \frac{3.00 \times 10^8 \text{ m/s}}{9.87 \times 10^{15} \text{ Hz}} \approx 3.04 \times 10^{-8} \text{ m} \].

Convert Wavelength to Nanometers

Since 1 nanometer (nm) is equal to \( 10^{-9} \) meters, we convert the wavelength from meters to nanometers by multiplying by \( 10^9 \):\[ 3.04 \times 10^{-8} \text{ m} \times 10^9 \text{ nm/m} = 30.4 \text{ nm} \].

Determine the Region of the Electromagnetic Spectrum

The electromagnetic spectrum is divided into regions based on wavelength. Ultraviolet light ranges approximately from 10 nm to 400 nm. Since our calculated wavelength is 30.4 nm, this radiation falls in the ultraviolet region of the electromagnetic spectrum.

Calculate the Energy of One Quantum

The energy of one quantum of electromagnetic radiation is given by Planck’s equation: \( E = h u \), where \( h \) (Planck's constant) is \( 6.626 \times 10^{-34} \text{ Js} \). Using the frequency provided, we calculate:\[ E = (6.626 \times 10^{-34} \text{ Js})(9.87 \times 10^{15} \text{ Hz}) \approx 6.54 \times 10^{-18} \text{ J} \].

Key concepts

Wavelength Calculation

When we talk about the wavelength of electromagnetic radiation, we're referring to the distance between two consecutive peaks or troughs in the wave. It's a critical property that helps classify different types of electromagnetic waves. To calculate the wavelength when the frequency is given, we use a well-known equation:

• The speed of light, denoted as \( c \), is \( 3.00 \times 10^8 \text{ m/s} \).
• Frequency (\( u \)) is the number of wave cycles per second, here \( 9.87 \times 10^{15} \text{ Hz} \).
• The formula \( \lambda = \frac{c}{u} \) allows us to find the wavelength \( \lambda \).

By plugging in the values, we find the wavelength in meters. In this case, the wavelength is approximately \( 3.04 \times 10^{-8} \text{ m} \). Converting wavelengths to different units, such as nanometers, is often useful, especially when classifying the wave in the electromagnetic spectrum. Remember, 1 meter equals \( 10^9 \) nanometers, so multiply accordingly to get \( 30.4 \text{ nm} \). This conversion helps when comparing different electromagnetic waves.

Electromagnetic Spectrum

The electromagnetic spectrum encompasses all types of electromagnetic radiation. It's a broad universe ranging from extremely low-energy radio waves to high-energy gamma rays. Each type of electromagnetic wave has characteristic wavelengths and frequencies.
In the visible spectrum, humans perceive colors. However, many ultraviolet waves, like the one calculated in the exercise, fall outside our visible range, despite being around us.

• Ultraviolet (UV) light ranges from about 10 to 400 nm in wavelength.
• The calculated wavelength of 30.4 nm situates this wave firmly in the UV region.

This distinction is important for various practical applications, including understanding solar radiation, electronics, and medical imaging technologies. Each region of the spectrum plays a crucial role in diverse technological and natural processes.

Energy of a Photon

The energy in a single quantum or photon of electromagnetic radiation can be directly correlated with its frequency through Planck's equation: \( E = h u \). Here:

• \( E \) stands for energy in joules (J).
• \( h \), Planck’s constant, is \( 6.626 \times 10^{-34} \text{ Js} \).
• \( u \) represents the frequency in hertz (Hz).

Multiplying these factors gives the energy per photon. In this scenario, the energy is approximately \( 6.54 \times 10^{-18} \text{ J} \). This might seem a small value, but consider the multitude of these photons acting together in light or radio waves. Understanding a photon's energy helps in fields like quantum mechanics, solar power, and even photography, where light energy determines image exposure.

Frequency to Wavelength Conversion

Understanding the conversion between frequency and wavelength can be challenging, but it's essential for analyzing electromagnetic waves. These two properties are inversely related, shown through the equation \( c = \lambda u \). As frequency increases, wavelength decreases, and vice versa.

• For instance, a higher frequency means more cycles per second, translating into shorter waves.
• In contrast, lower frequencies lengthen the waves.

When solving problems, it's crucial to appreciate this relationship to predict how waves will behave or interact with matter. This conversion is foundational in areas such as spectroscopy, wireless communication, and astrophysics. Combined with units of measurement, these concepts form the basis for comprehending a range of scientific phenomena.