Problem 20 - Chemistry

Question

A particular form of electromagnetic radiation has a frequency of \(8.11 \times 10^{14} \mathrm{~Hz}\). (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?

Step-by-Step Solution

Answer

The wavelength of this radiation in meters is approximately 0.37 meters, and in nanometers, it is approximately 370 nm. This places it in the ultraviolet category of the electromagnetic spectrum. The energy of one quantum of this radiation is approximately \(5.36 \times 10^{-19} Joules\).

Step 1: Calculate Wavelength in Meters

First, solve for wavelength in meters using the formula \(c = \lambda \times v\). Rearrange it to get \(\lambda = c/v\). Substitute \(v = 8.11 \times 10^{14} Hz\) and \(c = 3.0 \times 10^{8} m/s\) into the equation and solve to find the wavelength in meters.

Step 2: Convert Wavelength to Nanometers

The wavelength in meters can be converted to nanometers by multiplying by \(1 \times 10^{9}\), since there are one billion (or \(1 \times 10^{9}\)) nanometers in a meter. So, \(\lambda (in nm) = \lambda (in m) \times 1 \times 10^{9}\).

Step 3: Categorize into Electromagnetic Spectrum

With the wavelength calculated, it can be compared to the known ranges of the electromagnetic spectrum to determine which category the radiation belongs to. The electromagnetic spectrum ranges from radio waves with longest wavelengths to gamma rays with shortest wavelengths.

Step 4: Calculate Energy

The Energy of one quantum of this radiation can be calculated using the formula \(E = h \times v\), where \(h = 6.63 \times 10^{-34} Js\) (Planck's constant) and \(v = 8.11 \times 10^{14} Hz\) (given frequency). Multiply these values to get the energy in joules.

Key Concepts

Frequency

Frequency refers to the number of cycles of a wave that pass a point in one second. It is measured in hertz (Hz), where one hertz is equal to one cycle per second. In the context of electromagnetic radiation, frequency is crucial because it determines the properties of the radiation.

• A high frequency means more cycles per second, indicating higher energy.
• When frequency is high, the wavelength becomes shorter.

A particular form of electromagnetic radiation has a frequency of \(8.11 \times 10^{14}\, \mathrm{Hz}\). This high frequency is typical for visible light, which means it features shorter wavelengths compared to lower frequency waves such as radio waves.

Wavelength

Wavelength is the distance between successive crests of a wave. It is often denoted by the Greek letter \(\lambda\). For electromagnetic waves, wavelength and frequency are inversely related. As frequency increases, wavelength decreases. This relationship is given by the formula:\[ c = \lambda \times v \]where \(c\) is the speed of light, approximately \(3.0 \times 10^{8} \text{ m/s}\). To find the wavelength in meters:\[ \lambda = \frac{c}{v} = \frac{3.0 \times 10^{8} \text{ m/s}}{8.11 \times 10^{14} \text{ Hz}} \]Converting the wavelength from meters to nanometers involves multiplying by \(1 \times 10^{9}\) because there are that many nanometers in a meter. This conversion is essential for identifying which part of the electromagnetic spectrum the radiation belongs to.

Energy of a Quantum

The energy of a quantum of electromagnetic radiation is related to its frequency by Planck's equation:\[ E = h \times v \]where \(E\) is the energy in joules, \(h\) is Planck's constant \(6.63 \times 10^{-34} \text{ Js}\), and \(v\) is the frequency. Since frequency is given as \(8.11 \times 10^{14} \text{ Hz}\), the energy can be calculated by:\[ E = 6.63 \times 10^{-34} \text{ Js} \times 8.11 \times 10^{14} \text{ Hz} \]The resulting energy quantifies how much energy a single photon, or quantum of this electromagnetic radiation, carries. Quantum energy calculations are essential in fields like quantum mechanics and atomic physics, unraveling the behavior of matter and energy at microscopic scales.

Electromagnetic Spectrum

The electromagnetic spectrum represents all types of electromagnetic radiation. It ranges from radio waves, which have the longest wavelengths and lowest frequencies, to gamma rays, which have the shortest wavelengths and highest frequencies. Understanding where a particular wave falls on this spectrum allows you to determine its properties and potential uses.Some key regions include:

• Radio Waves: Long wavelengths, used in broadcasting.
• Microwaves: Shorter than radio waves, used in cooking and radar.
• Infrared: Associated with heat.
• Visible Light: The spectrum seen by the human eye, where \(8.11 \times 10^{14} \text{ Hz}\) falls into, indicating visible color light.
• Ultraviolet: Can cause sunburns.
• X-rays: Used in medical imaging.
• Gamma Rays: Produced by radioactive atoms and in nuclear explosions.

By identifying the radiation's wavelength, you can place it into the correct region of the electromagnetic spectrum, thereby predicting its interaction with matter and its potential applications.