Problem 127 - General Chemistry: Principles and Modern Applications

Question

For electromagnetic radiation transmitted through a vacuum, state whether each of the following properties is directly proportional to, inversely proportional to, or independent of the frequency: (a) velocity; (b) wavelength; (c) energy per mole. Explain.

Step-by-Step Solution

Answer

a) Velocity is independent of frequency. b) Wavelength is inversely proportional to frequency. c) Energy per mole is directly proportional to frequency.

Step 1: Analyzing the relationship with velocity

For electromagnetic radiation traveling through a vacuum, the velocity, often represented by 'c', is constant regardless of its frequency. 'c' stands for the speed of light in vacuum, which is always 3.0 x 10^8 m/s. Hence, the frequency is independent of the velocity.

Step 2: Analyzing the relationship with wavelength

The wavelength, often denoted by 'λ', of electromagnetic radiation is inversely proportional to its frequency. This fact can be understood by the wave equation, c=\( \lambda \cdot \nu \), where \( \nu \) is frequency. As c is constant, if \( \nu \) increases, \( \lambda \) decreases and vice-versa.

Step 3: Analyzing the relationship with energy per mole

The energy of electromagnetic radiation is directly proportional to its frequency. This relationship is denoted by Planck's equation, E=h \cdot \( \nu \), where h is Planck's constant. As the frequency \( \nu \) increases, energy E also increases, therefore they are directly proportional. To find energy per mole, multiply energy of one photon with Avogadro's number (6.022 x 10^23). As both the elements being multiplied are directly proportional to frequency, energy per mole is also directly proportional to frequency.

Key Concepts

Frequency Relationships

In the world of electromagnetic radiation, understanding how various properties relate to the frequency is key. These relationships help us comprehend the behavior of light and other types of radiation. One important property is how velocity and frequency interact.
Electromagnetic radiation travels through a vacuum at a constant speed, which is the speed of light. This value is universally recognized as approximately 3.0 x 10^8 meters per second. Since this speed is constant, the frequency of electromagnetic waves does not influence velocity in a vacuum. Therefore, we can say that frequency is independent of velocity.
Understanding these relationships allows us to predict and calculate how different forms of electromagnetic waves will behave in different environments. Knowing that frequency doesn't affect velocity simplifies calculations involving wave speed through a vacuum. It's crucial for designing and interpreting scientific experiments and various technological applications.

Wavelength and Frequency

Wavelength and frequency share an interesting and interconnected relationship. Picture a wave moving through space: as you increase the frequency, you actually decrease the wavelength, and vice versa. This relationship is known as inverse proportionality.
The specific formula used to explain this is the wave equation:

• \[ c = \lambda \cdot u \]
• Where \( c \) is the speed of light, \( \lambda \) is the wavelength, and \( u \) is the frequency.

Because the speed of light \( c \) remains constant in a vacuum, any increase in frequency must result in a decrease in wavelength. This notion is crucial in optics and communications technology among many other fields. If you know the frequency of a wave, you can easily find its wavelength using this formula, and the inverse is also true.
Thus, understanding how these properties interconnect allows us to build better tools for communication, like radios, cell phones, and optical fibers.

Planck's Equation

Planck's equation is a cornerstone in understanding the energy of electromagnetic radiation. It tells us how energy is related to the frequency of electromagnetic waves. In simple terms, higher frequency means higher energy.
Planck's equation is expressed as:

• \[ E = h \cdot u \]
• Where \( E \) stands for energy, \( h \) is Planck's constant (6.626 x 10^-34 J s), and \( u \) is the frequency.

According to this equation, energy is directly proportional to frequency. Meaning, as frequency goes up, energy increases too. This concept is not only meaningful in theoretical physics but also in many practical applications like calculating the energy carried by light or electromagnetic waves used in medicine or technology.
A noteworthy application involving Planck's equation is determining the energy per mole of photons. By multiplying by Avogadro's number (6.022 x 10^23), you can find how much energy is delivered when similar frequencies act on a larger scale, such as in reactions or astronomical processes.