Problem 94 - Chemistry Matter and Change

Question

How is the energy of a quantum of emitted radiation related to the frequency of the radiation? (Chapter 5\()\)

Step-by-Step Solution

Answer

The energy of a quantum of radiation is directly proportional to its frequency, given by the equation \( E = h \nu \).

Step 1: Understanding the Question

We need to find out how the energy of a single quantum of radiation is connected to the frequency of that radiation. This involves understanding key concepts in physics, specifically quantum mechanics.

Step 2: Recalling Planck's Equation

The relationship between energy and frequency is given by Planck's equation: \[ E = h u \]where \( E \) is the energy of the quantum (or photon), \( u \) is the frequency of the radiation, and \( h \) is Planck's constant, approximately equal to \( 6.626 \times 10^{-34} \text{ Js} \).

Step 3: Analyzing the Equation

According to Planck's equation, the energy of a quantum of radiation is directly proportional to its frequency. This means that as the frequency increases, the energy also increases, and vice versa.

Step 4: Equation Interpretation

If given a specific frequency of radiation, you can calculate the energy of a quantum of that radiation by multiplying the frequency by Planck's constant. This direct proportionality simplifies the relation significantly.

Key Concepts

Planck's equation

Planck's equation is a fundamental principle in quantum mechanics that describes the relationship between the energy of a quantum (or photon) and its frequency. The equation is represented as \( E = h u \). Here:

• \( E \) stands for the energy of the quantum, which is typically measured in joules (J).
• \( h \) is Planck's constant, a crucial physical constant with the value approximately \( 6.626 \times 10^{-34} \) joule-seconds (Js).
• \( u \) (nu) represents the frequency of the radiation, measured in hertz (Hz).

This equation was groundbreaking because it provided a quantitative way to link the wave and particle nature of light. One of Max Planck's motivations for this discovery was addressing the problem of "black-body radiation," which classical physics could not adequately explain. This relation paved the path for the development of quantum theory, changing the way scientists understood the physical universe.

Energy-frequency relationship

The energy-frequency relationship in quantum mechanics highlights the direct proportionality between energy and frequency. In simpler terms, higher frequency radiation means higher energy photons and vice versa. When you look at Planck's equation, \( E = h u \), it becomes evident:

• If the frequency \( u \) increases, then the energy \( E \) will increase if Planck's constant \( h \) remains unchanged.
• This relationship is linear, meaning doubling the frequency will double the energy.

This concept is significant in various applications, such as explaining the photoelectric effect, where light shining on certain materials can eject electrons. The energy of the electrons depends on the frequency of the incoming light, affirming the energy-frequency relationship. Such insights demonstrate the wave-particle duality of light and deepen our understanding of light's behavior.

Radiation quantum

A quantum of radiation refers to the smallest discrete amount of energy that can be emitted or absorbed as electromagnetic radiation. In the context of Planck's equation, each packet or photon of light embodies a quantum of energy with energy level determined by:

• Planck's constant \( h \)
• The frequency of the radiation \( u \)

This concept is central to quantum mechanics and distinguishes it from classical physics, which treated energy as a continuous quantity. The term "quantum" signifies a "packet" of energy, highlighting the idea that energy transfer on a microscopic scale occurs in discrete amounts. This understanding is why quantum mechanics aptly describes phenomena at the atomic and subatomic level, providing a more accurate depiction than classical theories.In practical terms, this also explains why the quantum nature of light allows for precise technological applications such as lasers and photovoltaics, which depend on the "all or nothing" energy levels associated with photons.