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 emitted radiation is directly proportional to the frequency of the radiation, as described by Planck's equation: \(E = h \times v\), where \(E\) represents the energy, \(h\) is Planck's constant (approximately \(6.6261 \times 10^{-34} Js\)), and \(v\) is the frequency of the radiation. As the frequency increases, so does the energy of the quantum of radiation, and vice versa.

Step 1: Understand Planck's equation

Planck's equation is a fundamental formula in quantum mechanics, which states the relationship between the energy (E) of a quantum of radiation and the frequency (v) of the radiation. The equation is given by: \(E = h \times v\), where \(h\) is Planck's constant,approximately equal to \(6.6261 \times 10^{-34} Js\) and \(v\) is the frequency of the radiation.

Step 2: Determine the relationship between energy and frequency

As mentioned before, Planck's equation states that the energy of a quantum of radiation is directly proportional to the frequency of the radiation. That is, \(E \propto v\), This equation shows that as the frequency of the radiation increases, so does the energy of the quantum of radiation, and vice versa.

Step 3: Express the proportionality as an equation

To express the proportionality between the energy and frequency as an equation, we introduce Planck's constant, \(h\), as the constant of proportionality. This gives us the relationship: \(E = h \times v\), The relationship between the energy of a quantum of emitted radiation and the frequency of the radiation, as described by Planck's equation, is that the energy is directly proportional to the frequency, with Planck's constant being the constant of proportionality.

Key Concepts

Quantum Mechanics

Quantum Mechanics is a fundamental theory in physics that describes the physical properties of nature at the smallest scales, typically at the level of atoms and subatomic particles. It differs significantly from classical physics, which explains the larger-scale phenomena we observe in everyday life. Quantum mechanics brings forward the concept of quantization, which suggests that properties such as energy are discretized, existing only in specific 'quantum' amounts rather than a continuous range.

This theory introduces several complex and unusual concepts such as wave-particle duality, where light and matter can exhibit properties of both waves and particles. Another important feature is the uncertainty principle, which posits limitations on the precision with which certain pairs of physical properties, like position and momentum, can be known simultaneously. Quantum mechanics lays the foundation for much of modern science and technologies, including semiconductors, lasers, and magnetic resonance imaging units.

• Deals with atomic and subatomic particles
• Energy levels are quantized
• Introduces the concept of wave-particle duality
• Gives rise to the uncertainty principle

In short, quantum mechanics provides a necessary framework for understanding the behavior of particles at the smallest scales, forming the backbone for explaining phenomena related to Planck's equation.

Energy-Frequency Relationship

The Energy-Frequency Relationship in quantum mechanics defines how the energy of a particle or wave is connected to its frequency. According to Planck's equation, this relationship is represented by the formula:\[ E = h \times v \]where \(E\) is the energy of a photon, a quantum of light or electromagnetic radiation, \(v\) is the frequency of the radiation, and \(h\) is Planck's constant. This equation reveals a direct proportionality between energy and frequency, meaning as one increases, so does the other.

This concept is crucial for understanding many phenomena in quantum physics and even helps explain why we see light in certain colors. For example:

• Higher frequency radiation, like ultraviolet light, has higher energy compared to lower frequency radiation like infrared light.
• The energy difference between levels in atoms yields specific frequencies, leading to emission or absorption spectra.
• This relationship is fundamental to technologies such as lasers, where controlled frequency and energy manipulation is necessary.

By understanding the energy-frequency relationship, we can better grasp how energy levels dictate certain physical properties, behaviors, and interactions of matter and radiation.

Planck's Constant

Planck's Constant is a fundamental constant in physics, key to the theory of quantum mechanics. Denoted by \(h\), it has a value of approximately \(6.6261 \times 10^{-34} J\cdot s\). This small but significant constant provides the scale of quantization, linking the energy of photons to their frequency within Planck's equation:
\[ E = h \times v \]
Planck's Constant represents the smallest action - the product of energy and time - that can occur in quantum mechanical systems. This finding shifted the classical understanding of physics, showing that energy is not continuous, but instead comes in discrete packets, or 'quanta'.

• It signifies that energy transactions at the atomic level occur in set quantities.
• Key in determining energy levels in atomic and molecular systems.
• Allows precise calculations in a wide range of scientific and technological applications, such as spectroscopy and the development of quantum computers.

In summary, Planck's constant serves as a bridge between the classical and quantum worlds, offering insight into the scale and nature of the quantum realm.