Problem 67 - Chemistry The Science in Context

Question

Which (if any) of the following statements about the frequency of a particle is true? a. Heavy, fast-moving objects have lower frequencies than those of lighter, faster-moving objects. b. Only very light particles can have high frequencies. c. Doubling the mass of an object and halving its velocity results in no change in its frequency.

Step-by-Step Solution

Answer

a. Heavy, fast-moving objects have lower frequencies than those of lighter, faster-moving objects. b. Only very light particles can have high frequencies. c. Doubling the mass of an object and halving its velocity results in no change in its frequency. Answer: c. Doubling the mass of an object and halving its velocity results in no change in its frequency.

Step 1: Statement a Analysis

The first statement says that heavy, fast-moving objects have lower frequencies than those of lighter, faster-moving objects. To test this, let's compare two different objects with different masses (M1 and M2) and velocities (V1 and V2), where M1 > M2 and V1 < V2. Using the de Broglie wavelength formula: λ1 = h / (M1 * V1) λ2 = h / (M2 * V2) Now, we will find their frequencies: f1 = c / λ1 f2 = c / λ2 Now if f1 < f2, the statement is true.

Step 2: Statement b Analysis

The second statement says that only very light particles can have high frequencies. To test this, let's compare two objects with different masses (M1 and M2), where M1 > M2. Using the de Broglie wavelength formula: λ1 = h / (M1 * V) λ2 = h / (M2 * V) Now, we will find their frequencies: f1 = c / λ1 f2 = c / λ2 Now if f1 < f2, the statement is true.

Step 3: Statement c Analysis

The third statement says that doubling the mass of an object and halving its velocity results in no change in its frequency. To test this, let's compare two objects with different masses (M and 2M) and velocities (V and V/2) respectively. Using the de Broglie wavelength formula: λ1 = h / (M * V) λ2 = h / (2M * (V/2)) Now let's find their frequencies: f1 = c / λ1 f2 = c / λ2 Now if f1 = f2, the statement is true.

Step 4: Final Conclusion

Upon analysis of the given statements and comparing their frequencies using de Broglie wavelength formula, we can conclude that the most accurate statement is: c. Doubling the mass of an object and halving its velocity results in no change in its frequency. This is because when we double the mass and halve the velocity, we find that the de Broglie wavelength remains the same, and hence, the frequency also remains the same.

Key Concepts

Particle Frequency

The concept of particle frequency can be a bit tricky but is fundamentally important in physics. Frequency refers to how often an event occurs, such as the oscillation of a wave. For particles, the frequency is linked to its wavelength, as described by the de Broglie wavelength formula. According to this principle, every particle has a wavelength that is inversely proportional to its momentum. The formula is expressed as:
\[ \lambda = \frac{h}{m \, v} \]

• \( \lambda \) is the de Broglie wavelength.
• \( h \) represents Planck’s constant.
• \( m \) is the mass of the particle.
• \( v \) stands for the velocity of the particle.

The frequency \( f \) of the particle is inversely related to its wavelength, i.e., \( f = \frac{c}{\lambda} \), where \( c \) is the speed of light. This means that higher momentum results in shorter wavelength and higher frequency.
Applying this to the statements in the exercise: a particle with greater mass and velocity might carry different frequencies based on their motion parameters. Therefore, examining whether only certain particles can have high or low frequencies, as seen in statements a and b, hinges heavily on their mass and velocity ratios.

Mass and Velocity Relationship

The relationship between mass and velocity plays a crucial role in understanding the behavior of particles. In classical physics, the momentum of a particle is simply the product of its mass and velocity \( p = m \, v \). However, when exploring particles on a smaller, quantum scale, this relationship determines the particle's wavelength and thus its frequency.
By manipulating the mass or velocity, we can directly affect the particle's de Broglie wavelength. If we double the mass and reduce the velocity by half, as considered in the third statement, the wavelength and consequently the frequency remain unchanged.
Let's break this down:

• For \(\lambda_1 = \frac{h}{m \, v} \), and \(\lambda_2 = \frac{h}{2m \, (v/2)} \), the simplification shows that \(\lambda_1 = \lambda_2\).
• This indicates that changes in mass and velocity, as specified, have no net effect on the frequency, corroborating statement c in the original exercise.

This mathematical exploration illustrates how particles maintain their frequency despite variations in mass and velocity under specific conditions.

Quantum Mechanics Concepts

Quantum mechanics introduces a fascinating paradigm where particles exhibit both wave-like and particle-like properties. This duality is captured beautifully by the de Broglie hypothesis, asserting that every particle, regardless of its mass, has a wavelength associated with its motion.

• Particles do not simply move; their paths can be described by a probability wave.
• This wave nature explains why different particles, depending on their mass and speed, show different wave behaviors, such as interference and diffraction.

In quantum mechanics, it's essential to understand that factors governing de Broglie wavelength and frequency interplay with other quantum characteristics, like energy quantization and uncertainty principles. Thus, conclusions drawn from more classical observations hint at the intrinsic unpredictabiliity and constraint by quantum principles.
This enriched understanding allows students to grasp the fundamental essence behind behaviors outlined in textbook exercises and highlight why they're crucial to wider quantum contexts.