Problem 68 - Chemistry: An Atoms-Focused Approach

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 speed result in no change in its frequency.

Step-by-Step Solution

Answer

Answer: Yes, statements A and C are true, while statement B is false.

Step 1: Statement A

Let's analyze the first statement: "Heavy, fast-moving objects have lower frequencies than those of lighter, faster-moving objects." We can calculate the frequency of a particle using the de Broglie wavelength formula. To do this, we first need to express frequency in terms of the wavelength. Since frequency (f) and wavelength (λ) are related through the equation f = c / λ, where c is the speed of light, we can write the frequency in terms of mass and velocity as: f = c * (m * v) / h Comparing two objects with different masses and velocities, we can see that if an object has a higher mass and lower velocity (heavy, fast-moving object), its frequency will be lower than that of an object with a lower mass and higher velocity (lighter, faster-moving object). Therefore, statement A is true.

Step 2: Statement B

Let's now analyze the second statement: "Only very light particles can have high frequencies." From the frequency equation derived above (f = c * (m * v) / h), we can observe that if the mass of a particle is very small, we can still achieve a high frequency by having a high velocity. However, it doesn't mean that only very light particles can have high frequencies. In some cases, even particles with higher masses can have high frequencies if their velocities are high enough. Therefore, statement B is false.

Step 3: Statement C

Finally, let's analyze the third statement: "Doubling the mass of an object and halving its speed result in no change in its frequency." Using the frequency equation derived previously (f = c * (m * v) / h), let's calculate the new frequency when the mass of a particle is doubled and its speed is halved: New mass: 2m New velocity: v/2 New frequency: f' = c * ((2m) * (v/2)) / h = c * (m * v) / h Comparing the new frequency (f') with the original frequency (f), we can see that both are equal because the change in mass and velocity cancels out in the equation. Therefore, statement C is true. To conclude, statements A and C are true, while statement B is false.

Key Concepts

de Broglie wavelength

The concept of de Broglie wavelength connects the wave properties of particles with their physical characteristics, such as mass and velocity. According to Louis de Broglie, every moving particle can exhibit wave-like behavior, and the associated wavelength is known as the de Broglie wavelength.

The de Broglie wavelength \( \lambda \) of a particle is given by the formula:

• \( \lambda = \frac{h}{mv} \)

Here, \( h \) is the Planck's constant, \( m \) is the mass of the particle, and \( v \) is its velocity.
Thus, the de Broglie wavelength inversely depends on the mass and velocity of the object. When either the mass or velocity of the particle increases, its de Broglie wavelength becomes shorter.
This principle is crucial in quantum mechanics, signifying that objects with smaller mass, like electrons, have more noticeable wave behavior compared to heavier masses.

mass and velocity relationship

The relationship between mass and velocity of a particle plays a significant role in determining its quantum characteristics. The mass and velocity of a particle influence both its de Broglie wavelength and frequency.
To understand the relationship:

• As the mass \( m \) increases with constant velocity, the frequency \( f \) decreases because \( f = \frac{c}{\lambda} = \frac{c}{\frac{h}{mv}} \) simplifies to \( \frac{mv \cdot c}{h} \).
• Similarly, with an increase in velocity \( v \) and constant mass \( m \), the frequency \( f \) increases.

This indicates that lighter and faster-moving particles tend to have higher frequencies and smaller de Broglie wavelengths, which also demonstrates wave properties more evidently than heavier ones.
In particle physics, this relationship further aids in analyzing high-energy particles, where often relativistic speeds are considered.

high frequency conditions

High frequency conditions in particles are achieved by manipulating the mass and velocity, as shown in their relationship to the de Broglie wavelength.
High frequency implies either very high velocities or relatively low mass:

• Light particles, such as electrons, naturally have higher frequencies due to their smaller mass. Their speed can be less than or close to the speed of light in appropriate conditions.
• Heavier particles can achieve high frequencies if they gain enough velocity, potentially through accelerator technology where particles are driven at high speeds.

In practical applications, the principles of high frequency conditions are vital for technologies such as electron microscopes, where the wave-like properties of electrons allow visualization of incredibly small objects with high resolution. Understanding these conditions is crucial for advancements in quantum technologies and high-energy particle research.