Problem 67 - Chemistry: An Atoms-Focused Approach

Question

Two objects are moving at the same speed. Which (if any) of the following statements about them are true? a. The de Broglie wavelength of the heavier object is longer than that of the lighter one. b. If one object has twice as much mass as the other, then its wavelength is one-half the wavelength of the other. c. Doubling the speed of one of the objects will have the same effect on its wavelength as doubling its mass.

Step-by-Step Solution

Answer

Explain your answer. a) The heavier object has a longer de Broglie wavelength than the lighter object. b) If one object has twice the mass of the other, its de Broglie wavelength is one-half the wavelength of the other object. c) The effect of doubling the mass or speed of an object on its de Broglie wavelength is the same, which is halving the wavelength. Answer: a) False - The heavier object has a shorter de Broglie wavelength than the lighter object. b) True - The de Broglie wavelength of the object with twice the mass is indeed one-half the wavelength of the other object. c) True - Doubling the speed or the mass of an object both result in halving the de Broglie wavelength.

Step 1: Statement a: Comparing de Broglie wavelengths of objects with different masses

We are given two objects moving at the same speed. Let mass of the objects be m1 and m2, such that m1 > m2. The de Broglie wavelengths λ1 and λ2 can be calculated using the formula: λ1 = h / (m1 * v) λ2 = h / (m2 * v) Since m1 > m2 and v is the same for both objects, λ1 < λ2. Thus, the de Broglie wavelength of the heavier object is shorter than that of the lighter one. So, statement a is false.

Step 2: Statement b: Comparing de Broglie wavelengths when one object has twice the mass of the other

Let's assume that the mass of the first object is m and the mass of the second object is 2m. Both objects are moving at the same speed v. The de Broglie wavelengths λ1 and λ2 can be calculated using the formula: λ1 = h / (m * v) λ2 = h / (2m * v) We can rewrite the equation for λ2 as follows: λ2 = h / (2m * v) = (1/2) * (h / (m * v)) = (1/2) * λ1 Thus, the wavelength of the object with twice the mass is indeed one-half the wavelength of the other. So, statement b is true.

Step 3: Statement c: Effects of doubling the speed or mass on the de Broglie wavelength

First, let's analyze the effect of doubling the speed of an object on its de Broglie wavelength. The initial wavelength λ1 is given by: λ1 = h / (m * v) Now, let's double the speed of the object to 2v. The new wavelength λ2 is given by: λ2 = h / (m * 2v) = (1/2) * (h / (m * v)) = (1/2) * λ1 Now, let's analyze the effect of doubling the mass of the object on its de Broglie wavelength. The initial wavelength λ1 is given by: λ1 = h / (m * v) Now, let's double the mass of the object to 2m. The new wavelength λ2 is given by: λ2 = h / (2m * v) = (1/2) * (h / (m * v)) = (1/2) * λ1 In both cases, doubling the speed or the mass resulted in the same effect on the de Broglie wavelength, which is halving the wavelength. So, statement c is true.

Key Concepts

Quantum Mechanics

Quantum mechanics is a branch of physics that deals with the behavior of very small particles, like atoms and subatomic components, under specific conditions. Unlike classical physics, which works well to describe the motion of macroscopic objects, quantum mechanics delves into probabilities.
It provides a framework for understanding how matter and energy behave on atomic and subatomic scales.

One of the landmarks of quantum mechanics is the introduction of the concept of the wave function, which describes the probability of finding a particle in a particular state. This concept is different from the deterministic nature of classical physics.
In quantum mechanics, particles have wave-like properties that lead to phenomena like interference and diffraction.

• Observations are not certain but probabilistic.
• Energy levels in atoms are quantized, meaning they can only take on discrete values.
• The observer effect suggests that measuring a system can affect its state.

Understanding quantum mechanics is crucial for exploring the nature of particles like electrons and for supporting modern technologies such as semiconductors and lasers.

Mass and Speed Relationship

The mass and speed relationship is a critical factor in determining the de Broglie wavelength of an object. According to the de Broglie hypothesis, every moving particle or object has an associated wavelength, called the de Broglie wavelength. This relationship is governed by the formula:\[\lambda = \frac{h}{m \cdot v}\]where \(\lambda\) is the wavelength, \(h\) is Planck's constant, \(m\) is the mass of the object, and \(v\) is its speed.

This formula shows how the wavelength of an object decreases with increasing mass or speed. For instance, a heavier object moving at the same speed as a lighter one will have a shorter wavelength.

• Higher mass means smaller wavelength.
• Greater speed leads to a shorter wavelength.

The inverse relationship with mass and speed explains why macroscopic objects appear not to show wavelike behavior—because their wavelengths are too small to observe in daily life.

Wave-Particle Duality

Wave-particle duality is an essential concept in quantum mechanics that defines how every particle exhibits both wave and particle characteristics. This duality is evident in the behavior of particles like photons and electrons.
For example, photons exhibit wave characteristics in interference patterns, yet they also behave as particles when exchanging energy with matter.

The de Broglie wavelength is a direct manifestation of this principle, illustrating that particles traditionally thought of as matter (like electrons) have wavelength properties.

• Light demonstrates both wave-like and particle-like properties.
• Electrons can create interference patterns (a wave property) when passing through double slits.

This concept broke the classical view of particles and waves as distinct entities, highlighting the need for quantum theory to accurately describe the physical world. Understanding wave-particle duality is crucial for comprehension of phenomena such as the behavior of electrons in an atom.