Problem 53 - Chemistry: Structure and Properties

Question

Calculate the de Broglie wavelength of a 143-g baseball traveling at 95 mph. Why is the wave nature of matter not important for a baseball?

Step-by-Step Solution

Answer

The de Broglie wavelength of the baseball is significantly smaller than the sizes of things we interact with daily, making its wave nature unimportant and undetectable in everyday life.

Step 1: Convert the Baseball's Speed to SI Units

First, convert the baseball's speed from miles per hour (mph) to meters per second (m/s). 1 mph is equal to 0.44704 m/s. So, multiply the baseball's speed in mph by 0.44704 to convert to m/s.

Step 2: Calculate the Momentum of the Baseball

Momentum (p) is the product of mass (m) and velocity (v). The mass should be in kilograms (kg), so convert the baseball's mass from grams to kilograms by dividing by 1000. Then, calculate the momentum by multiplying the mass in kg by the velocity in m/s.

Step 3: Calculate the de Broglie Wavelength

The de Broglie wavelength (\(\lambda\)) is found using the equation \(\lambda = \frac{h}{p}\), where \(h\) is Planck's constant (\(6.626 \times 10^{-34} \)Js). Substitute the calculated momentum into this equation to find the de Broglie wavelength.

Step 4: Discuss the Significance of the de Broglie Wavelength for a Baseball

The calculated de Broglie wavelength for a macroscopic object like a baseball is extremely small, and therefore, the wave nature of the baseball is imperceptible and not significant in practical terms. Quantum mechanical effects, including wave-particle duality, are typically significant only at the atomic or subatomic level.

Key Concepts

Wave-Particle Duality

Wave-particle duality is a foundational concept in quantum mechanics that every particle exhibits both wave-like and particle-like properties. This dual nature becomes evident when we conduct experiments at the microscopic level, such as with electrons or photons. For example, light can behave like a wave, as demonstrated by the interference patterns in the famous double-slit experiment, or like particles, known as photons, when its energy is being transferred.

When it comes to larger objects, like a baseball, their wave-like properties are not observable. This is because their wavelengths are incredibly tiny compared to their size, so the wavelike behavior is essentially 'hidden' from our ability to measure or observe it. The overall concept emphasizes that the behavior of quantum objects is unlike anything we experience in everyday life.

Momentum

Momentum in physics is typically defined as the product of an object's mass and its velocity. It is a vector quantity, which means it has both magnitude and direction. In classical mechanics, momentum conservation is one of the key principles; it tells us that the total momentum of a closed system remains constant unless acted upon by external forces.

In quantum mechanics, however, momentum also plays a crucial role, as it is directly related to the wavelength of a particle. This relationship is given by de Broglie's hypothesis, which posits that every moving particle or object has an associated wave, with its wavelength inversely proportional to its momentum. This concept significantly departs from classical intuitions and is central to understanding quantum phenomena.

Quantum Mechanics

Quantum mechanics is a fundamental theory in physics that describes nature at the smallest scales of energy levels of atoms and subatomic particles. Unlike classical physics, which deals with the macroscopic world, quantum mechanics operates on the principles that on a very small scale, energy is quantized, and physical properties like position, speed, and color of objects are probabilistic rather than deterministic.

One of the bizarre implications of quantum mechanics is the concept of superposition, where particles can exist in multiple states simultaneously. Its predictions often contradict our everyday experiences and intuitions about how the world operates, which can make quantum mechanics a challenging concept to grasp. This theory is the backbone of various modern technologies, including semiconductors, lasers, and quantum computing.

Planck's Constant

Planck's constant is a physical constant that plays a vital role in quantum mechanics. Denoted by the symbol \( h \), it has a value of approximately \( 6.62607015 \times 10^{-34} \text{Js} \). It was first introduced by Max Planck, and it signifies the quantization of energy. This constant is utilized in the computation of a photon's energy from its frequency, according to the formula \( E = h u \), where \( E \) is the energy and \( u \) is the frequency.

Planck's constant is also a key element of the de Broglie hypothesis, relating the momentum of a particle to its wavelength. It shows that the energy of wave-like objects (like photons) is quantized, further highlighting the surprising link between the energy and momentum of particles and their wave-like characteristics.

SI Units Conversion

Conversion of units to the International System of Units (SI) is essential for consistency and accuracy in scientific calculations. SI units provide a standard for measurements, ensuring that experiments and calculations are reproducible and universally understood. In the context of de Broglie wavelength calculations, mass needs to be in kilograms (kg), and the velocity in meters per second (m/s).

When dealing with problems like calculating the wavelength of a baseball, it is crucial to convert the mass from grams to kilograms by dividing by 1000 and the velocity from miles per hour to meters per second by multiplying with a conversion factor, in this case, 0.44704. Proper unit conversion ensures that the calculations are correct and the resulting measurements accurately reflect the quantities involved.