Problem 64 - Chemistry: An Atoms-Focused Approach

Question

Why do matter waves not add significantly to the challenge of hitting a bascball thrown at 99 mph \((44 \mathrm{m} / \mathrm{s}) ?\)

Step-by-Step Solution

Answer

Answer: The effect of matter waves is insignificant in hitting a baseball thrown at 99 mph because the de Broglie wavelength associated with the baseball is extremely small compared to its size. This means that the challenge of hitting the baseball lies mainly in its speed and trajectory, rather than any influence from its extremely small matter wave properties.

Step 1: De Broglie Wavelength Formula

The de Broglie wavelength is the wavelength associated with a particle's matter wave. The formula to calculate the de Broglie wavelength is given by: \(\lambda = \dfrac{h}{mv}\) Where \(\lambda\) is the de Broglie wavelength, \(h\) is the Planck's constant (\(6.626 \times 10^{-34} \mathrm{J.s}\)), \(m\) is the mass of the particle, and \(v\) is the velocity of the particle.

Step 2: Calculate the de Broglie Wavelength of Baseball

First, we need the mass and velocity of the baseball. A typical baseball weighs about 145 grams (0.145 kg) and the velocity is given as 44 m/s. Plug these values into the de Broglie wavelength formula: \(\lambda = \dfrac{6.626 \times 10^{-34} \, \mathrm{J} \cdot \mathrm{s}}{(0.145 \, \mathrm{kg})(44 \, \mathrm{m} / \mathrm{s})}\) Now, calculate the de Broglie wavelength: \(\lambda \approx 1.08 \times 10^{-34} \, \mathrm{m}\)

Step 3: Compare the de Broglie Wavelength to the Baseball's Diameter

The de Broglie wavelength we calculated is extremely small compared to the size of a baseball. The diameter of a baseball is approximately 0.07 m. We can find the ratio between the de Broglie wavelength and the baseball's diameter: \(\dfrac{1.08 \times 10^{-34} \, \mathrm{m}}{0.07 \, \mathrm{m}} \approx 1.54 \times 10^{-36}\) This ratio is incredibly small, which means the de Broglie wavelength of the baseball is insignificant compared to its size.

Step 4: Concluding Remarks

The wavelength of matter waves of a baseball moving at 99 mph (44 m/s) is extremely small compared to the size of the baseball, which implies that the effect of matter waves is insignificant in this case. The challenge of hitting the baseball lies primarily in its speed and trajectory, rather than any influence from its extremely small matter wave properties.

Key Concepts

Planck's Constant

Planck's constant is a fundamental quantity in quantum mechanics, symbolized by the letter \(h\). It plays a key role in processes at the microscopic scale, providing the link between the energy of a photon and the frequency of its corresponding electromagnetic wave. The value of Planck's constant is \(6.626 \times 10^{-34} \mathrm{J.s}\). This tiny number reflects how energy exchanges at the atomic and subatomic levels are quantized, which means they occur in small, discrete packets that are governed by this constant quantity.
Understanding Planck's constant is essential for delving into concepts like the photoelectric effect and the wave-particle duality of matter. Within the context of de Broglie wavelengths, Planck's constant relates to the matter wave properties of particles, such as their associated wavelengths.

Matter Waves

Matter waves, also known as de Broglie waves, are a fascinating consequence of quantum mechanics. They arose from Louis de Broglie's hypothesis, asserting that all particles could exhibit wave-like behavior. This idea suggests that every piece of matter, regardless of size, carries a wave characteristic called the de Broglie wavelength.
De Broglie's insight bridged the gap between particle physics and wave mechanics, proposing that particles move not only as discrete particles but also with an associated wave pattern. This wave property becomes significant primarily at the atomic and subatomic scales. Larger objects, like a baseball, have much smaller and practically negligible de Broglie wavelengths, making their wave-like behavior invisible in everyday life.

Particle Velocity

Particle velocity plays an essential role in calculating the de Broglie wavelength. As per the formula \(\lambda = \frac{h}{mv}\), velocity is inversely proportional to the wavelength. This means that as the velocity \(v\) of a particle increases, its associated de Broglie wavelength \(\lambda\) decreases, provided the mass \(m\) and Planck's constant \(h\) remain constant.
For a baseball traveling at 44 m/s, the velocity contributes to the calculation of its negligible de Broglie wavelength. Despite its high speed in human terms—99 mph—the baseball's matter wave property is minuscule due to its relatively large mass.

Baseball Mass

Baseball mass is another crucial element in the calculation of its de Broglie wavelength. A typical baseball has a mass of about 0.145 kg (145 grams). When calculating the de Broglie wavelength, mass directly influences the size of the wavelength: \(\lambda = \frac{h}{mv}\).
The heavier the particle, the shorter its de Broglie wavelength becomes, when velocity and Planck's constant remain static. In the case of common everyday objects such as a baseball, their considerable mass results in a de Broglie wavelength that is so tiny it is practically impossible to detect.
As a result, the wave properties associated with massive objects like baseballs are insignificant, which is why they do not affect activities such as hitting a baseball in a noticeable way.