Problem 42 - General Chemistry: Principles and Modern Applications

Question

What is the wavelength, in nanometers, associated with a \(1000 \mathrm{kg}\) automobile traveling at a speed of \(25 \mathrm{m} \mathrm{s}^{-1},\) that is, considering the automobile to be a matter wave? Comment on the feasibility of an experimental measurement of this wavelength.

Step-by-Step Solution

Answer

The wavelength of an automobile having a velocity of 25 m/s and a mass of 1000 kg is around \(2.65 \times 10^{-38}\) m, or \(2.65 \times 10^{-29}\) nm. This wavelength is extremely small, far smaller than feasible to observe or measure with current technology, if not impossible. Thus, an experimental measurement of this wavelength is not feasible.

Step 1: Apply de Broglie's equation

Firstly, it will be necessary to apply the de Broglie's equation, using the provided mass (m=1000 kg) and velocity (v=25 m/s). The Planck's constant (h) value is approximately \(6.626 \times 10^{-34}\) Joule-second.

Step 2: Calculation

The calculation is performed as follow: \(\lambda = h / (m \cdot v) = (6.626 \times 10^{-34}) / (1000 \cdot 25)\).

Step 3: Convert result into nanometers

The obtained result will be in meters. However, it's necessary to convert this to nanometers (1 meter = \(1 \times 10^{9}\) nanometers). Multiply the result by \(1 \times 10^{9}\).

Step 4: Comment on the feasibility

Finally, comment on the feasibility of observing this wavelength. Considering the size of the calculated wavelength, discuss whether it is possible to measure it in a realistic experimental setup.

Key Concepts

Matter Waves

In the world of quantum mechanics, every moving particle is associated with a wave-like behavior known as a "matter wave". This revolutionary idea was introduced by Louis de Broglie in 1924.

De Broglie proposed that not only does light exhibit both particle and wave characteristics, but matter, such as electrons or even automobiles, can too. To the classical physicist, the concept of a car behaving like a wave might sound baffling. However, matter waves become significant when dealing with tiny particles, like electrons, where their wavelengths are comparable to their size.

For objects with substantial mass, such as a 1000 kg automobile traveling at 25 m/s, the associated matter wave is impractically tiny. The smaller the mass and higher the velocity, the more noticeable these waves become. In larger everyday objects, this wave behavior doesn't show up due to their incredible smallness relative to the object size.

Wavelength Calculation

The calculation of a matter wave's wavelength is fundamental in quantum mechanics, especially when applying de Broglie's equation. This equation states:

• \( \lambda = \frac{h}{m \cdot v} \)

Here, \( \lambda \) represents the wavelength, \( h \) is the Planck's constant, \( m \) is mass, and \( v \) is velocity. Applying this formula determines the wave-like properties of a piece of matter.

For our problem, with a mass of 1000 kg and a speed of 25 m/s, we substitute these values in:

• \( \lambda = \frac{6.626 \times 10^{-34}}{1000 \times 25} \)
• \( \lambda = 2.65 \times 10^{-38} \) meters

This result is initially in meters. Converting this to nanometers involves multiplying by \( 1 \times 10^{9} \), producing an even smaller magnitude.

• Wavelength \( = 2.65 \times 10^{-29} \) nanometers

Such extremely short wavelengths are much beyond current experimental capabilities to measure.

Planck's Constant

The Planck's constant is a pivotal number in the realm of quantum physics. Represented by \( h \), it plays a vital role in linking the energy of a photon with the frequency of its wave. Expressed roughly as \( 6.626 \times 10^{-34} \) Joule-seconds, it signals the quantization of energy.

In de Broglie's matter wave formula, Planck's constant serves as a bridge connecting mass and velocity to determine a particle's wavelength:

• \( E = h \cdot f \)

where \( E \) is the energy, and \( f \) is the frequency.

This constant is exceptionally tiny, making its effects negligible for macroscopic objects, but prominent for atomic-scale phenomena.
It illustrates why matter waves aren't detectable in everyday life. For instance, a Planck's constant-sized impact on a large mass implies an invisible wavelength, as seen in our previous mammoth automobile example.