Question

A chemistry student observed each of the following objects: (a) a \(10,000 \mathrm{~kg}\) truck moving at \(27.8 \mathrm{~m} / \mathrm{sec}\), (b) a \(50 \mathrm{mg}\) flea flying at \(1 \mathrm{~m} / \mathrm{sec}\) and (c) a water molecule moving at 500 \(\mathrm{m} / \mathrm{sec}\). The student proceeded to calculate the wavelength in centimeters of each object. What were these wavelength? \(\mathrm{h}=6.626 \times 10^{-34} \mathrm{~J} \mathrm{sec}\)

Short answer

The wavelengths of each object in centimeters are: (a) truck: \(\lambda \approx 2.39 \times 10^{-36} \, \mathrm{cm}\) (b) flea: \(\lambda \approx 3.98 \times 10^{-26} \, \mathrm{cm}\) (c) water molecule: \(\lambda \approx 4.42 \times 10^{-10} \, \mathrm{cm}\)

Step-by-step solution

(Step 1: Calculate the mass and velocity in SI units for each object)

Convert the given mass and velocity of the objects into SI units (kilograms and meters per second) before performing the calculations. (a) truck: \(m = 10,000 \, \mathrm{kg}\), \(v = 27.8 \, \mathrm{m/s}\) (b) flea: \(m = 50 \times 10^{-6} \, \mathrm{kg}\), \(v = 1 \, \mathrm{m/s}\) (c) water molecule: We know the mass of a water molecule (\(H_2O\)) is approximately \(3 \times 10^{-26}\, \mathrm{kg}\), while the given value of velocity is \(v = 500 \, \mathrm{m/s}\)

(Step 2: Use the de Broglie equation to find the wavelength for each object)

Use the de Broglie equation \(\lambda = \frac{h}{mv}\) and the given Planck's constant \(h = 6.626 \times 10^{-34} \, \mathrm{J} \cdot \mathrm{s}\) to find the wavelengths for each object. (a) truck: \(\lambda = \frac{6.626 \times 10^{-34} \, \mathrm{J} \cdot \mathrm{s}}{(10,000 \, \mathrm{kg}) (27.8 \, \mathrm{m/s})}\) (b) flea: \(\lambda = \frac{6.626 \times 10^{-34} \, \mathrm{J} \cdot \mathrm{s}}{(50 \times 10^{-6} \, \mathrm{kg}) (1 \, \mathrm{m/s})}\) (c) water molecule: \(\lambda = \frac{6.626 \times 10^{-34} \, \mathrm{J} \cdot \mathrm{s}}{(3 \times 10^{-26} \, \mathrm{kg}) (500 \, \mathrm{m/s})}\)

(Step 3: Calculate the wavelength for each object in meters)

Plug in the values of mass and velocity into the de Broglie equation and solve for the wavelength in meters for each object. (a) truck: \(\lambda \approx 2.39 \times 10^{-38} \, \mathrm{m}\) (b) flea: \(\lambda \approx 3.98 \times 10^{-28} \, \mathrm{m}\) (c) water molecule: \(\lambda \approx 4.42 \times 10^{-12} \, \mathrm{m}\)

(Step 4: Convert the wavelength to centimeters)

Convert the wavelengths from meters to centimeters using the conversion factor \(1 \, \mathrm{m} = 100 \, \mathrm{cm}\). (a) truck: \(\lambda \approx 2.39 \times 10^{-36} \, \mathrm{cm}\) (b) flea: \(\lambda \approx 3.98 \times 10^{-26} \, \mathrm{cm}\) (c) water molecule: \(\lambda \approx 4.42 \times 10^{-10} \, \mathrm{cm}\) The wavelengths of each object in centimeters are: (a) truck: \(\lambda \approx 2.39 \times 10^{-36} \, \mathrm{cm}\) (b) flea: \(\lambda \approx 3.98 \times 10^{-26} \, \mathrm{cm}\) (c) water molecule: \(\lambda \approx 4.42 \times 10^{-10} \, \mathrm{cm}\)

Key concepts

Planck's constant

Planck's constant, denoted as \(h\), is a fundamental quantity in quantum mechanics. It plays a crucial role in the de Broglie wavelength formula. With a value of \(6.626 \times 10^{-34}\, \text{J} \cdot \text{s}\), Planck's constant connects the energy carried by a photon to its frequency. In essence, it helps bridge the gap between the wave and particle nature of matter. This constant is indispensable when calculating the wavelength of particles with mass, revealing quantum behaviors that are usually only observable at the microscopic level.

Mass and velocity conversion

Before using the de Broglie equation, it is essential to ensure that the mass and velocity of each object are expressed in compatible SI units.

• The mass should be in kilograms (\(\text{kg}\)).
• Velocity should be in meters per second (\(\text{m/s}\)).

For instance, a flea with a mass of \(50\, \text{mg}\) needs to be converted to \(\text{kg}\) as \(50 \times 10^{-6}\, \text{kg}\). Similarly, double-check velocities to ensure they’re in \(\text{m/s}\) for proper application in physics equations.Proper conversion ensures accuracy in calculating other physical properties like wavelengths.

Quantum mechanics

Quantum mechanics is the branch of physics dealing with phenomena at microscopic scales, such as atomic and subatomic levels. It operates under the principles where the classic laws of physics do not always apply. One of the intriguing concepts in quantum mechanics is wave-particle duality. This principle states that particles, like electrons and even larger objects, have properties of both waves and particles. The de Broglie wavelength is a direct application of this duality, allowing us to calculate the wavelength of any object with a given mass and velocity using quantum mechanics principles.

Wavelength calculation

Calculating the wavelength of an object from its mass and velocity is accomplished using the de Broglie equation: \[\lambda = \frac{h}{mv}\] This equation requires:

• \(h\) - Planck's constant \(6.626 \times 10^{-34} \text{J} \cdot \text{s}\)
• \(m\) - mass of the object in kilograms
• \(v\) - velocity in meters per second

After inserting these values into the formula, solve for the wavelength \(\lambda\). This gives the wavelength in meters. If needed, as in the original exercise, you can convert it into centimeters by multiplying by 100.The result reveals fascinating insights into how wave properties manifest, even in large objects, under quantum mechanics.