Question

Use the de Broglie relationship to determine the wavelengths of the following objects: (a) an 85 -kg person skiing at \(50 \mathrm{~km} / \mathrm{hr}\), (b) a \(10.0-\mathrm{g}\) bullet fired at \(250 \mathrm{~m} / \mathrm{s},(\mathrm{c})\) a lithium atom moving at \(2.5 \times 10^{5} \mathrm{~m} / \mathrm{s}\), (d) an ozone \(\left(\mathrm{O}_{3}\right)\) molecule in the upper atmosphere moving at \(550 \mathrm{~m} / \mathrm{s}\).

Short answer

The wavelengths of the objects are: (a) \(5.63 \times 10^{-38} \mathrm{m}\) for the 85-kg person skiing, (b) \(2.65 \times 10^{-35} \mathrm{m}\) for the 10.0-g bullet, (c) \(2.26 \times 10^{-11} \mathrm{m}\) for the lithium atom, and (d) \(1.49\times10^{-11} \mathrm{m}\) for the ozone molecule.

Step-by-step solution

Convert Velocity to SI Units

First, we must convert the given velocity from km/hr to m/s: \(v = 50 \frac{\mathrm{km}}{\mathrm{hr}} \times \frac{1000 \mathrm{m}}{1 \mathrm{km}} \times \frac{1 \mathrm{hr}}{3600 \mathrm{s}} = 13.89 \frac{\mathrm{m}}{\mathrm{s}}\)

Calculate the de Broglie Wavelength

Now, we can plug the values into the de Broglie formula: \(λ = \frac{6.626 \times 10^{-34} \mathrm{Js}}{(85 \mathrm{kg})(13.89 \frac{\mathrm{m}}{\mathrm{s}})} = 5.63 \times 10^{-38} \mathrm{m}\) (b) 10.0-g bullet fired at 250 m/s

Convert Mass to SI Units

First, we must convert the mass of the bullet from g to kg: \(m = 10.0 \mathrm{g} \times \frac{1 \mathrm{kg}}{1000 \mathrm{g}} = 0.01 \mathrm{kg}\)

Calculate the de Broglie Wavelength

Now, we can plug the values into the de Broglie formula: \(λ = \frac{6.626 \times 10^{-34}\mathrm{Js}}{(0.01 \mathrm{kg})(250 \frac{\mathrm{m}}{\mathrm{s}})} = 2.65 \times 10^{-35} \mathrm{m}\) (c) Lithium atom moving at \(2.5 \times 10^5\) m/s

Find Mass of Lithium Atom

A lithium atom has an atomic mass of 7. The mass can be calculated as: \(m=\frac{7\times1.66\times10^{-27}kg}{1}\) \(m = 1.162\times10^{-26} kg\)

Calculate the de Broglie Wavelength

Now, we can plug the values into the de Broglie formula: \(λ = \frac{6.626 \times 10^{-34} \mathrm{Js}}{(1.162 \times 10^{-26}\mathrm{kg})(2.5 \times 10^5 \frac{\mathrm{m}}{\mathrm{s}})} = 2.26 \times 10^{-11} \mathrm{m}\) (d) Ozone (O3) molecule moving at 550 m/s

Find Mass of Ozone Molecule

An ozone molecule consists of three oxygen atoms, each with an atomic mass of 16. The mass can be calculated as: \(m_\text{ozone}=\frac{3 \times 16 \times 1.66 \times 10^{-27} \mathrm{kg}}{1}\) \(m_\text{ozone} = 7.989\times10^{-26}kg\)

Calculate the de Broglie Wavelength

Now, we can plug the values into the de Broglie formula: \(λ = \frac{6.626 \times 10^{-34} \mathrm{Js}}{(7.989\times10^{-26}\mathrm{kg})(550 \frac{\mathrm{m}}{\mathrm{s}})} = 1.49\times10^{-11} \mathrm{m}\)

Key concepts

de Broglie Equation

Imagine everything around you, from the tiniest electron to large vehicles, as both particles and waves. This dual nature emerges from the de Broglie equation. French physicist Louis de Broglie proposed that particles of matter also exhibit wave-like properties. The de Broglie wavelength, denoted as \( \lambda \) in physics, describes the wavelength of a “matter wave” associated with a moving particle. The de Broglie equation, expressed as \( \lambda = \frac{h}{mv} \) where \(h \) is Planck’s constant, \(m \) is the mass, and \(v \) is the velocity of the particle, allows us to calculate this wavelength.

Understanding the de Broglie wavelength is crucial for fields like quantum mechanics, where it helps explain phenomena that classical physics can’t, such as the functioning of a scanning tunneling microscope which uses the wave nature of electrons to create images of atomic landscapes.

Matter Wave

The concept of a matter wave is intrinsic to quantum mechanics and is a fundamental part of understanding the de Broglie equation. A matter wave represents the wave characteristics of particles, presenting a way to better understand the behavior of particles at microscopic scales, such as electrons and other subatomic particles. Unlike waves on a string or sound waves, matter waves are probability waves, which mathematical functions are used to describe the likelihood of finding a particle in a particular location. While we cannot see these waves, we can measure their effects through quantum experiments.

For instance, the patterns formed during electron diffraction experiments confirm the wave nature of particles, illustrating the concepts represented by the de Broglie equation. These interpretations help us delve into the bizarre and non-intuitive realm of quantum physics, where particles and waves are no longer distinct entities.

Atomic Mass Unit

An atomic mass unit (amu or u) is a standard unit of mass that quantifies mass on an atomic or molecular scale. One atomic mass unit is defined as one twelfth of the mass of an unbound neutral atom of carbon-12 at rest, which is approximately \(1.660539040 \times 10^{-27} \mathrm{kg}\). When working with atomic and subatomic particles, using amu makes calculations more manageable and easier to comprehend.

In our exercise, to compute the de Broglie wavelength for a lithium atom or an ozone molecule, we first convert the mass from amu to kilograms, the standard SI unit of mass. This allows us to input the mass into the de Broglie equation. By using amu, we can easily relate microscopic masses to the more tangible scale of kilograms, linking the atomic scale to everyday units of measure.

Velocity Conversion

Velocity conversion involves changing the units of velocity to match the unit system of other variables used in an equation. In classical physics and daily life, we often encounter velocity units like kilometers per hour (km/h) or miles per hour (mph). However, in scientific calculations, especially in physics, we use the SI unit of velocity, which is meters per second (m/s).

Converting velocity is crucial because the de Broglie equation requires that the mass and velocity be in SI units to calculate the wavelength correctly. For example, a ‘person skiing at 50 km/h’ is converted to ‘13.89 m/s’ before using it in our calculation. Precision in these conversions ensures the accuracy of scientific work and enables scientists to compare and replicate experiments universally.

SI Units

SI units, or the International System of Units, form the foundation of modern science and technology. They are the globally agreed-upon standard units which allow for the consistent representation and comparison of measurements across different regions and scientific disciplines. SI units include the kilogram (kg) for mass, meter (m) for length, and second (s) for time, among others.

In the context of our exercise, working with SI units is vital for obtaining accurate and meaningful results when calculating the de Broglie wavelength. To determine the wavelengths for various moving objects, we ensure all variables such as mass, velocity, and Planck's constant are expressed in kilograms, meters per second, and Joule seconds (Js), respectively. This adherence to SI units simplifies the calculation process and allows for a coherent application of the de Broglie equation.