Problem 40 - Chemistry

Question

Protons can be accelerated to speeds near that of light in particle accelerators. Estimate the wavelength (in \(\mathrm{nm}\) ) of such a proton moving at \(2.90 \times\) \(10^{8} \mathrm{~m} / \mathrm{s} .\) (Mass of a proton \(\left.=1.673 \times 10^{-27} \mathrm{~kg} .\right)\)

Step-by-Step Solution

Answer

The estimated wavelength of the proton is approximately 0.399 nm.

Step 1: Understanding the Problem

First, identify the given values: speed of light \(v = 2.90 \times 10^{8} \, \mathrm{m/s}\) and mass of a proton \(m = 1.673 \times 10^{-27} \, \mathrm{kg}\). We have to estimate the wavelength \(\lambda\) in nanometres. We'll find this using the de Broglie's formula \(\lambda = \frac{h}{mv}\), where \(h\) is the Planck’s constant and equals \(6.63 \times 10^{-34} \, \mathrm{Js}\).

Step 2: Applying the De Broglie's Formula

Using the de Broglie's formula \(\lambda = \frac{h}{mv}\), substitute the given values: \(\lambda = \frac{6.63 \times 10^{-34}\, \mathrm{Js} }{(1.673 \times 10^{-27} \, \mathrm{kg} ) (2.90 \times 10^{8} \, \mathrm{m/s}) }\).

Step 3: Calculating the Wavelength

Next, perform the necessary calculations. Note that the unit of the wavelength will be in meters. For the final answer, we need to convert it to nanometers (1m = \(10^{9}\)nm ).

Step 4: Conversion of Units

Finally, convert the calculated wavelength from meters to nanometers by multiplying by \(10^{9}\).

Key Concepts

Particle Accelerators

Particle accelerators are huge machines that boost particles to incredibly high speeds and energies. They are used to explore the fundamental particles of the universe. Inside a particle accelerator, particles travel down a long tube, often circling in loops thanks to magnetic fields. Because of the high speeds, particles like protons can be accelerated to nearly the speed of light.

• These machines help us understand more about the subatomic world, revealing new particles and interactions.
• They also provide insights into the forces that hold matter together, helping scientists uncover the universe's origins.
• Particle accelerators have practical applications such as in medicine, where they help in treating cancer through radiation therapy.

Using particle accelerators, scientists can examine the properties of particles, like protons, when they move near light speed. This is crucial for measuring properties, such as the de Broglie's wavelength, which directly ties into fields like quantum mechanics.

Planck’s Constant

Planck's constant is a fundamental constant in physics, vital for quantum mechanics. Represented by the symbol \( h \), its value is \( 6.63 \times 10^{-34} \, \text{Js} \). This constant links the energy of a photon to its frequency, highlighting the particle-like behavior of light.

• Needs to be small because it measures the quantization of energy at the atomic scale, where energy comes in tiny, discrete packets called quanta.
• Provides insight into how microscopic objects, like particles, behave very differently from macroscopic objects.
• Is used in formulas like de Broglie's to calculate the wavelength of particles given their momentum.

When you understand Planck's constant, it opens up the world of quantum mechanics, revealing how particles can exhibit both wave-like and particle-like properties. It is essential for calculations in many quantum experiments and technologies.

Mass of a Proton

The mass of a proton is a critical parameter in a wide range of physics calculations. It is approximately \( 1.673 \times 10^{-27} \mathrm{~kg} \). Protons are one of the building blocks of atoms, found in every atomic nucleus. Their mass is a consistent measure of the force required to accelerate protons in particle accelerators.

• Protons, despite being minuscule, have substantial mass compared to other subatomic particles like electrons.
• The protons mass plays a vital role in determining the characteristics and stability of atoms.
• It is crucial in calculations involving atomic interactions, nuclear reactions, and the study of materials at the atomic level.

When protons reach high speeds in particle accelerators, understanding their mass helps scientists predict their behavior and interaction outcomes, shedding light on the forces that shape our universe.