Problem 45 - Chemistry The Central Science

Question

Neutron diffraction is an important technique for determining the structures of molecules. Calculate the velocity of a neutron needed to achieve a wavelength of \(0.955 \AA .\) (Refer to the inside cover for the mass of the neutron).

Step-by-Step Solution

Answer

The velocity of the neutron needed to achieve a wavelength of \(0.955 \AA\) is approximately \(4.15\times 10^{3} ms^{-1}\).

Step 1: Find the de Broglie wavelength formula

The de Broglie wavelength formula is used to relate the wavelength and momentum of a particle. It is given by: \(λ = \frac{h}{p}\) where, \(λ\) = wavelength of the particle h = Planck's constant (6.63 x 10^{-34} Js) p = momentum of the particle, given by (mass x velocity).

Step 2: Rearrange the formula to solve for velocity

We need to find the velocity of the neutron. So, we rearrange the de Broglie wavelength formula to solve for velocity (v). \(\lambda = \frac{h}{m_{n}v}\) \(v = \frac{h}{m_{n} \lambda}\) where, \(m_{n}\) = mass of the neutron.

Step 3: Substitute the given values and mass of neutron

Now we will substitute the given values in the formula. The mass of the neutron can be found on the inside cover of the book and is equal to \(1.675\times 10^{-27} kg\). The given wavelength is \(0.955\times 10^{-10} m\). \(v = \frac{6.63\times 10^{-34} Js}{(1.675 \times 10^{-27} kg)(0.955 \times 10^{-10} m)}\)

Step 4: Calculate the velocity

Next, we will calculate the velocity of the neutron using a calculator. \(v = \frac{6.63\times 10^{-34}}{(1.675 \times 10^{-27})(0.955 \times 10^{-10})} = \frac{6.63\times 10^{-34}}{1.598 \times 10^{-37}}\) \(v = 4.15\times 10^{3} ms^{-1}\)

Step 5: State the final answer

The velocity of the neutron needed to achieve a wavelength of 0.955 Angstrom is approximately \(4.15\times 10^{3} ms^{-1}\).

Key Concepts

de Broglie Wavelength

The concept of de Broglie wavelength is a fundamental principle in quantum mechanics, introduced by physicist Louis de Broglie. It asserts that all particles exhibit wave-like properties. The de Broglie wavelength (abla) is given by the equation:

• \(\lambda = \frac{h}{p}\)

Here, \(\lambda\) represents the wavelength, \(h\) is Planck's constant, and \(p\) represents the momentum of the particle. This equation suggests that lighter particles such as neutrons, when moving at high velocities, demonstrate significant wave-like behavior, making neutron diffraction a viable method for probing molecular structures.
Neutron diffusion utilizes this principle by allowing neutrons to interact with materials, providing insights into their atomic arrangements. The effectiveness of this technique hinges on matching the neutron's wavelength with the spacing between atoms, a factor adjusted by altering the neutron's velocity.

Planck's Constant

Planck's constant (denoted as \(h\)) is a pivotal constant in physics, crucial to the world of quantum mechanics. This constant defines the scale at which quantum effects occur, with a value of approximately \(6.63 \times 10^{-34} Js\).
Named after Max Planck, it's foundational to the de Broglie wavelength equation, bridging the relationship between a particle's wave and particle characteristics.

• Planck's constant allows us to connect the wavelength and momentum through the equation: \(\lambda = \frac{h}{p}\)
• It acts as a conversion factor between the frequency of a wave and its energy in Planck's famous equation: \(E = hf\)

Understanding Planck's constant helps us comprehend how tiny particles like neutrons can display behavior typically associated with larger wave-like phenomena.

Momentum of Particles

Momentum is a central concept in both classical and quantum physics. It’s a measure of the quantity of motion an object has and is defined by the equation:

• \(p = mv\)

where \(p\) is momentum, \(m\) stands for mass, and \(v\) is velocity. In the context of quantum physics and neutron diffraction, momentum plays a crucial role in determining wave properties.
Since the de Broglie wavelength (\(\lambda\)) is inversely proportional to the momentum \(p\), smaller particles moving at higher velocities will have short wavelengths. This property is essential for the functioning of neutron diffraction techniques, which rely on manipulating momentum to match the wavelengths needed for accurate measurement of atomic and molecular structures.

• Lowering the velocity of a neutron increases its wavelength, beneficial for examining structures with larger atomic spacing.
• Conversely, increasing velocity reduces the wavelength, suitable for closely packed atomic arrangements.

Understanding momentum and its role in particle behavior provides insights into the wave-particle duality, a cornerstone of modern physics.