Problem 43 - 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

To find the velocity of a neutron needed to achieve a wavelength of \( 0.955 \AA \), we use the de Broglie wavelength formula \( v = \frac{h}{mλ} \). After converting the wavelength to meters (\( λ = 9.55 \times 10^{-11} \) m) and plugging in the values for Planck's constant (h) and the mass of the neutron (m), we get \( v = \frac{6.626 \times 10^{-34}}{(1.6749 \times 10^{-27})(9.55 \times 10^{-11})} \approx 4190.19 \, m/s \).

Step 1: Identify the given information and de Broglie wavelength formula

We are given the following information: - Wavelength (λ) of neutron: 0.955 Å (angstroms) - Mass (m) of the neutron: Refer to the inside cover (the mass of a neutron is approximately \(1.6749 \times 10^{-27}\) kg) We will use the de Broglie wavelength formula: \[λ = \frac{h}{mv}\]

Step 2: Convert the given wavelength from angstroms to meters

First, we need to convert the given wavelength from angstroms to meters, so that all units are consistent. Since 1 Å is equal to \( 1 \times 10^{-10} \) meters: 0.955 Å = \( 0.955 \times 10^{-10} \) meters = \( 9.55 \times 10^{-11} \) meters Now, the given wavelength (λ) is \( 9.55 \times 10^{-11} \) meters.

Step 3: Rearrange the de Broglie wavelength formula to solve for velocity (v)

We need to rearrange the equation to solve for v: \[λ = \frac{h}{mv} \Rightarrow v = \frac{h}{mλ}\]

Step 4: Plug in the given values and Planck's constant into the equation

Now, we plug in the given values and the value of Planck's constant (h) which is approximately \(6.626 \times10^{-34}\) Js: \[v = \frac{6.626 \times 10^{-34}}{(1.6749 \times 10^{-27})(9.55 \times 10^{-11})}\]

Step 5: Calculate the velocity of the neutron

Now, we calculate the velocity by performing the operations: \[v = \frac{6.626 \times 10^{-34}}{(1.6749 \times 10^{-27})(9.55 \times 10^{-11})} = 4190.19 m/s\] So, the velocity of the neutron needed to achieve a wavelength of 0.955 Å is approximately 4190.19 m/s.

Key Concepts

de Broglie Wavelength

The concept of de Broglie wavelength is fundamental in quantum mechanics. It explains how particles such as neutrons can exhibit wave-like properties. This is key in understanding neutron diffraction techniques. In simple terms, any moving particle has an associated wavelength, thanks to its wave-like characteristics. The de Broglie wavelength (\( \lambda \)) is calculated by the formula \( \lambda = \frac{h}{mv} \), where:

• \( h \) is Planck's constant
• \( m \) is the mass of the particle (such as a neutron, approximately \(1.6749 \times 10^{-27}\) kg)
• \( v \) is the velocity of the particle

To apply this formula effectively, it's crucial to ensure all units match, like converting wavelengths from angstroms to meters. Understanding the de Broglie wavelength helps in interpreting the results of neutron diffraction experiments, which reveal detailed molecular structures.

Planck's Constant

Planck's constant is a central quantity in quantum mechanics, denoted as \( h \). It connects the amount of energy a photon carries with the frequency of its electromagnetic wave. When examining particle wave characteristics like those in the de Broglie wavelength equation, Planck's constant emerges to link wave nature and particle behavior. Its value is approximately \( 6.626 \times 10^{-34} \) Joule seconds (Js).

Planck's constant is named after Max Planck, who introduced it while working on black-body radiation problems. It represents the smallest action or energy transfer measurable in quantum mechanics, highlighting the discrete nature of quantum phenomena. In neutron diffraction, its inclusion in the equation \( \lambda = \frac{h}{mv} \) is crucial for calculating the wavelength of particles like neutrons, efficiently unveiling molecular structures.

Neutron Velocity

Neutron velocity is a critical factor when calculating de Broglie wavelengths. It dictates how particles such as neutrons exhibit wave properties as they travel. Velocity in physics refers to how fast something is moving in a specified direction. For neutrons used in diffraction, achieving the right velocity is essential to obtain accurate wavelengths that match experimental conditions.

Using the rearranged de Broglie equation \( v = \frac{h}{m\lambda} \), one can solve for velocity when the mass and desired wavelength are known:

• \( h \) is Planck's constant
• \( m \) represents the mass of the neutron
• \( \lambda \) is the wavelength we aim to achieve

By plugging these values into the equation, you can find out how fast a neutron must travel to achieve a specific wavelength. Neutron velocity manipulation is pivotal in setting up neutron diffraction experiments, which plays a substantial role in studying and understanding material structures.