Problem 49 - 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 \(1.25 \AA\). (Refer to the inside cover for the mass of the neutron.)

Step-by-Step Solution

Answer

To calculate the velocity of a neutron needed to achieve a wavelength of \(1.25 \AA\), we can use the De Broglie relation: \(\lambda = \frac{h}{mv}\). First, convert the given wavelength to meters: \(\lambda = 1.25 \times 10^{-10} m\). The mass of the neutron is \(m_{n} = 1.675 \times 10^{-27} kg\). Rearrange the De Broglie equation to solve for the velocity: \(v = \frac{h}{m \lambda}\). Plug in the values and solve for the velocity: \(v = \frac{6.626 \times 10^{-34} Js}{(1.675 \times 10^{-27} kg)(1.25 \times 10^{-10} m)}\). Thus, the velocity of the neutron is approximately \(3.155 \times 10^5 \ m/s\).

Step 1: 1. Recall the De Broglie relation

The De Broglie relation states that: \[\lambda = \frac{h}{mv}\] where \(\lambda\) is the wavelength, \(h\) is the Planck's constant (\(6.626 \times 10^{-34} \ Js\)), \(m\) is the mass, and \(v\) is the velocity of the particle.

Step 2: 2. Convert the given wavelength to meters

We are given the wavelength of the neutron as \(1.25 \AA\). Let's convert this to meters: \[\lambda = 1.25 \AA = 1.25 \times 10^{-10} m\]

Step 3: 3. Find the mass of the neutron

Referring to the inside cover of your textbook (or any credible source of such information), find the mass of the neutron: \(m_{n} = 1.675 \times 10^{-27} kg\)

Step 4: 4. Rearrange the De Broglie equation to solve for the velocity

We want to find the velocity (\(v\)) of the neutron. Rearrange the De Broglie relation to solve for \(v\): \[v = \frac{h}{m \lambda}\]

Step 5: 5. Plug in the values and solve for the velocity

Now, plug in the values for the Planck's constant (\(h\)), the neutron's mass (\(m_{n}\)), and the wavelength (\(\lambda\)) into the equation: \[v = \frac{6.626 \times 10^{-34} Js}{(1.675 \times 10^{-27} kg)(1.25 \times 10^{-10} m)}\] Calculate the result: \[v = 3.155 \times 10^5 \frac{m}{s}\] So, to achieve a wavelength of \(1.25 \AA\), a neutron must have a velocity of approximately \(3.155 \times 10^5 \ m/s\).

Key Concepts

De Broglie relation

The De Broglie relation is a fundamental concept in quantum mechanics that bridges the gap between classical and quantum physics. It establishes that particles such as electrons, protons, and neutrons exhibit both particle and wave-like properties. The relation is given by:
\[\lambda = \frac{h}{mv}\]
where \(\lambda\) is the wavelength associated with the particle, \(h\) is Planck's constant, \(m\) is the particle's mass, and \(v\) is its velocity. This equation implies that every particle has an associated wavelength, which becomes significant at the quantum scale, such as in neutron diffraction experiments. Interestingly, as the particle's velocity increases, its wavelength decreases, and vice versa. This foundational principle is crucial when studying phenomena like diffraction patterns produced by particles passing through a crystal lattice.

Wavelength calculation

Calculating the wavelength of a particle involves manipulating the De Broglie relation. For certain applications such as neutron diffraction, the calculation of a neutron's wavelength is essential to understand the diffraction patterns that emerge when neutrons interact with materials at the atomic scale. To make these calculations tangible, we first convert the desired wavelength into SI units, meters, which is a straightforward step but crucial for consistency in equations. For instance, converting an angstrom to meters gives us:
\[1 \AA = 10^{-10} m\]
Once we have the wavelength in the correct units, we use the De Broglie relation to find other particle properties, like velocity, creating a clear connection between the abstract wave-like behavior of particles and their more familiar physical properties.

Neutron velocity

The velocity of a neutron is an important factor when conducting experiments like neutron diffraction. Understanding the neutron's velocity enables scientists to predict and analyze the interaction of neutrons with different materials at an atomic level. Using the De Broglie relation, we can rearrange it to solve for the neutron's velocity if we know its mass and desired wavelength:
\[v = \frac{h}{m\lambda}\]
This equation shows the inverse relationship between velocity and wavelength, reinforcing the wave-particle duality concept. Plugging in the values for Planck's constant, the neutron's mass, and the wavelength, we can solve for the velocity, which is essential for achieving the correct diffraction pattern needed to decipher the structure of molecules. Moreover, the accuracy of these values is critical; using precise constants and unit conversion ensures our calculation of the neutron's velocity is correct, facilitating the advancement of material science and molecular biology fields.