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 A. (Refer to the inside cover for the mass of the neutron.)

Short answer

The velocity of a neutron needed to achieve a wavelength of 1.25 Å is approximately \(3.16 \times 10^5 ms^{-1}\).

Step-by-step solution

Write down the given values and constants

We are given: - Wavelength \(\lambda = 1.25\) Å (note: 1 Å = \(10^{-10}\) m, so we need to convert the wavelength to meters) - Mass of neutron \(m = 1.675 \times 10^{-27}\) kg (given on the inside cover) - Planck's constant \(h = 6.626 \times 10^{-34}\) Js

Convert the wavelength from Å to meters

Convert the wavelength from Å to meters by using the conversion factor: \(\lambda = 1.25\) Å \(\times 10^{-10}\frac{m}{Å} = 1.25 \times 10^{-10}\) m

Set up the de Broglie wavelength formula

Using the de Broglie wavelength formula, substitute the given values and constants, and solve for the velocity (v): \[\lambda = \frac{h}{p} = \frac{h}{mv}\]

Solve for the velocity of the neutron

Rearrange the de Broglie wavelength formula to solve for the velocity: \[v = \frac{h}{m \lambda}\] Plug in the values: \[v = \frac{6.626 \times 10^{-34} Js}{(1.675 \times 10^{-27} kg)(1.25 \times 10^{-10} m)}\]

Calculate the velocity

Perform the calculation: \[v \approx 3.16 \times 10^5 ms^{-1}\] So, the velocity of a neutron needed to achieve a wavelength of 1.25 Å is approximately \(3.16 \times 10^5 ms^{-1}\).

Key concepts

de Broglie wavelength

The de Broglie wavelength is a fundamental concept that describes the wave-like behavior of particles. According to de Broglie's hypothesis, any moving particle or object has a wave associated with it. This wave is described by the wavelength denoted as \(\lambda\). The formula for the de Broglie wavelength is:\[\lambda = \frac{h}{p}\]where \(h\) is Planck's constant \((6.626 \times 10^{-34} Js)\) and \(p\) is the momentum of the particle.For particles with mass (like neutrons), momentum \(p\) is expressed as \(mv\), the product of mass \(m\) and velocity \(v\). Plugging this into the de Broglie equation gives:\[\lambda = \frac{h}{mv}\]This equation allows for calculation of the wavelength of a moving particle when its velocity is known, or vice versa. It demonstrates the fascinating duality in quantum mechanics, highlighting that matter can exhibit both particle-like and wave-like properties. If you know the desired wavelength, the equation can be rearranged to solve for the particle's velocity.

wavelength conversion

When working on problems involving wavelengths, unit conversion is crucial, especially in scientific calculations. In this exercise, the wavelength of 1.25 Ångströms (Å) must be converted to meters because scientific calculations rely on SI units.1 Å is equivalent to \(10^{-10}\) meters, allowing for a straightforward conversion:- Given: \(\lambda = 1.25 \text{ Å}\)- Conversion: \(1 \text{ Å} = 10^{-10} m\)- Therefore: \(\lambda = 1.25 \times 10^{-10} \text{ m}\)This conversion helps us use the wavelength in the de Broglie equation appropriately. Having everything in standard units not only aligns with scientific norms but also reduces errors when performing calculations. Converting units consistently and correctly is a fundamental skill in handling real-world physics problems.

calculation of velocity

To find the velocity of a neutron given a specific wavelength, we use the de Broglie wavelength formula rearranged for velocity:\[v = \frac{h}{m\lambda}\]Using this formula, step-by-step, we can compute the velocity:- Planck's constant \(h = 6.626 \times 10^{-34} Js\)- Mass of neutron \(m = 1.675 \times 10^{-27} kg\)- Wavelength in meters \(\lambda = 1.25 \times 10^{-10} m\)Plug these values into the equation:\[v = \frac{6.626 \times 10^{-34} Js}{(1.675 \times 10^{-27} kg)(1.25 \times 10^{-10} m)}\]Perform the calculation to find:\[v \approx 3.16 \times 10^5 ms^{-1}\]This velocity \(v\) represents the speed a neutron must travel to have a wavelength of 1.25 Å, which is key in neutron diffraction studies. Calculating such precise physical quantities is essential in the field of quantum mechanics and aids in understanding the wave properties of microscopic particles.