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Chapter 6: Problem 49

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

Expert verified
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

01

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
02

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
03

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): \[\lambda = \frac{h}{p} = \frac{h}{mv}\]
04

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: \[v = \frac{6.626 \times 10^{-34} Js}{(1.675 \times 10^{-27} kg)(1.25 \times 10^{-10} m)}\]
05

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 \(3.16 \times 10^5 ms^{-1}\).

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