Question

Neutron diffraction is used in determining the structures of molecules. a. Calculate the de Broglie wavelength of a neutron moving at \(1.00 \%\) of the speed of light. b. Calculate the velocity of a neutron with a wavelength of \(75 \mathrm{pm}\left(1 \mathrm{pm}=10^{-12} \mathrm{~m}\right)\)

Short answer

a. The de Broglie wavelength of a neutron moving at 1% of the speed of light is approximately \(1.319 \times 10^{-13} m\). b. The velocity of a neutron with a wavelength of \(75 pm\) is approximately \(5.281\times10^4 m/s\).

Step-by-step solution

Part a: Calculate the de Broglie wavelength

To calculate the de Broglie wavelength, we first need the formula: \(λ = \dfrac{h}{mv}\) Where λ is the de Broglie wavelength, h is Planck's constant (\(6.626 \times 10^{-34} Js\)), m is the mass of a neutron (\(1.675 \times 10^{-27}kg\)), and v is the velocity of the neutron. The velocity of the neutron is 1% of the speed of light, so we have to calculate that first: \(v = 0.01c\) Where c is the speed of light (\(3.00 \times 10^8 m/s\)) and 0.01 represents the 1% of the speed of light.

Calculate the velocity of the neutron

First, let's find the velocity of the neutron: \(v = 0.01\times(3.00 \times 10^8 m/s) = 3.00 \times 10^6 m/s\)

Compute the de Broglie wavelength

Now, we can use the given formula to calculate the de Broglie wavelength: \(λ = \dfrac{6.626 \times 10^{-34} Js}{(1.675\times10^{-27} kg)(3.00\times10^6 m/s)}\) \(λ = \dfrac{6.626 \times 10^{-34}}{5.025\times10^{-21}}\) \(λ = 1.319 \times 10^{-13} m\) Therefore, the de Broglie wavelength of the neutron moving at 1% of the speed of light is approximately \(1.319 \times 10^{-13} m\).

Part b: Calculate the velocity of a neutron with a given wavelength

In this part, we are given the wavelength and need to find the velocity. We can rearrange the de Broglie wavelength formula to calculate the velocity: \(v = \dfrac{h}{mλ}\) We are given the wavelength as \(75 pm\), which is equal to \(75\times10^{-12} m\).

Calculate the velocity of the neutron

Now, we can plug in the values to calculate the velocity of the neutron: \(v = \dfrac{6.626 \times 10^{-34} Js}{(1.675\times10^{-27} kg)(75\times10^{-12}m)}\) \(v = \dfrac{6.626 \times 10^{-34}}{1.256\times10^{-38}}\) \(v = 5.281\times10^4 m/s\) Therefore, the velocity of a neutron with a wavelength of \(75 pm\) is approximately \(5.281\times10^4 m/s\).