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Chapter 2: Problem 54

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

Expert verified
a. The de Broglie wavelength of the neutron moving at \(1\%\) of the speed of light is approximately \(1.32 \times 10^{-13} m\). b. The velocity of a neutron with a wavelength of \(75 pm\) is approximately \(5.29 \times 10^3 m/s\).

Step by step solution

01

a. Calculating the de Broglie wavelength of a neutron moving at 1% of the speed of light

First, we need the de Broglie wavelength formula, which is: \[\lambda = \frac{h}{p}\] where lambda \(\lambda\) represents the wavelength, h is Planck's constant, and p is the momentum of the particle. Now we need to find the momentum of the neutron, which can be determined by using the formula: \[p = mv\] where m is the mass of the neutron and v is its velocity. The mass of a neutron is approximately \(1.67 \times 10^{-27}\) kg, and its velocity can be found by taking \(1\%\) of the speed of light (\(3 \times 10^{8} m/s\)). So the velocity will be: \[v = 0.01 \times 3 \times 10^8 = 3 \times 10^6 m/s\] Now we can use this velocity to determine the momentum, and subsequently the de Broglie wavelength.
02

Calculate the momentum

Use the mass of the neutron and the calculated velocity to find its momentum: \[p = (1.67 \times 10^{-27})(3 \times 10^6) = 5.01 \times 10^{-21} kg \cdot m/s\]
03

Calculate the de Broglie wavelength

Use the momentum we just found and plug it into the de Broglie wavelength formula. Planck's constant is approximately \(6.626 \times 10^{-34} Js\): \[\lambda = \frac{6.626 \times 10^{-34}}{5.01 \times 10^{-21}} = 1.32 \times 10^{-13} m\] Hence, the de Broglie wavelength of the neutron moving at \(1\%\) of the speed of light is approximately \(1.32 \times 10^{-13} m\).
04

b. Calculating the velocity of a neutron with a wavelength of 75 pm

Now, we need to calculate the velocity of a neutron with a given de Broglie wavelength. For this, we already know the wavelength \(\lambda\) and can use the same de Broglie wavelength formula to find the velocity.
05

Calculate the momentum from the given wavelength

Rearrange the de Broglie wavelength formula to solve for the momentum: \[p = \frac{h}{\lambda}\] We are given that the wavelength is \(75 pm = 75 \times 10^{-12} m\). Now, using this value for \(\lambda\) and Planck's constant, we have: \[p = \frac{6.626 \times 10^{-34}}{75 \times 10^{-12}} = 8.835 \times 10^{-24} kg \cdot m/s\]
06

Calculate the velocity using the momentum

Now, we can use the momentum to find the velocity of the neutron, with the same formula: \[v = \frac{p}{m}\] We already have the mass of a neutron, so using the calculated momentum, we can solve for the velocity: \[v = \frac{8.835 \times 10^{-24}}{1.67 \times 10^{-27}} = 5.29 \times 10^3 m/s\] Therefore, the velocity of a neutron with a wavelength of \(75 pm\) is approximately \(5.29 \times 10^3 m/s\).

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