Question

A particle cannot generally be localized to distances much smaller than its de Broglie wavelength. This fact can be taken to mean that a slow neutron appears to be larger to a target particle than does a fast neutron in the sense that the slow neutron has probabilities of being found over a larger volume of space. For a thermal neutron at room temperature of 300 K , find (c) State how this effective size compares with both nuclear and atomic dimensions.

Answer 1

A particle cannot generally be localized to distances much smaller than its de Broglie wavelength. This fact can be taken to mean that a slow neutron appears to be larger to a target particle than does a fast neutron in the sense that the slow neutron has probabilities of being found over a larger volume of space. For a thermal neutron at room temperature of 300 K, the effective size compares with both nuclear and atomic dimensions.

We will first find the kinetic energy at room temperature

We will be using the equation:

$E=\frac{3}{2}K_BT=\frac{3}{2} \times(1.38 \times 10^{-23}J/k)(300k)=6.21 \times 10^{-21} J$

Relation between Kinetic energy and momentum is given by:

$E=\frac{p^2}{2m}$

$E$ is the kinetic energy

$p$ is the linear momentum

$m$ is the mass of the neutron

We can find the value of $p$ by rearranging the equation

The mass of the neutron is$1.67 \times 10^{-27} kg$  The value of E as calculated above is $6.21 \times 10^{-21} J$.

Therefore,

$p = \sqrt{2\times 1.67 \times 10^{-27} \times 6.21 \times 10^{-21} }\\p=4.56\times 10^{-24}kg \cdot m/s$

The relation between de Broglie wavelength and the momentum is given by:

$\lambda = \frac{h}{p}$

where, $h$ is the Plank's constant and other terms have their usual meaning.

Substituting the value of $h$ and $p$, we get:

$\lambda = \frac{6.626 \times 10^{-34}}{4.56 \times 10^{-24}} =1.45 \times 10^{-10}m$

Therefore, $\lambda = 145pm$

We  observed that the effective size of the neutron is larger than the size of the nucleus with the order of $10^{-15}$, and its size is comparable with the size of the atom with the order of $10^{-10}$.

To study more about momentum. refer:

brainly.com/question/7538238

#SPJ4