Question

What fraction of the $\alpha$ particles in Rutherford's gold foil experiment are scattered at large angles? Assume the gold foil is two layers thick, as shown in Figure $2.9,$ and that the approximate diameters of a gold atom and its nucleus are 270 $\mathrm{pm}$ and $1.0\times {10}^{-2}\mathrm{pm}$, respectively. Assume that the gold nuclei in each layer are offset from each other.

Short answer

The fraction of alpha particles scattered at large angles in Rutherford's gold foil experiment is approximately $2.7\times {10}^{-9}$.

Step-by-step solution

Determine the number of gold nuclei and alpha particles

To calculate the fraction of alpha particles scattered at large angles, we need to know the number of gold nuclei in the foil and the number of alpha particles incident on the foil. The gold foil is two layers thick, and the nuclei are offset from each other. Let's assume N represents the number of gold nuclei in the foil and M represents the number of alpha particles incident on the foil.

Calculate the probability of an alpha particle interacting with a gold nucleus

We will now compute the probability of an alpha particle interacting with a gold nucleus. The probability of interaction can be determined by considering the cross-sectional area of a gold nucleus and comparing it to the area occupied by a gold atom. The cross-sectional area of a gold nucleus can be approximated as a circle with radius $r_n=1.0\times {10}^{-2}\mathrm{pm}$, and the area of a gold atom as a circle with radius $r_a=270\mathrm{pm}$. The probability of interaction can be determined as the ratio of the cross-sectional area of the nucleus to the area of the atom: $P_{interaction}=\frac{\pi {r_n}^2}{\pi {r_a}^2}$ Notice that the pi term cancels out: $P_{interaction}=\frac{{r_n}^2}{{r_a}^2}$

Calculate the probability of interaction

Now, plug in the values for the radii of the gold atom and its nucleus: $P_{interaction}=\frac{(1.0\times {10}^{-2}\mathrm{pm}{)}^2}{(270\mathrm{pm}{)}^2}$ Calculate the probability of interaction: $P_{interaction}=1.37\times {10}^{-9}$

Determine the fraction of alpha particles scattered at large angles

Now we need to take into account that the alpha particles have the possibility to interact with two layers of gold nuclei. The probability that an alpha particle does not interact with either layer can be calculated as: $P_{nointeraction}=(1-P_{interaction}{)}^2$ Calculate the probability of no interaction: $P_{nointeraction}=(1-1.37\times {10}^{-9}{)}^2\approx 0.9999999973$ Now, the fraction of alpha particles scattered at large angles is equal to the complement of the probability of no interaction: $P_{largeangles}=1-P_{nointeraction}$ Calculate the fraction of alpha particles scattered at large angles: $P_{largeangles}=1-0.9999999973\approx 2.7\times {10}^{-9}$ The fraction of alpha particles scattered at large angles in Rutherford's gold foil experiment is approximately $2.7\times {10}^{-9}$.

Key concepts

Alpha Particles

Alpha particles, often represented as $\alpha$ particles, are a type of ionizing radiation consisting of two protons and two neutrons tightly bound together. This makes them identical to a helium-4 nucleus. - **Radioactive Decay:** Alpha particles are typically emitted by heavy elements like uranium and radium as they undergo radioactive decay.- **Charge and Mass:** They have a positive charge due to the presence of protons without any surrounding electrons, and a significant mass relative to other forms of radiation like beta particles or gamma rays.- **Interactions:** When encountering other materials, like in Rutherford's experiment, alpha particles can cause various interactions depending on the thickness and type of foil they penetrate.In the context of Rutherford's experiment, alpha particles were crucial because their positive charge allowed them to be deflected by the gold nuclei. Their interactions and scatter patterns provided groundbreaking insight into the structure of the atom, suggesting the presence of a small, dense, positively charged nucleus at the center of atoms.

Gold Foil

In Rutherford's famous experiment, a thin sheet of gold foil was used to study the scattering of alpha particles. - **Material Characteristics:** Gold was chosen for its malleability, which enables it to be hammered into extremely thin sheets just a few atoms thick. This thinness was necessary to allow high precision in detecting how alpha particles behaved as they passed through. - **Two-Layer Thickness:** For this specific exercise, the gold foil is considered to be two layers thick. These layers were offset, meaning the second layer did not sit directly behind the first, ensuring better detection of scattered paths by detecting the paths of alpha particles. - **Role in Experiment:** The foil was a crucial component in revealing how particles were deflected, thus indicating the distribution of charge and mass within the atom. Using gold foil allowed Rutherford to observe not just straightforward atom collisions but deflections occurring at varying angles, helping unravel the mystery of atomic structure.

Scattering Angles

Scattering angles refer to the directions and degrees to which the paths of alpha particles change as they pass through the gold foil. - **Definition:** The angle at which an alpha particle is deflected from its initial path after interacting with a nucleus. - **Large Angle Scattering:** A significant fraction of the alpha particles were observed to scatter at large angles, indicating that they had encountered a dense and positively charged nucleus powerful enough to alter their paths dramatically. - **Implications:** These observations led Rutherford to propose the nuclear model of the atom, where most of an atom's mass and all of its positive charge are concentrated within a tiny central nucleus. The study of these angles was essential in understanding atomic structure, revealing complexities that simply wouldn’t be apparent with small angle scatterings. This demonstrated the non-uniform nature of the atom, contrary to the previously accepted "plum pudding" model of atomic structure.