Ch.2 - Atoms, Molecules, and Ions

Problem 17

Chapter 2, Problem 17

What fraction of the a particles in Rutherford's gold foil experimentare scattered at large angles? Assume the gold foilis two layers thick, as shown in Figure 2.9, and that the approximatediameters of a gold atom and its nucleus are 270pm and 1.0 * 10–2 pm, respectively. Hint: Calculate the crosssectional area occupied by the nucleus as a fraction of thatoccupied by the atom. Assume that the gold nuclei in eachlayer are offset from each other.

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Identify the key concept: Rutherford's gold foil experiment involved alpha particles being scattered by the nuclei of gold atoms. The fraction of particles scattered at large angles is related to the size of the nucleus compared to the atom.

Calculate the cross-sectional area of a gold atom: Use the formula for the area of a circle, \( A = \pi r^2 \), where \( r \) is the radius of the atom. Given the diameter of a gold atom is 270 pm, the radius is half of that.

Calculate the cross-sectional area of a gold nucleus: Similarly, use the formula \( A = \pi r^2 \) for the nucleus, where the diameter is given as \( 1.0 \times 10^{-2} \) pm, and find the radius.

Determine the fraction of the area occupied by the nucleus: Divide the cross-sectional area of the nucleus by the cross-sectional area of the atom to find the fraction of the atom's area that is occupied by the nucleus.

Consider the two-layer thickness of the gold foil: Since the gold nuclei in each layer are offset, the probability of scattering at large angles is related to the fraction of the area occupied by the nucleus in both layers.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Rutherford's Gold Foil Experiment

Rutherford's gold foil experiment demonstrated that atoms consist of a small, dense nucleus surrounded by a cloud of electrons. When alpha particles were directed at a thin foil of gold, most passed through, but a small fraction were deflected at large angles. This led to the conclusion that the nucleus is positively charged and occupies a tiny volume compared to the overall size of the atom.

Cross-Sectional Area

The cross-sectional area in this context refers to the effective area that a nucleus presents to incoming alpha particles. It is calculated based on the radius of the nucleus and is crucial for determining the likelihood of scattering events. By comparing the cross-sectional area of the nucleus to that of the entire atom, one can estimate the fraction of alpha particles that will interact with the nucleus.

Scattering Angles

Scattering angles in particle physics refer to the angles at which particles are deflected after colliding with a target. In the context of Rutherford's experiment, large-angle scattering indicates a significant interaction with the nucleus, while small-angle scattering suggests that the particles have passed through the electron cloud. Understanding these angles helps in quantifying the fraction of particles that experience strong interactions with the nucleus.

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