Question

If the gold foil in a Rutherford type experiment is \(1 / 10,000\) in. thick, what is a probable minimum number of gold atoms an alpha particle passed through before hitting the fluorescent screen? 1 in \(=2.54 \mathrm{~cm}, 1 \mathrm{~cm}=10^{8} \AA\). The radius of a gold atom is \(1.5 \AA\)

Short answer

The probable minimum number of gold atoms an alpha particle passes through before hitting the fluorescent screen is approximately 8,467 atoms.

Step-by-step solution

Convert gold foil thickness to Angstrom (Å)

First, we need to convert the given thickness of the gold foil, which is \(1 / 10,000\) inches, into Angstroms (Å): 1. Convert inches to centimeters: \(1/10,000 \ \text{in} = (1/10,000) * 2.54 \ \text{cm}\) 2. Convert centimeters to Angstrom: \((1/10,000) * 2.54 \text{ cm} = (1/10,000) * 2.54 * 10^8 \ Å\)

Calculate the number of gold atoms expected in a single layer

Now, let's calculate the number of gold atoms expected in one single layer of the gold foil. Using the diameter of a gold atom, we can find the thickness of a single layer of gold atoms: 1. Calculate the diameter of a gold atom: \(1.5 \ Å * 2 = 3 \ Å\)

Determine the probable minimum number of gold atoms

We will divide the gold foil's thickness (in Å) by the thickness of a single layer of gold atoms (in Å) to find the probable minimum number of gold atoms an alpha particle passes through. Let's call this number N. \[ N = \frac{(1/10,000) * 2.54 * 10^{8} \ Å}{3 \ Å} \] Now simplify: \[ N = (1/10,000) * 2.54 * \frac{10^{8}}{3} \] \[ N = \frac{2.54 * 10^{8}}{30,000} \] \[ N = 8,467\] From this calculation, the probable minimum number of gold atoms an alpha particle passes through before hitting the fluorescent screen is approximately 8,467 atoms.

Key concepts

Alpha Particles

Alpha particles are tiny, positively charged particles made up of two protons and two neutrons. They essentially form the nucleus of a helium atom. These particles play a crucial role in nuclear physics and are commonly emitted during radioactive decay processes. During experiments like Rutherford's, alpha particles are used because their positive charge and relatively large mass (compared to other atomic particles) make them perfect projectiles for probing atomic structures. When guided towards thin foils, such as gold, they can scatter after impacting atomic nuclei, providing vital information about the inner workings and arrangement of atoms.

Gold Foil Experiment

The gold foil experiment, also known as Rutherford scattering, was a groundbreaking experiment conducted by Ernest Rutherford in 1909. This experiment aimed to uncover the atomic structure of matter.

• In this setup, alpha particles were directed at a thin sheet of gold foil.
• Surprisingly, though most particles passed straight through, some were deflected at various angles, and a few even bounced back.

This unexpected scattering pattern led to the conclusion that atoms are mostly empty space with a tiny, dense nucleus at the center. The gold foil experiment significantly changed the atomic model by replacing the "plum pudding" model with Rutherford's nuclear model, illustrating that the atom consists of a dense core surrounded by orbiting electrons.

Atomic Structure

Atomic structure refers to the arrangement of subatomic particles – electrons, protons, and neutrons – within an atom. This concept is essential for understanding chemical reactions and the behavior of elements. In the Rutherford model of the atom, derived from the gold foil findings:

• The atom is mostly empty space, allowing alpha particles to pass through with little deflection.
• A dense, positively charged nucleus resides at the center, which deflects a few alpha particles.

Electrons orbit around this nucleus in different energy levels. This model laid the groundwork for more advanced atomic theories, influencing how we perceive matter at an atomic level.

Thickness Conversion

Converting thickness from one unit to another is crucial in experiments and calculations, such as the Rutherford scattering experiment. To analyze the number of layers that alpha particles penetrate, it's essential to convert the thickness of materials accurately into the units used in calculations. In the problem statement, the thickness of the gold foil is given in inches. For practical physical calculations:

• First, convert inches to centimeters. Since 1 inch equals 2.54 cm, the conversion scales accordingly.
• Then, convert centimeters to Angstroms (Å), where 1 cm equals 10^8 Å.

Converting these measures allows us to compute the number of atom layers an alpha particle passes through. This specific skill of converting measurements is valuable in scientific calculations to ensure precision and accuracy.