Question

It is good practice to actively read the textbook and to try to verify claims that are made when you can. The following claim is made in your textbook: " ... if the nucleus were the size of a grape, the electrons would be about one mile away on average." Provide mathematical support for this statement.

Short answer

The statement "if the nucleus were the size of a grape, the electrons would be about one mile away on average" is mathematically valid as an approximation. We found the ratio between the size of a grape (0.02 meters) and the size of an atomic nucleus (1.2 x 10^-14 meters) to be approximately 1.67 x 10^15. Using the Bohr radius, the average distance of an electron from the nucleus (5.29 x 10^-11 meters), we calculated the scaled distance to be approximately 8.84 x 10^4 meters. Converting this to miles, the scaled distance is about 0.55 miles. Despite not being exactly 1 mile, this still illustrates the significant distance between the nucleus and the electrons within an atom.

Step-by-step solution

Identify the size of a nucleus and a grape

We know that the size of an atomic nucleus is approximately 1.2 x 10^-14 meters for a medium-sized atom, and a grape is about 2 centimeters (0.02 meters) in diameter.

Find the ratio of the size of a grape to the size of a nucleus

We will now find the ratio between the size of a grape and the size of an atomic nucleus, which will help us in finding the scaled distance of electrons from the nucleus. Ratio = (Size of grape) / (Size of nucleus) Ratio = (0.02 meters) / (1.2 x 10^-14 meters) Ratio ≈ 1.67 x 10^15

Identify the average distance of electrons from the nucleus

The average distance of an electron from the nucleus in a hydrogen atom is about 5.29 x 10^-11 meters, which is also called the Bohr radius (denoted by \(a_0\)). It is an approximation as the actual electron distance varies depending on the atomic size and electron's energy level.

Calculate the scaled distance of electrons from the nucleus

Now that we have the ratio between the size of a grape and the size of an atomic nucleus, we will multiply this ratio by the Bohr radius to get the scaled distance of electrons from the nucleus when the nucleus is the size of a grape. Scaled distance = Ratio x Bohr radius Scaled distance = (1.67 x 10^15) x (5.29 x 10^-11 meters) Scaled distance ≈ 8.84 x 10^4 meters

Convert the scaled distance to a familiar unit (miles)

To make the calculated distance more relatable, we will convert the scaled distance from meters to miles. 1 mile ≈ 1609 meters Scaled distance in miles = (8.84 x 10^4 meters) / (1609 meters) Scaled distance in miles ≈ 0.55 miles Although the result is not exactly 1 mile, we can observe that the approximation in the textbook statement is valid, considering differences in atomic sizes and electron distances. The statement can be considered as an easy-to-relate presentation of the significant distance between the nucleus and the electrons within an atom. With the nucleus being the size of a grape, electrons would be more than half a mile away.

Key concepts

Bohr Radius

The concept of the Bohr radius, denoted as \( a_0 \), is central to understanding the architecture of an atom. It represents the mean distance between the nucleus and the electron in the ground state of a hydrogen atom, which equals approximately \(5.29 \times 10^{-11}\) meters. This value is a fundamental constant in atomic physics and serves as a reference point for atomic scale measurements.

To help visualize this incredibly small distance, let's analogize it with something familiar. Imagine if a standard track on a football field was scaled down so that the fifty-yard line represented the nucleus of an atom. In this scenario, the Bohr radius would place the electron nearly at the borders of the field. However, in reality, the electron does not orbit the nucleus in perfect circles like lanes on a track. It is better described by a cloud-like region defining the probable locations where an electron might be found, which quantum mechanics predicts.

Nuclear Size

The nucleus of an atom, while immensely dense, is tiny in comparison to the rest of the atom. Its size is about \(1.2 \times 10^{-14}\) meters for a medium-sized atom. To put this in perspective, let's use an everyday object—a grape, which is approximately 2 centimeters (or 0.02 meters) in diameter. The nuclear size is dwarfed by the size of a grape to such an extent that if the nucleus were to be scaled up to the size of a grape, the proportions would be radically different than what we observe.

Let's consider the following h4 subheadings to break down this concept:

A Lesson in Scale

When comparing objects of vastly different sizes, it is often beneficial to use ratios. For instance, if you were to scale up the size of the nucleus to match a grape, you would be magnifying it by a factor of around \(1.67 \times 10^{15}\). This extreme difference illustrates just how small the nucleus is in comparison to the electron's cloud around it.

The Core of the Matter

Despite its size, the nucleus carries nearly all the mass of an atom. Protons and neutrons, which make up the nucleus, are collectively called nucleons. Their mass dwarfs that of the orbiting electrons, giving the atom its substance and general properties.

Atomic Scale Modeling

Understanding objects at an atomic scale requires abstract thinking and, often, models to assist in visualization. Atomic scale modeling is a technique used by scientists and educators to illustrate the relationships between the nucleus and the electrons of an atom. It often involves scaling up atomic distances to more relatable, human-scale measurements. This is precisely what was done when comparing the size of an atomic nucleus to a grape and then determining the corresponding scaled distance of electrons.

The process of atomic scale modeling not only aids in visualization but also supports the learning of fundamental concepts of atomic physics. For instance, it helps to convey the vast emptiness of an atom, emphasizing that most of an atom's volume is actually unoccupied space. Such models also illustrate principles of quantum mechanics, like electron probability distributions, since electrons do not maintain fixed orbits like planets around the sun, but rather exist in areas of probability called orbitals.

Effective atomic scale modeling can leave a lasting impression, enabling students to grasp the scale and structure of the atomic world, which is not perceptible to the naked eye. Through these models, we can communicate the elegance and complexity within the seemingly simple building blocks of matter.