Question

A chemistry instructor makes the following claim: "Consider that if the nucleus were the size of a grape, the electrons would be about 1 mile away on average." Is this claim reasonably accurate? Provide mathematical support.

Short answer

The chemistry instructor's claim is not reasonably accurate. Based on our calculations, when the nucleus is the size of a grape, the electrons would be approximately \(441,000 m\) away on average, which is far greater than one mile (\(1,609 m\)).

Step-by-step solution

Find the actual size of an atomic nucleus

An atomic nucleus is composed of protons and neutrons. The approximate size of an atomic nucleus can be represented as \(r = r_0 A^{1/3}\), where \(r\) is the radius of the nucleus, \(r_0\) is a constant of approximately \(1.2 \times 10^{-15} m\), and \(A\) is the atomic mass number (number of protons and neutrons). For simplicity, let's consider a hydrogen nucleus (a single proton) so that \(A=1\). Therefore, the nucleus can be approximated to have a radius of \(r = (1.2 \times 10^{-15})m\).

Find the average distance of an electron from the nucleus

The average distance of an electron from the nucleus in a hydrogen atom is equal to the Bohr radius, which is approximately \(5.29 \times 10^{-11}m\).

Calculate the scale factor

Now we must calculate the scale factor between the size of a grape and the size of the nucleus. Let's assume that a grape has a diameter of approximately 2 cm or a radius of 1 cm (\(0.01 m\)). The scale factor can then be found by dividing the grape radius by the nucleus radius: \(\frac{0.01}{1.2 \times 10^{-15}} \approx 8.33 \times 10^{15}\).

Scale the distance between the electron and the nucleus

To find the scaled distance of the electron from the nucleus, multiply the average electron-nucleus distance by the scale factor: \((5.29 \times 10^{-11})m \times (8.33 \times 10^{15}) \approx 441,000 m\).

Compare the scaled distance to one mile

Finally, we need to compare our scaled distance to one mile to see if the chemistry instructor's claim is reasonably accurate. One mile is equal to \(1,609 m\). The scaled distance between the electron and the nucleus in our model is approximately \(441,000 m\), which is significantly greater than one mile (\(1,609 m\)). In conclusion, based on our calculations, the chemistry instructor's claim is not reasonably accurate as the calculated distance is far greater than one mile when the nucleus is the size of a grape.