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Chapter 2: Problem 122

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

Expert verified
The chemistry instructor's claim is not reasonably accurate. The actual size of a nucleus is approximately \(1.0 \times 10^{-14}\) meters, and the average distance of an electron from the nucleus is about \(5.3 \times 10^{-11}\) meters for a hydrogen atom. Scaling up the size of a nucleus to a grape (0.02 meters), we find that the scaled-up distance between the nucleus and electrons would be \(1.06 \times 10^5\) meters, which is approximately 65.85 times larger than the claimed distance (1 mile, or 1609.34 meters).

Step by step solution

01

Find the actual size of a nucleus and the distance between the nucleus and electrons

The size of a nucleus is approximately \(1.0 \times 10^{-14}\) meters, and the average distance of an electron from the nucleus is about \(5.3 \times 10^{-11}\) meters for a hydrogen atom.
02

Scale up the size of a nucleus to a grape and calculate the corresponding scale factor

Let's consider the size of a grape to be approximately 2 cm (0.02 meters). The scale factor is the ratio of the grape size to the actual nucleus size: \[\text{Scale factor} = \frac{\text{Size of grape}}{\text{Actual size of nucleus}}\] Now, let's calculate the scale factor: \[\text{Scale factor} = \frac{0.02}{1.0 \times 10^{-14}} = 2 \times 10^{15}\]
03

Determine the distance of the electrons using the scale factor

Now, let's multiply the actual distance between the nucleus and electrons by the scale factor to find the scaled-up distance: \[\text{Scaled-up distance} = \text{Actual distance} \times \text{Scale factor}\] Solving for the scaled-up distance: \[\text{Scaled-up distance} = (5.3 \times 10^{-11}) (2 \times 10^{15}) = 1.06 \times 10^5 \ \text{meters}\]
04

Compare the calculated distance with the claimed distance (1 mile)

To compare our calculated scaled distance with the claimed distance (1 mile), let's convert 1 mile to meters: 1 mile = 1609.34 meters Now let's see how close our calculated distance is to the claimed 1 mile away: \[\frac{\text{Calculated distance}}{\text{Claimed distance}} = \frac{1.06 \times 10^5}{1609.34} = 65.85\]
05

Analyze and conclude

The calculated scaled distance between the nucleus and the electrons is approximately 65.85 times larger than the claimed distance (1 mile). It shows that the claim "if the nucleus were the size of a grape, the electrons would be about 1 mile away on average" is not reasonably accurate.

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