Question

If the nucleus of an atom were the size of a golf ball (4-cm diameter), what would be the diameter of the atom?

Short answer

The diameter of the atom would be 400 m.

Step-by-step solution

Understand the Scale

In most atoms, the nucleus is very small compared to the overall size of the atom. Typically, the nucleus is about 10,000 times smaller than the atom itself.

Identify the Given Size

In this problem, the nucleus is represented as having a 4 cm diameter (like a golf ball).

Calculate the Atom's Diameter

Using the scale factor from Step 1, multiply the diameter of the nucleus (4 cm) by 10,000 to find the diameter of the atom: \[ 4 ext{ cm} \times 10,000 = 40,000 ext{ cm} \]

Conversion to Meters

Convert the diameter of the atom from centimeters to meters by dividing by 100 (since 1 m = 100 cm):\[ 40,000 ext{ cm} \div 100 = 400 ext{ m} \]

Key concepts

Nuclear Scale

Atoms are incredibly tiny, yet they are composed of even smaller substructures. At the heart of each atom lies the nucleus, a dense region comprised mainly of protons and neutrons. Despite its mass and significance, the nucleus occupies only a very small part of the atom's volume. Imagine the nucleus like a tiny seed within a much larger fruit; this central seed represents the nucleus, while the rest of the fruit corresponds to the vast space occupied by the atom's electron cloud.
The size comparison between the nucleus and the atom is quite striking. Typically, a nucleus is about 10,000 times smaller than the entire atom. This means if you scaled up a nucleus to the size of a golf ball, the whole atom would expand to a size that's 10,000 golf balls lined up in a row. This illustrates the astonishingly empty space within atoms, where electrons orbit at relatively large distances from the nucleus. Understanding this nuclear scale helps us appreciate how compact and dense the nucleus is compared to the rest of the atom.

Atomic Diameter Calculation

Calculating the diameter of an atom involves understanding the relationship between the nucleus and the entire atomic structure. Once we acknowledge the scale factor that dictates the nucleus is about 10,000 times smaller than the atom, we can use it to calculate the atom's diameter when the nucleus is represented with a known size.
In our problem, we imagine a nucleus as a golf ball, giving it a diameter of 4 cm. To find out how large the atom would be, multiply this diameter by the scale factor:

• Given nucleus diameter: 4 cm
• Scale factor: 10,000 (nucleus size to atom size)

This gives us a calculation:\[4 \text{ cm} \times 10,000 = 40,000 \text{ cm}\]This shows that if our nucleus is 4 cm, the entire atom would have a diameter of 40,000 cm. This example provides a clear visualization of atomic proportions and helps to understand the vast space atoms occupy at the atomic level.

Unit Conversion in Measurements

Converting measurements from one unit to another is crucial across various scientific computations. In our exercise, after calculating the atom's diameter in centimeters, the final step involves converting that measurement into meters for a clearer understanding in scientific contexts.
Often, we need unit conversions to communicate magnitudes consistently, especially when dealing with such large numbers. To convert the diameter from centimeters to meters, use the conversion factor where 1 meter is equivalent to 100 centimeters.
This can be represented mathematically as:\[40,000 \text{ cm} \div 100 = 400 \text{ m}\]Here, we divide the total centimeters by 100, which efficiently scales the measurement into meters. This simple conversion makes it easier to grasp the atom's diameter, which turns out to be an extensive 400 meters when visualized on this scale. Unit conversion not only aids in solving problems accurately but also ensures that the results are meaningful and easy to interpret.