Question

An aluminum atom has an average diameter of about \(3.0 \times 10^{-8} \mathrm{~cm}\). The nucleus has a diameter of about \(2.0 \times 10^{-13} \mathrm{~cm}\). Calculate the ratio of the atom's diameter to its nucleus.

Short answer

The ratio of the atom's diameter to its nucleus is \(1.5 \times 10^{5}\).

Step-by-step solution

Write down the given diameters

The diameter of the aluminum atom is given as: \(3.0 \times 10^{-8} \mathrm{~cm}\).The diameter of the nucleus is given as: \(2.0 \times 10^{-13} \mathrm{~cm}\).

Set up the ratio

The ratio of the diameter of the atom to the nucleus is the diameter of the atom divided by the diameter of the nucleus. \[\text{Ratio} = \frac{\text{Diameter of atom}}{\text{Diameter of nucleus}}\]

Plug in the values and calculate the ratio

Substitute the given values into the ratio formula: \[\text{Ratio}= \frac{3.0 \times 10^{-8}}{2.0 \times 10^{-13}}\]Perform the division by subtracting the exponents of 10 because the bases are the same.

Calculate the final answer

Subtracting the exponents: \[\text{Ratio} = \frac{3.0}{2.0} \times 10^{(-8) - (-13)}\]\[\text{Ratio} = 1.5 \times 10^{5}\]So, the ratio of the atom's diameter to its nucleus is \(1.5 \times 10^{5}\).

Key concepts

Atomic Diameter

The diameter of an atom can be envisioned as the distance between one edge of the atom through its center to the opposite edge. Atoms are building blocks of matter, incredibly small particles that constitute every physical object in the universe. For example, an aluminum atom, a common element found in many materials, has an average diameter of about \(3.0 \times 10^{-8}\) centimeters.

Understanding the scale of atomic diameters is fundamental in chemistry and physics, as they provide insights into the structure of substances and help to explain properties like electrical conductivity and heat transfer. In the context of exercises, the diameter of an atom is usually provided to compare its size to that of its nucleus, or to calculate related properties such as the atom's radius or volume.

Nuclear Diameter

In contrast to the overall size of an atom, the nuclear diameter refers to the size of the nucleus alone. The nucleus is the dense central core of an atom where protons and neutrons are compactly arranged. For an aluminum atom, the nuclear diameter is about \(2.0 \times 10^{-13}\) centimeters, much smaller than the atom's overall diameter. This highlights the fact that most of the atom's volume is composed of empty space where electrons 'orbit' the nucleus.

The concept of nuclear diameter is pivotal when discussing nuclear physics and atomic scale phenomena. While the diameter of an entire atom is essential for understanding chemical interactions, the nuclear diameter is crucial for comprehending nuclear reactions, forces, and properties.

Ratio Calculation

Ratio calculations are a way of comparing different quantities to see their relative sizes or amounts. The ratio of the diameter of an atom to its nucleus is calculated by taking the atomic diameter and dividing it by the nuclear diameter. This calculation can reveal the astonishing fact that the nucleus is much smaller compared to the total size of the atom.

Such ratios can also have implications for understanding the empty space within an atom and the potential energy that results from the arrangement of particles within an atom. When faced with ratio problems, it's important to ensure that units are consistent and to use correct mathematical operations, as demonstrated in our textbook exercise solution.

Scientific Notation

Scientific notation is a way of expressing very large or very small numbers more succinctly. It's particularly useful in fields like physics and chemistry where such extreme values are common. In scientific notation, a number is written as the product of a number between 1 and 10 and a power of 10. For instance, the diameter of an aluminum atom is expressed as \(3.0 \times 10^{-8}\) centimeters.

When calculating ratios involving scientific notation, as shown in the textbook solution, we can simplify the process by subtracting the exponents of the powers of 10 when dividing. This method is a powerful tool for handling complex calculations that would otherwise be difficult to perform with standard decimal numbers.