Question

Approximately how many times greater is the diameter of an atom than the diameter of its nucleus? Knowing that most of an atom’s mass is contained in the nucleus, what can you conclude about the density of the nucleus?

Short answer

The diameter of an atom is approximately \(10^4\) to \(2 \times 10^5\) times greater than the diameter of its nucleus. Given that most of the atom's mass is contained in the nucleus, which occupies a much smaller volume compared to the entire atom, we can conclude that the density of the nucleus is significantly higher than the average density of the atom.

Step-by-step solution

Understand the size of an atom and its nucleus

An atom consists of a central nucleus, which contains protons and neutrons, and electrons orbiting around the nucleus. The diameter of an atom is typically in the range of 1-2 Å (1 Å = \(10^{-10}\) meters), while the diameter of its nucleus is much smaller, about 1-10 fm (1 fm = \(10^{-15}\) meters).

Calculate the ratio

We will now calculate the ratio between the diameter of an atom and the diameter of its nucleus. Using the given size ranges for both, we can calculate the ratio as: \[\frac{\text{diameter of atom}}{\text{diameter of nucleus}} = \frac{1-2 \ Å}{1-10 \ \text{fm}}\] We can now convert the unit of the atom's diameter from Å to fm by using the conversion factor: \(1 \ Å = 10^5 \ \text{fm}\). So the ratio becomes: \[\frac{10^5-2 \times 10^5 \ \text{fm}}{1-10 \ \text{fm}}\]

Estimate the ratio

To get an approximate ratio between the diameters, we can consider the smallest possible atom diameter (1 Å) and the largest nucleus diameter (10 fm): \[\frac{10^5\ \text{fm}}{10\ \text{fm}} = 10^4\] Similarly, using the largest possible atom diameter (2 Å) and the smallest possible nucleus diameter (1 fm): \[\frac{2 \times 10^5 \ \text{fm}}{1 \ \text{fm}} = 2 \times 10^5\] Based on these calculations, we can approximately say that the diameter of an atom is 10^4 to 2 x 10^5 times greater than the diameter of its nucleus.

Density conclusion

Since most of the atom's mass is concentrated in the nucleus, while the nucleus occupies a much smaller volume compared to the entire atom, we can conclude that the density of the nucleus is significantly higher than the average density of the atom. This is because density is defined as mass divided by volume, and given the fact that most of the mass is in the nucleus, which has a much smaller volume, the density of the nucleus must be much greater than the atom itself.

Key concepts

Atomic Diameter

The diameter of an atom gives us a measure of how large the atom appears from one end to another. We're talking about scales that are incredibly small. Typically, the diameter of an atom falls within a range of 1-2 Å (angstroms), where 1 Å equals \(10^{-10}\) meters. This is a useful measure, because it helps us to imagine the size of the atom when considering it in different situations like chemical bonding or interactions with surfaces.

The contrastingly tiny nucleus within the atom typically spans about 1-10 fm (femtometers), with 1 fm equating to \(10^{-15}\) meters. This indicates that even though the nucleus is central to the atom, it comprises only a small fraction of the atom's overall diameter. The extremely small size of the nucleus compared to the whole atom is crucial to understanding atomic behavior, particularly in nuclear reactions where changes do not affect the atom's larger atomic structure.

Nuclear Density

In the context of atomic structure, nuclear density refers to how tightly packed the components within the nucleus are. Most of an atom's mass resides in this minuscule nucleus, which hosts protons and neutrons. The density is calculated as mass divided by volume. Because the nucleus occupies a substantially smaller volume than the atom itself, yet contains most of its mass, the density of a nucleus is astoundingly high compared to the rest of the atom.

Material density is usually discussed in bulk terms, but nuclear density is so much greater that it gives interesting insight into the peculiarities of atomic composition. For example, this elevated density explains why nuclear interactions release a massive amount of energy compared to chemical reactions, which primarily involve the much less dense electron cloud surrounding the nucleus. This is why nuclear processes are at the core of modern energy solutions such as nuclear power.

Proton and Neutron

Protons and neutrons are the building blocks of the nucleus. A proton is a positively charged particle, while a neutron carries no charge, making it neutral. Together, these particles account for most of the mass of an atom. Typically, the number of protons in an atom's nucleus defines the element, distinguishing, for instance, gold from silver or hydrogen from helium.

The presence of neutrons plays a pivotal role as they add to the mass while neutrally affecting the atom's electrical character. This balance ensures the stability of the atom. A balanced number of neutrons compared to protons is often crucial for the stability of the nucleus. In "isotopes," atoms of the same element may have different numbers of neutrons, which can lead to unstable configurations that are prone to radioactive decay. Through this decomposition, isotopes release various particles and energy that are significant in processes like radiocarbon dating and nuclear power generation.

Nucleus Size

The nucleus of an atom, although extremely small compared to the overall atom, plays a crucial role in defining the atom's properties and behaviors. Measuring between 1-10 femtometers, the nucleus holds almost all of the atom's mass despite its diminutive size. It is within this small core that protons and neutrons reside, and through their interactions, the various isotopes of elements are created.

The size of the nucleus varies slightly among different elements, but its minute scale is largely consistent. The energetic implications of nuclear size are significant, allowing for nuclear fission and fusion. These processes, respectively involving the splitting or merging of nuclei, underlie modern technologies like nuclear power plants and have profound implications for theoretical and applied physics.

• Nuclear size affects radioactive decay rates.
• It's central to understanding nuclear stability and reactions.