Question

Carbon- 12 contains 6 protons and 6 neutrons. The radius of the nucleus is approximately \(2.7 \mathrm{fm}\), and the radius of the atom is approximately \(70 \mathrm{pm}\). Calculate the volume of the nucleus and the volume of the atom. What percentage of the carbon atom's volume does the nucleus occupy? (Hint: Convert both given radii to \(\mathrm{m}\), and then calculate their volumes using the formula for the volume of a sphere, which is \(V={ }_{3}^{4} \pi r^{3}\).)

Short answer

Nucleus volume: \(2.07004 \times 10^{−41} \text{m}^3\), Atom volume: \(1.436756 \times 10^{-30} \text{m}^3, Nucleus occupies approximately \)1.44167 \times 10^{-11}%$ of the carbon atom's volume.

Step-by-step solution

Convert Radii to Meters

First, convert the given radii from femtometers and picometers to meters. Since there are \(10^{15}\) femtometers (fm) in 1 meter and \(10^{12}\) picometers (pm) in 1 meter, the conversions are as follows:Nucleus radius in meters: \(2.7\) fm \(= 2.7 \times 10^{-15}\) mAtom radius in meters: \(70\) pm \(= 70 \times 10^{-12}\) m

Calculate the Volume of the Nucleus

Use the volume formula for a sphere, \(V_{nucleus} = \frac{4}{3}pi r_{nucleus}^3\), where \(r_{nucleus} = 2.7 \times 10^{-15}\) m:\(V_{nucleus} = \frac{4}{3} pi (2.7 \times 10^{-15})^3\) cubic meters.

Calculate the Volume of the Atom

Similarly, use the volume formula for a sphere to calculate the atom's volume, \(V_{atom} = \frac{4}{3} pi r_{atom}^3\), where \(r_{atom} = 70 \times 10^{-12}\) m:\(V_{atom} = \frac{4}{3} pi (70 \times 10^{-12})^3\) cubic meters.

Determine the Percentage Volume Occupied

Calculate the percentage of the atom's volume that the nucleus occupies using the formula: Percentage =\(\left(\frac{V_{nucleus}}{V_{atom}} \right) \times 100\). Substitute the previously calculated volumes into this formula to get the result.

Key concepts

Nuclear Volume Calculation

Understanding the volume of an atomic nucleus, despite its minuteness, plays a crucial role in nuclear physics and chemistry. Here, we're specifically discussing the methodology for calculating the volume of a Carbon-12 nucleus. Like most atomic nuclei, a Carbon-12 nucleus can be approximated as a sphere. To calculate the volume of such a spherical nucleus, we use the formula for the volume of a sphere:
\[V_{nucleus} = \frac{4}{3}\pi r_{nucleus}^3\]
where \(r_{nucleus}\) is the radius of the nucleus. For Carbon-12, given the radius as 2.7 femtometers (\(2.7 \times 10^{-15} \) meters), the volume calculation becomes a straightforward substitution into this formula. The challenge often lies in managing the very small numbers and understanding their implications on the scale of atoms.

Atomic Volume Calculation

We can similarly calculate the volume of an entire atom, which is significantly larger than that of the nucleus. The assumption that an atom is spherical allows for a simple calculation using the same formula for the volume of a sphere:
\[V_{atom} = \frac{4}{3}\pi r_{atom}^3\]
The radius here is much larger; for Carbon-12, the atomic radius is given as 70 picometers, or \(70 \times 10^{-12}\) meters. Remember that an atom is mostly empty space, so its volume will be many orders of magnitude larger than the nucleus's volume. Calculating the atomic volume offers insights into atomic structure, electron cloud density, and other fundamental properties in chemistry.

Radius Conversion in Chemistry

When dealing with atomic structures, we often need to convert measurements between different scales. The reason for these conversions lies in the extremely small scales at which atomic and nuclear dimensions are given—typically femtometers for nuclei and picometers or angstroms for whole atoms. Since the units of measurements in physics and chemistry are standard (meters in the International System of Units), it is essential to convert the given radii to meters before proceeding with volume calculations. This step ensures compatibility with the formula for volume and facilitates the comparison with macroscopic objects.

Percentage Volume of Nucleus

Making sense of just how minuscule a nucleus is compared to the entire atom can be profoundly enlightening. After calculating the separate volumes of the nucleus and atom in the previous steps, we find the percentage which the nuclear volume occupies of the atomic volume using the equation:
\[\text{Percentage Volume} = \left(\frac{V_{nucleus}}{V_{atom}} \right) \times 100\%\]
This percentage represents the dense, central part of the atom relative to the largely empty space where the electron cloud exists. The number is often surprisingly small, reflecting the vast difference in scale within an atom. Understanding this percentage helps in visualizing atomic structure and grasping concepts related to nuclear density and atomic size.