Question

Carbon-12 contains 6 protons and 6 neutrons. The radius of the nucleus is approximately 2.7 fm (femtometers), and the radius of the atom is approximately 70 pm (picometers). Calculate the volume of the nucleus and the volume of the atom. What percentage of the carbon atom's volume is occupied by the nucleus? (Assume two significant figures.)

Short answer

The volume of the nucleus is approximately \(6.48 \times 10^{-41}\) m^3, the volume of the atom is approximately \(1.44 \times 10^{-30}\) m^3, and the nucleus occupies approximately \(0.0045%\) of the atom's volume.

Step-by-step solution

Calculate the volume of the nucleus

Use the formula for the volume of a sphere, which is \( V = \frac{4}{3} \pi r^3 \), where \( r \) is the radius of the nucleus. The radius of the carbon-12 nucleus is given as 2.7 femtometers (fm). Convert this value to meters by noting that 1 fm = \( 1 \times 10^{-15} \) meters. Then calculate the volume.

Calculate the volume of the atom

Again, use the formula for the volume of a sphere. The radius of the atom is given as 70 picometers (pm). Convert the value to meters by noting that 1 pm = \( 1 \times 10^{-12} \) meters. Then compute the volume.

Calculate the percentage of the atom's volume occupied by the nucleus

Divide the volume of the nucleus by the volume of the atom and multiply by 100 to get the percentage.

Key concepts

Nuclear Volume

The nuclear volume is a measure of the space occupied by the nucleus of an atom, which comprises protons and neutrons. Since the nucleus is approximately spherical, we can calculate its volume using the formula for the volume of a sphere:

\[ V_{nucleus} = \frac{4}{3} \pi r^3 \]
where \( r \) is the radius of the nucleus. For a carbon-12 atom, with a nucleus radius of approximately 2.7 fm, converting femtometers to meters is essential since standard volume calculations are in cubic meters (m³). One femtometer is \( 1 \times 10^{-15} \) meters. Thus, the calculation starts with converting 2.7 fm into meters, raising this converted radius to the third power, multiplying by pi (\( \pi \)), and finally, multiplying by \( \frac{4}{3} \). The result gives us the nuclear volume, which is minuscule compared to the overall atomic volume.

Atomic Volume

The atomic volume relates to the space within the boundary defined by the atom's electron cloud. The electron cloud extends far beyond the nucleus, reaching radii measured in picometers (pm). As with the nucleus, the atomic volume is also calculated using the sphere's volume formula:

\[ V_{atom} = \frac{4}{3} \pi r^3 \]
For our carbon-12 atom, the radius is roughly 70 pm. Since 1 pm equals \( 1 \times 10^{-12} \) meters, we need to convert picometers to meters before performing the volume calculation. After converting, we follow the same steps as in calculating the nuclear volume: cube the radius, multiply by \( \pi \), and then the result by \( \frac{4}{3} \) to get the atomic volume in cubic meters. The atomic volume will demonstrably be much larger due to the significantly bigger radius of the electron cloud.

Volume of a Sphere

The volume of a sphere is determined by the amount of three-dimensional space it occupies, and it's calculated with a mathematical formula that incorporates the sphere's radius. The radius is the distance from the center of the sphere to any point on its surface.

The formula we use to calculate a sphere's volume is:
\[ V = \frac{4}{3} \pi r^3 \]
This equation tells us that the volume is directly proportional to the cube of the radius—meaning small changes in radius lead to significant changes in volume. This fact is particularly evident when comparing the volume of atomic nuclei to the whole atoms’ volumes; a slight increase in radius results in a much larger volume due to the cubed term.

Percentage Volume Occupied

To conceptualize how much space the nucleus takes up inside the atom, we compare the nuclear volume to the atomic volume. This comparison tells us the percentage of the atom's volume that is occupied by its nucleus. The calculation is a straightforward division followed by a conversion to a percentage:

\[ \text{% Volume} = \left( \frac{V_{nucleus}}{V_{atom}} \right) \times 100 \]
In this instance, we take the previously calculated volumes for both the nucleus and the atom of carbon-12, insert those values into the formula, and calculate. Generally, the nucleus occupies a tiny fraction of the atom's volume, a testament to the vast empty spaces within an atom. Understanding this concept is crucial in grasping the scale and structure of atoms in the context of atomic physics and chemistry.