Question

An average dimension for the radius of an atom is \(1.0 \times 10^{-8} \mathrm{~cm}\), and the average radius of the nucleus is \(1.0 \times 10^{-13} \mathrm{~cm}\). Determine the ratio of atomic volume to nuclear volume. Assume that the atom is spherical \(V=(4 / 3) \pi r^{3}\) for a sphere].

Short answer

The ratio of atomic volume to nuclear volume is \( (10^{5})^3 = 10^{15} \).

Step-by-step solution

Calculate the Volume of the Atom

Use the formula for the volume of a sphere, which is given by \( V_{atom} = \frac{4}{3} \pi r_{atom}^3 \), where \( r_{atom} \) is the radius of the atom. Substitute \( r_{atom} = 1.0 \times 10^{-8} \mathrm{cm} \) into the formula to find the atomic volume.

Calculate the Volume of the Nucleus

Similarly, calculate the volume of the nucleus using the same formula \( V_{nucleus} = \frac{4}{3} \pi r_{nucleus}^3 \), where \( r_{nucleus} \) is the radius of the nucleus. Substitute \( r_{nucleus} = 1.0 \times 10^{-13} \mathrm{cm} \) into the formula to find the nuclear volume.

Calculate the Ratio of Atomic Volume to Nuclear Volume

To find the ratio of the atomic volume to the nuclear volume, divide the atomic volume by the nuclear volume: \( \text{Ratio} = \frac{V_{atom}}{V_{nucleus}} \). Simplify the expression by canceling terms if possible.

Simplify the Ratio to Solve for the Answer

Upon dividing the two volumes, the constants \( \frac{4}{3} \pi \) will cancel out, and you will be left with the ratio of \( \left( \frac{r_{atom}}{r_{nucleus}} \right)^3 \). Substitute the given radius values and calculate the ratio to get the final answer.

Key concepts

Atomic Structure

Understanding the structure of the atom is crucial to grasping concepts in nuclear physics and chemistry. At its most basic level, an atom consists of a nucleus made up of protons and neutrons, around which electrons orbit. The nucleus is incredibly small compared to the overall size of the atom, which is mostly empty space.

Consider the atom as a minuscule solar system, with the nucleus being the sun and the electrons acting as planets in vast orbits. Despite its size, the nucleus contains most of the atom's mass since protons and neutrons are far heavier than electrons. Atomic radius typically involves the space from the nucleus to the outer boundary of the surrounding cloud of electrons.

Volume Calculation

The volume of a sphere, which is what we assume the shape of an atom and its nucleus to be, can be calculated using the formula \( V = \frac{4}{3} \pi r^{3} \), where \( r \) is the radius of the sphere. When we're calculating the ratio of atomic volume to nuclear volume, it's almost like comparing the size of a beach ball to a grain of sand.

Once we have the radii of the atom and its nucleus, we compute their individual volumes. It's essential to note that when calculating these volumes, the cubic nature of the radius in the formula means that small changes in the radius result in significant changes in the calculated volume. This property leads to the astonishingly large differences between the atomic and nuclear volumes.

Nuclear Physics

Nuclear physics explores the constituents and structure of atomic nuclei. The discovery that the atom is not indivisible but rather has a small, dense nucleus has profound implications. It explains why the atomic volume is vastly greater than the nuclear volume.

The ratio of atomic volume to nuclear volume is a staggering figure due to the extreme difference in scale. This ratio underscores the central concept in nuclear physics and chemistry: the nucleus occupies an almost negligible amount of space within the atom, but it plays a critical role due to its mass and the interactions it undergoes, such as nuclear fission or fusion.