Question

The hydrogen atom contains 1 proton and 1 electron. The radius of the proton is approximately \(1.0 \mathrm{fm}\) (femtometer), and the radius of the hydrogen atom is approximately \(53 \mathrm{pm}\) (picometers). Calculate the volume of the nucleus and the volume of the atom for hydrogen. What percentage of the hydrogen 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

The nucleus occupies approximately \(1.0 \times 10^{-5}\%\) of the volume of the hydrogen atom.

Step-by-step solution

- Convert the radii to meters

To perform calculations involving volume, first convert the given radii from femtometers and picometers to meters. For the proton, convert the radius of 1.0 fm to meters by multiplying with \(10^{-15}\). For the hydrogen atom, convert the radius of 53 pm to meters by multiplying with \(10^{-12}\).

- Calculate the volume of the proton nucleus

Use the volume formula for a sphere, \(V=\frac{4}{3}\pi r^3\), with the radius in meters to calculate the volume of the proton.

- Calculate the volume of the hydrogen atom

Again, use the volume formula for a sphere, \(V=\frac{4}{3}\pi r^3\), with the radius in meters to calculate the volume of the hydrogen atom.

- Calculate the percentage of the volume occupied by the nucleus

Divide the volume of the nucleus by the volume of the hydrogen atom and multiply by 100 to find the percentage of the atom's volume that the nucleus occupies.

Key concepts

Nuclear Radius Conversion

Understanding the scale of atomic structures requires us to get comfortable with very small units, like femtometers (fm) and picometers (pm). One femtometer is equal to \(10^{-15}\) meters, while one picometer is equal to \(10^{-12}\) meters. To make these quantities more manageable, we employ dimensional analysis—a technique that involves the multiplication of physical quantities by conversion factors to express them in different units. For instance, converting the proton's radius from femtometers to meters enables us to perform volume calculations using a standardized metric system unit, ensuring accuracy and comprehensibility in nuclear physics calculations.

When approaching a problem like calculating the volume of a hydrogen atom, converting the nuclear radius of a proton from 1.0 fm to meters sets the foundational stage for subsequent computational steps. We multiply the given radius by \(10^{-15}\), translating the abstractly small measurement into a concrete value expressed in meters, which is vital for applying formulas involving volume.

Volume of a Sphere Formula

The formula for calculating the volume of a sphere is pivotal in a variety of scientific disciplines, particularly in atomic physics when determining the space occupied by atomic particles. This formula is expressed as
\[V = \frac{4}{3}\pi r^3\],
with \(V\) representing the volume and \(r\) denoting the radius of the sphere. It's crucial that radius is in meters to ensure the consistency of units throughout the calculation. This cubic relationship underscores how a small change in the radius of a sphere translates into a large disparity in volume, a fact that becomes especially significant when considering the vast differences between the size of the nucleus and the entire atom in an atomic model.

Percentage Volume Calculation

Contrasting the volumes of different components of an atom, such as the nucleus versus the whole atom, helps us comprehend the scale and structure of atoms. The percentage volume calculation serves this exact purpose. Once we determine the volumes of both the nucleus and the atom, we calculate what fraction of the atom's volume is actually occupied by the nucleus by dividing the smaller volume by the larger one. Multiplying this ratio by 100 converts it into a percentage. This computation gives us insight into the fact that despite the atom's nucleus carrying almost all the atomic mass, it occupies a minuscule portion of the atom's entire volume—illuminating the vast empty spaces that characterize atom structures.

Atomic Structure

At the heart of such calculations is the atomic structure, a fundamental concept in chemistry and physics. An atom consists of a dense nucleus made up of protons and neutrons, surrounded by a cloud of electrons. The classic Bohr model depicts the hydrogen atom, the simplest of all, as having a single electron orbiting a single proton nucleus. This nucleus, while comprising nearly the entire mass of the atom, actually occupies a remarkably small portion of the total volume. Calculations like the one above vividly illustrate this disparity and emphasize the intricate nature of atomic makeup, pivotal for understanding chemical properties and reactions.

Dimensional Analysis

Dimensional analysis is a mathematical technique used frequently in physics and chemistry for unit conversion and problem-solving. It involves the use of conversion factors to switch from one unit of measurement to another. This is essential when working with units like femtometers and picometers, which are orders of magnitude smaller than daily units of measurement. Dimensional analysis ensures that all the units are correctly aligned to prevent discrepancies in calculations. It also helps when checking the plausibility of an answer, as consistent units across a calculation can confirm that the applied methods and obtained results are logically sound.