Question

A sample of gaseous neon atoms at atmospheric pressure and \(0^{\circ} \mathrm{C} \mathrm{con}\) tains \(2.69 \times 10^{22}\) atoms per liter. The atomic radius of neon is 69 \(\mathrm{pm} .\) What fraction of the space is occupied by the atoms themselves? What does this reveal about the separation between atoms in the gaseous phase?

Short answer

A very small fraction of the space is occupied by Neon atoms, which reveals that there is a considerable amount of empty space between atoms in the gaseous phase.

Step-by-step solution

Calculate the volume of a single Neon atom

Assuming the Neon atom can be modeled as a sphere, use the formula for the volume of a sphere, which is \(V = \frac{4}{3}\pi r^3\), where \(r\) is the radius. Given the radius of a Neon atom as 69 pm (or 69 x 10^{-12} m), the volume of one Neon atom is \(V_{atom} = \frac{4}{3}\pi (69 \times 10^{-12})^3\).

Calculate the total volume of Neon atoms per liter

Multiply the volume of a single Neon atom by the number of atoms per liter to obtain the total volume occupied by Neon atoms. With the number of atoms given as \(2.69 \times 10^{22}\), the total volume is \(V_{total} = 2.69 \times 10^{22} \times V_{atom}\).

Convert 1 liter to cubic meters

To compare the volumes, they need to be in the same units. Convert 1 liter to cubic meters. Since 1 liter equals \(1 \times 10^{-3} m^3\), the volume of the container is \(V_{container} = 1 \times 10^{-3} m^3\).

Calculate the fraction of space occupied by Neon atoms

Divide the total volume of Neon atoms by the volume of the container to find the fraction of space occupied: \(\text{Fraction occupied} = \frac{V_{total}}{V_{container}}\).

Interpret the results

If the fraction is significantly less than 1, it indicates that most of the space in the sample is empty, revealing that atoms in a gas are widely separated compared to their size., which is characteristic of the gaseous phase.

Key concepts

Volume of a Sphere

The volume of a sphere is an important concept when it comes to understanding the space occupied by spherical objects, such as atoms in their most simplified model. The formula to calculate the volume of a sphere is given by \( V = \frac{4}{3}\pi r^3 \), where \( r \) is the radius of the sphere. Imagine a tiny ball, where every point on the surface is equally distant from its center - that's our sphere, and every bit of space inside that sphere counts towards its volume.
For a neon atom modeled as a sphere, we consider its radius (in this case, 69 picometers) and apply it to the formula. This calculation gives us the volume of a single Neon atom, which is essential in determining how much space in a container is actually occupied by gas particles.

Avogadro's Law

Avogadro's Law is a fundamental principle in chemistry that relates to the volume occupied by gases. It states that equal volumes of all gases, at the same temperature and pressure, contain the same number of molecules. This number is known as Avogadro's number, and it gives us a direct link between the amount of substance and the volume it occupies when in the form of a gas.
When we know the volume of the individual gas particle (in this case, a Neon atom), we can use Avogadro's law to understand the behavior of the gas as a whole. It tells us that regardless of the type of gas, if we have one mole of the gas (which is \(6.022 \times 10^{23}\) particles), it will occupy the same volume under the same conditions of temperature and pressure, approximately \(22.4 \, L\) at standard temperature and pressure (STP).

Particle Separation in Gases

Particle separation in gases is a concept that describes the average distance between individual gas particles in a sample. In gases, atoms or molecules move freely and in random motion, leading to a significant amount of space between them compared to their size. This is unlike solids or liquids where particles are closely packed together.
With gaseous neon, as given in our exercise, the fraction of space occupied by the atoms is a clear indicator of particle separation. When we calculate that the atoms themselves occupy only a small fraction of the space, it confirms that there is a substantial distance between them. This separation allows gas particles to move around easily, which is why gases can flow and take the shape of their containers. Understanding this property of gases helps to elucidate why they compress and expand much more easily than solids or liquids.

Gas Volume Calculation

To calculate the volume occupied by a gas, we can follow the steps given in our Neon example. After determining the volume of an individual Neon atom, we multiply it by the total number of atoms to get the volume they occupy altogether. However, we also need to consider the volume of the container itself which, in this case, is 1 liter or \(1 \times 10^{-3} \, m^3\).
By finding the ratio of the total volume occupied by the Neon atoms to the volume of the container, we obtain the fraction of space actually taken up by the Neon atoms. If, for instance, the fraction is very small, it implies that the Neon atoms, even though they are countless, occupy just a tiny part of the total space—a characteristic indicative of the gas phase. This calculation is not just theoretical; it has practical implications in chemistry and engineering where the volume of gases needs to be accurately measured for various applications.