The radius of a xenon atom is 1.3×10^{– 8} cm. A 100-mL flask is filled with Xe at a pressure of 1.0 atm and a temperature of 273 K. Calculate the fraction of the volume that is occupied by Xe atoms. (Hint: The atoms are spheres.)

Verified step by step guidance

1. Calculate the volume of a single xenon atom using the formula for the volume of a sphere: $V = \frac{4}{3} \pi r^3$, where $r$ is the radius of the xenon atom.
2. Convert the radius from centimeters to meters to ensure consistency in units.
3. Determine the number of moles of xenon gas in the flask using the ideal gas law: $PV = nRT$, where $P$ is pressure, $V$ is volume, $n$ is the number of moles, $R$ is the ideal gas constant, and $T$ is temperature.
4. Calculate the number of xenon atoms using Avogadro's number, which relates moles to the number of atoms.
5. Multiply the volume of a single xenon atom by the total number of atoms to find the total volume occupied by xenon atoms, and then divide by the volume of the flask to find the fraction of the volume occupied.

Key Concepts

Atomic Radius and Volume of a Sphere

The atomic radius is the distance from the nucleus of an atom to the outer boundary of its electron cloud. For spherical atoms, the volume can be calculated using the formula for the volume of a sphere, V = (4/3)πr³, where r is the radius. Understanding this concept is essential for determining the volume occupied by individual xenon atoms in the given problem.

Ideal Gas Law

The Ideal Gas Law, represented as PV = nRT, relates the pressure (P), volume (V), temperature (T), and number of moles (n) of a gas. This law is crucial for understanding the behavior of gases under various conditions and helps in calculating the total number of xenon atoms in the flask, which is necessary for finding the fraction of volume occupied by the atoms.

Volume Fraction

Volume fraction is a measure of the volume occupied by a component relative to the total volume of the mixture. It is calculated by dividing the volume occupied by the xenon atoms by the total volume of the flask. This concept is key to solving the problem, as it allows for quantifying how much space the xenon atoms occupy compared to the entire volume of the flask.