It turns out that the van der Waals constant b equals four times the total volume actually occupied by the molecules of a mole of gas. Using this figure, calculate the fraction of the volume in a container actually occupied by Ar atoms (b) at 20.27 MPa pressure and 0 °C. (Assume for simplicity that the ideal-gas equation still holds.)

Verified step by step guidance

1. Identify the van der Waals constant b for argon (Ar) from a reference source.
2. Use the relationship that b equals four times the total volume occupied by the molecules of a mole of gas to find the volume occupied by the Ar atoms: \( V_{molecules} = \frac{b}{4} \).
3. Calculate the volume of the container using the ideal gas law: \( PV = nRT \), where \( P = 20.27 \text{ MPa} \), \( n = 1 \text{ mole} \), \( R = 8.314 \text{ J/(mol K)} \), and \( T = 273.15 \text{ K} \).
4. Convert the pressure from MPa to Pa for consistency in units: \( 1 \text{ MPa} = 10^6 \text{ Pa} \).
5. Calculate the fraction of the volume in the container actually occupied by Ar atoms using the formula: \( \text{Fraction} = \frac{V_{molecules}}{V_{container}} \).

Key Concepts

Van der Waals Equation

The Van der Waals equation is an adjustment of the ideal gas law that accounts for the volume occupied by gas molecules and the attractive forces between them. It introduces two constants, 'a' and 'b', where 'b' represents the volume occupied by one mole of gas molecules. This equation is crucial for understanding real gas behavior under various conditions, especially at high pressures and low temperatures.

Molar Volume

Molar volume is the volume occupied by one mole of a substance, typically measured in liters per mole. In the context of gases, it helps to determine how much space the gas molecules occupy compared to the total volume of the container. Understanding molar volume is essential for calculating the fraction of volume occupied by gas atoms in a given scenario.

Fraction of Volume Occupied

The fraction of volume occupied by gas atoms is a ratio that compares the volume occupied by the gas molecules to the total volume of the container. This concept is important for understanding the efficiency of gas packing and the behavior of gases under different conditions. It can be calculated using the molar volume and the total volume of the container, providing insight into the physical properties of the gas.