Calculate the pressure exerted by 1 mol of an ideal gas in a box that is 0.500 L and at 298 K. Have each group member calculate the pressure of 1 mol of the following real gases in the same box at the same temperature: He, Ne, H2, CH4, and CO2 using the van der Waals equation. Compare group members' answers and compare all answers with the pressure of an ideal gas. Assuming that the van der Waals equation predictions are accurate, account for why the pressure of each gas is higher or lower than that predicted for an ideal gas.

The ideal gas law is a fundamental principle in chemistry and physics, providing a simple relationship between the pressure (P), volume (V), the number of moles (n), and temperature (T) for a hypothetical gas that perfectly adheres to these specific proportions. Mathematically, it is expressed as

\[ PV = nRT \]

where R represents the ideal gas constant (0.0821 L·atm/(mol·K)). According to this law, if you calculate the pressure of an ideal gas held in a 0.500 L volume at 298 K, as shown in the exercise, you will find a standardized value, assuming no intermolecular forces and that the individual gas particles occupy no space. This principle is important because it sets a baseline from which the behavior of real gases can be compared and understood.

The ideal gas law assumes that the particles are in constant, random motion, and they elastically collide with each other and the walls of the container. It is a good approximation for gases under high temperature and low pressure. However, in the real world, gases show deviations from this ideal behavior due to intermolecular forces and the finite size of gas molecules.

Real gases deviate from the ideal gas law due to two main factors: the volume occupied by their molecules and the attraction or repulsion between them. These deviations become significant when the gas is under high pressure, low temperature, or high density.

The van der Waals equation is a more nuanced formula that accounts for these real-world behaviors of gases. It modifies the ideal gas law with two constants, a and b:

\[ \left( P + \frac{an^2}{V^2} \right)(V - nb) = nRT \]

The constant a corrects for the attractive forces between gas molecules, which tend to bring them closer together, thereby reducing the actual volume they occupy and hence the pressure exerted. The constant b accounts for the fact that gas particles have volume, therefore taking up space and not allowing gases to be compressed indefinitely. When students calculate the pressures using the van der Waals equation, they can observe that the pressures for real gases like He, Ne, H2, CH4, and CO2 will often differ from those anticipated by the ideal gas law, providing insight into each gas's unique physical properties.

Intermolecular forces are the forces of attraction or repulsion between neighboring particles (atoms, molecules, or ions). They significantly influence a substance's properties, such as boiling point, melting point, and solubility; and in the context of gases, these forces affect how closely molecules will be to one another, impacting the pressure a gas exerts on its container.

In an ideal gas scenario, it is assumed that these forces do not exist; however, in reality, molecules do attract or repel each other to some extent. This is integrated into the van der Waals equation through the a constant: the larger the value of a, the stronger the intermolecular forces. For instance, the larger value of a for CO2 in comparison to He reflects CO2's stronger intermolecular attraction, which has direct consequences on the behavior of the gas under various conditions.

By comparing the pressures calculated using the ideal gas law and the van der Waals equation, students can deepen their understanding of intermolecular forces. When the pressure of a real gas is lower than the ideal prediction, it can be inferred that the attractive forces between particles are significant, causing the gas to be less compressible than expected. Conversely, if the pressure is higher, it may indicate that repulsive forces have a greater effect.