Calculate the pressure exerted by 1 mol of an ideal gas in a box that is \(0.500 \mathrm{~L}\) and \(298 \mathrm{~K}\). Have each group member calculate the pressure of \(1 \mathrm{~mol}\) of the following gases in the same box at the same temperature: He, Ne, \(\mathrm{H}_{2}, \mathrm{CH}_{4},\) and \(\mathrm{CO}_{2} .\) Compare group members' answers as well as 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 Van der Waals equation provides a more realistic mathematical model for describing the behavior of real gases, as it accounts for two significant factors that the ideal gas law does not: intermolecular forces and the volume that gas particles occupy. The equation is represented as:
\[\begin{equation}\frac{an^2}{V^2}\right)RT\frac{nRT}{V}\]One might wonder how this equation improves upon the ideal gas law. The factor includes the strength of attraction between gas molecules, while considers the volume occupied by the molecules themselves. Unlike an ideal gas where these factors are ignored (molecules do not attract each other and have no volume), the Van der Waals equation provides correction terms that bring us closer to the real behavior of gases in the physical world.
These adjustments are essential when dealing with gases at high pressure or low temperature, where deviations from ideal behavior are more pronounced. The Van der Waals constants, and for each specific gas offer insights into their molecular characteristics, such as size and intermolecular forces, which help in understanding and predicting their behavior under different conditions.

Molecular intermolecular forces are the attractive and repulsive forces between molecules that affect their physical and chemical properties. These forces arise from various sources such as dipole-dipole interactions, hydrogen bonding, and London dispersion forces.

London Dispersion Forces

These are weak forces that exist between all molecules, resulting from the temporary dipoles that occur when electrons within a molecule are unevenly distributed.

Dipole-Dipole Interactions

Occur between molecules that have permanent dipoles, meaning there's an uneven distribution of charge within the molecule itself.

Hydrogen Bonding

Is a special case of dipole-dipole interaction that occurs when hydrogen is bonded to a highly electronegative atom like oxygen, nitrogen, or fluorine.
Each of these forces plays a role in the real gases behavior, affecting their physical state, boiling and melting points, and their deviation from ideal gas law behavior. The Van der Waals equation attempts to take these interactions into account through the parameter, which can help explain why different gases behave in distinct ways under the same conditions.

The gas constant, symbolized as '', is a fundamental parameter in various equations of state, including the ideal gas law. Its value in the context of the ideal gas law is approximately L·atm/(K·mol), and it serves as a bridge linking pressure (P), volume (V), and temperature (T), with the amount of substance in moles (n).
This constant is the same for all ideal gases and allows us to understand, through the ideal gas law, how changes in one state variable (such as temperature or volume) will affect another (like pressure) when the quantity of gas and the other state variables remain constant. Essentially, encapsulates all the kinetic energy-related aspects of molecules in an ideal gas, assuming no intermolecular forces and that the molecules occupy no space.
The specific value of varies depending on the units used for pressure, volume, and temperature and must be consistent with these units to apply the ideal gas law correctly. It's crucial to choose the correct value for calculations, ensuring that the calculated pressure or volume of a gas is accurate.

Real gases differ from ideal gases primarily because of the existence of molecular intermolecular forces and the finite volume that gas particles occupy. The ideal gas law assumes none of these factors affect the behavior of the gas, an assumption that holds reasonably well under standard conditions of temperature and pressure. However, as those conditions deviate significantly from the standard (particularly at very high pressures or very low temperatures), the behavior of real gases diverges from the ideal prediction.
For example, at high pressures, gas particles are closer together, and the effect of intermolecular forces becomes significant. These forces reduce the energy with which particles hit the walls of a container, resulting in lower pressure than an ideal gas would exhibit. Additionally, the actual volume of gas particles becomes a larger portion of the total volume, further affecting the pressure and volume relationship stated by the ideal gas law.
Understanding this behavior is crucial in industrial applications where gases are often used under non-ideal conditions, and the Van der Waals equation is one tool that helps engineers and scientists predict and account for the non-ideal behavior of gases. By applying the correct 'a' and 'b' constants, one can calculate more accurate pressures and volumes for real gases, facilitating safer and more efficient designs and processes.