Calculate the pressure exerted by \(0.5000\) mole of \(\mathrm{N}_{2}\) in a 10.000-L container at \(25.0^{\circ} \mathrm{C}\) a. using the ideal gas law. b. using the van der Waals equation. c. Compare the results. d. Compare the results with those in Exercise 115 .

The Van der Waals equation improves upon the Ideal Gas Law by accounting for the finite size of molecules and the intermolecular forces present in real gases. While the Ideal Gas Law presumes that gas particles have no volume and do not attract or repel each other, real gases clearly do not follow this assumption. The equation is expressed as:
\[\begin{equation}\left(P + \frac{n^2a}{V^2} \right) \left(V - nb\right) = nRT\end{equation}\]
where:

• \(P\) is the pressure of the gas,
• \(n\) is the number of moles of the gas,
• \(V\) is the volume of the gas,
• \(T\) is the temperature in Kelvin,
• \(R\) is the universal gas constant,
• \(a\) and \(b\) are Van der Waals constants that are specific to each gas, accounting for the gas particle's size and the strength of intermolecular forces, respectively.

In practical use, the Van der Waals equation can provide a more accurate description of gas behavior, particularly under conditions of high pressure and low temperature where deviations from ideal behavior are most pronounced.

The measure of force exerted by gas particles as they collide with the walls of a container is known as gas pressure. This fundamental concept is crucial in thermodynamics and can be calculated using several different equations, depending on the assumptions made about the gas.
For an ideal gas, the equation is simple, as shown in the Ideal Gas Law:
\(P = \frac{nRT}{V}\)
This equation states that the pressure (P) of the gas is directly proportional to the number of moles (n), the gas constant (R), and the temperature (T), and inversely proportional to the volume (V).
However, as seen with the Van der Waals equation, for real gases these calculations are adjusted to factor in intermolecular forces and the volume occupied by the gas particles themselves. The ability to accurately calculate gas pressure is critical in various fields, from chemistry to engineering, and it allows us to predict how gases will behave under changing conditions.

Real gases display behaviors that deviate from the Ideal Gas Law at high pressures and low temperatures. These deviations occur due to the volume occupied by the gas molecules and the forces that they exert on each other. Under typical laboratory conditions, real gases often behave similarly to ideal gases, allowing scientists to use the simpler Ideal Gas Law for calculations.
However, when conditions are far from standard temperature and pressure, or when high precision is necessary, corrections like those provided by the Van der Waals equation are used. The real gas behavior considers factors such as non-zero particle volume and attractions between particles, which can significantly affect how the gas expands, compresses, and how much pressure it exerts. Understanding real gas behavior is essential not only in academic settings but also in industries that handle gases under a variety of pressures and temperatures, ensuring safe and efficient operations.