Problem 142 - Chemistry The Science in Context

Question

Under what conditions is the pressure exerted by a real gas less than that predicted for an ideal gas?

Step-by-Step Solution

Answer

Answer: The pressure exerted by a real gas is less than that predicted by the ideal gas law when the additional pressure term an^2/V^2 in the Van der Waals equation is greater than zero. This condition is generally met when the gas particles have strong intermolecular attractive forces, and the volume containing the gas is relatively small.

Step 1: Understand the ideal gas law

The ideal gas law is a mathematical equation that describes the relationship between pressure, volume, temperature, and the number of moles in an ideal gas. The ideal gas law can be written as: PV = nRT where P is the pressure, V is the volume, n is the number of moles, R is the ideal gas constant, and T is the temperature in Kelvin.

Step 2: Understand the Van der Waals equation

The Van der Waals equation is a modification of the ideal gas law that takes into account the size of real gas particles and the existence of intermolecular forces between them. The Van der Waals equation is given by: (P + an^2/V^2)(V - nb) = nRT where a and b are Van der Waals constants that depend on the specific gas, and all other variables are the same as in the ideal gas equation.

Step 3: Compare ideal and real gases

Under certain conditions, the pressure exerted by real gases is less than that predicted by the ideal gas law. We will now compare the ideal gas law and the Van der Waals equation to identify these conditions.

Step 4: Identify the conditions for less pressure in real gases

To determine when the pressure of a real gas is less than that predicted by the ideal gas law, we can rewrite the Van der Waals equation and find when the pressure term P in the ideal gas law is greater than that in the Van der Waals equation: P_ideal = P_real + an^2/V^2 The pressure of the real gas, P_real, is less than the predicted pressure, P_ideal, when the additional pressure term an^2/V^2 is greater than zero. This additional term represents the effect of intermolecular forces between gas particles.

Step 5: Conclusion

The pressure exerted by a real gas is less than that predicted by the ideal gas law when the additional pressure term an^2/V^2 in the Van der Waals equation is greater than zero. This condition is generally met when the gas particles have strong intermolecular attractive forces, and the volume containing the gas is relatively small.

Key Concepts

Understanding the Ideal Gas Law

The ideal gas law is a cornerstone of thermodynamics, offering a simplified model for the behavior of gases. This equation is symbolically represented as PV = nRT, where P stands for pressure, V is the volume, n reflects the number of moles of the gas, R is the ideal gas constant, and T represents the absolute temperature in Kelvin. The law postulates that gases consist of randomly moving particles that do not interact with one another and that occupy no volume. Consequently, it's a valuable tool in conditions where gases behave ideally—high temperature and low pressure—where intermolecular forces are negligible.

To fully tap into the essence of this law, envision a balloon expanding as it is heated; the increase in temperature correlates with an increase in volume, assuming pressure is constant. The ideal gas law provides a straightforward equation to predict this outcome, which depends on the constants of the gas constant and the numbers of moles, which remain consistent as long as the gas’s identity doesn’t change.

Real Gases and the Van der Waals Equation

Real gases deviate from the ideal behavior predicted by the ideal gas law under certain conditions such as high pressure or low temperature. In these scenarios, the volume of the gas particles and the forces between them—the intermolecular forces—cannot be dismissed. The Van der Waals equation rectifies this by modifying the ideal gas law to account for particle volume and intermolecular interactions. It's expressed as (P + an^2/V^2)(V - nb) = nRT, where a and b are substance-specific constants that quantify the strength of intermolecular forces and the volume occupied by gas particles, respectively.

Consider the 'a' term in this context: higher values of 'a' imply stronger attractive forces between particles, leading to lower pressure than what the ideal gas law would predict. Similarly, 'b' relates to the volume of particles, suggesting a smaller free volume (V - nb) for particles to move, hence, they are not as free as an ideal gas would be.

Intermolecular Forces and Gas Behaviour

Intermolecular forces are the attractive and repulsive forces between molecules that influence the behavior of real gases. These forces come in various forms, including London dispersion forces, dipole-dipole interactions, and hydrogen bonds. The Van der Waals equation captures the effect of these forces within the additional pressure term an^2/V^2. When the gas particles are close together, as in a compressed gas, these forces become significant. Consequently, gases under these conditions deviate from the ideal gas law.

For example, at high pressures or low temperatures, molecules are pushed closer together, making the attractive forces between them more prominent. These forces tend to pull the molecules collectively, which can be visualized as if they were trying to stick together, hence decreasing the pressure exerted on the container walls compared to what would be predicted by the ideal gas law. This is precisely why the pressure of real gases can be substantially less than what we expect, especially when the intermolecular attractions are strong, as outlined by the additional term in the Van der Waals equation.