Problem 91 - Chemistry: The Central Science in SI Units, Global Edition

Question

Which statement concerning the van der Waals constants \(a\) and \(b\) is true? (a) The magnitude of \(a\) relates to molecular volume, whereas \(b\) relates to attractions between molecules. (b) The magnitude of \(a\) relates to attractions between molecules, whereas \(b\) relates to molecular volume. (c) The magnitudes of \(a\) and \(b\) depend on pressure. (d) The magnitudes of \(a\) and \(b\) depend on temperature.

Step-by-Step Solution

Answer

(b) The magnitude of \(a\) relates to attractions between molecules, whereas \(b\) relates to molecular volume.

Step 1: Recall the van der Waals Equation

The van der Waals equation is given by: \[ \left(P + a\frac{n^2}{V^2}\right) (V-nb) = nRT \] Here, P is the pressure, V is the volume, n is the number of moles, R is the universal gas constant, and T is the temperature.

Step 2: Understand the van der Waals Constants

In the van der Waals equation, the constants 'a' and 'b' have specific roles: - Constant 'a' is related to the strength of the attractive forces between the molecules. The larger the 'a' value, the stronger the attractive forces between the molecules. - Constant 'b' is related to the effective volume occupied by the molecules. The larger the 'b' value, the larger the molecular size (volume) of the gas.

Step 3: Evaluate the Given Statements

Now let's evaluate each statement: (a) The magnitude of \(a\) relates to molecular volume, whereas \(b\) relates to attractions between molecules. This statement swaps the roles of the van der Waals constants. So, it is incorrect. (b) The magnitude of \(a\) relates to attractions between molecules, whereas \(b\) relates to molecular volume. This statement correctly represents the roles of constants 'a' and 'b' in the van der Waals equation. So, this statement is true. (c) The magnitudes of \(a\) and \(b\) depend on pressure. The values of 'a' and 'b' are determined by the nature of the gas molecules, not by the pressure. So, this statement is incorrect. (d) The magnitudes of \(a\) and \(b\) depend on temperature. The values of 'a' and 'b' are determined by the nature of the gas molecules, not by the temperature. So, this statement is incorrect.

Step 4: Conclusion

Based on our understanding of the van der Waals constants and evaluating the given statements, the correct answer is: (b) The magnitude of \(a\) relates to attractions between molecules, whereas \(b\) relates to molecular volume.

Key Concepts

Intermolecular Forces

Intermolecular forces are the forces of attraction that exist between molecules. These forces play a crucial role in determining the physical properties of substances, such as boiling points and solubilities. In the context of the van der Waals equation, these forces are represented by the constant **a**.
Constant **a** quantifies how strongly molecules attract one another.
- A higher value of **a** indicates stronger intermolecular attractions, which can lead to greater deviations from ideal behavior for a gas.
- This is because stronger attractions cause molecules to stick together more closely, reducing the kinetic energy available for movement and expansion.
In practical terms, gases with high **a** values tend to liquify more easily as the attractive forces overcome the kinetic energy of the molecules. Knowing about intermolecular forces helps us understand why some gases behave more ideally than others, especially under conditions of high pressure and low temperature.

Molecular Volume

Molecular volume refers to the space that gas molecules occupy. When using the van der Waals equation, this concept is taken into account by the constant **b**. The value of **b** is directly related to the size of the gas molecules.
- A larger **b** implies that the molecules occupy more space.
- This is significant as it affects how closely packed the molecules are within a container, influencing the volume available for movement.
In ideal gas law assumptions, gas particles are considered to have no volume and do not interact. However, in reality, gas molecules do take up space. This is why the van der Waals equation includes **b** to adjust the ideal gas equation for more accurate predictions. Recognizing this adjustment helps in understanding how real gases differ from ideal ones, especially at high pressures where molecular volumes become more significant.

Gas Laws

Gas laws are a set of laws that describe the behavior of gases. They provide mathematical relationships between pressure, volume, temperature, and the number of moles of a gas.
- In their simplest form, gas laws assume that gas molecules do not interact and occupy no volume.
- The ideal gas law, for instance, is expressed as PV = nRT, representing an ideal scenario that does not occur perfectly in nature.
The van der Waals equation is one of the modifications made to the ideal gas law to account for real-world conditions. It corrects for the intermolecular forces and the finite molecular volume by incorporating constants **a** and **b**. Shifting from the ideal gas law to including these corrections helps us more accurately predict the behavior of real gases, especially under less-than-ideal conditions, like those of high pressure and low temperature. Understanding these adaptations prepares students to analyze and solve problems regarding gas behaviors more accurately.