Problem 91 - Chemistry The Central Science In Si Units

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: Understanding the van der Waals Equation

The van der Waals equation is \( \left( P + \frac{a}{V_m^2} \right) (V_m - b) = RT \), where \( P \) is pressure, \( V_m \) is molar volume, \( R \) is the gas constant, \( T \) is temperature, \( a \) accounts for intermolecular forces, and \( b \) accounts for the volume occupied by gas molecules.

Step 2: Identifying the Role of Constant a

The constant \( a \) in the van der Waals equation corrects for the intermolecular attractions between gas molecules. Stronger attractions lead to a larger \( a \) value, thus it relates to attractions between molecules.

Step 3: Identifying the Role of Constant b

The constant \( b \) corrects for the finite volume occupied by gas molecules, reducing the available volume for the molecules' motion. This means that \( b \) relates to the molecular volume, or the size of the gas molecules themselves.

Step 4: Analyzing the Statements

Review the options: - (a) Incorrect: It incorrectly reverses the roles of \( a \) and \( b \).- (b) Correct: It correctly states \( a \) relates to attractions and \( b \) to molecular volume.- (c) Incorrect: Constants \( a \) and \( b \) are independent of pressure.- (d) Incorrect: Constants \( a \) and \( b \) are independent of temperature.

Key Concepts

Intermolecular Forces

Intermolecular forces are the forces that act between molecules and influence many physical properties of substances, including boiling points, melting points, and solubilities. In the context of the van der Waals equation, the constant \( a \) accounts for these forces. Unlike ideal gases, real gases have attractions between molecules that affect their behavior.Types of Intermolecular Forces
Differences in molecular interactions arise from types of intermolecular forces, such as:

• Dipole-Dipole Interactions: These occur between molecules that have permanent dipoles, meaning they have a partial positive and partial negative charge.
• London Dispersion Forces: All molecules exhibit these forces, arising due to temporary dipoles induced in molecules.
• Hydrogen Bonds: A strong type of dipole interaction, present especially in molecules where hydrogen is bonded to more electronegative elements like nitrogen, oxygen, or fluorine.

The presence and strength of these forces can explain the non-ideal behavior of gases, and the need for correction in the van der Waals equation.

Molecular Volume

Molecular volume refers to the physical space that the molecules of a substance occupy. In the van der Waals equation, the constant \( b \) is used to account for the volume occupied by gas molecules, acknowledging that unlike ideal gas, these particles do take up a portion of the total volume in which they are contained.Understanding the Molecular Volume
When assessing real gases, it's important to consider:

• Finite Volume of Molecules: Each molecule occupies a specific space, which reduces the volume available for gas molecule movement.
• Proportional to Size: The value of \( b \) is related to the size of a molecule. Larger molecules will result in a higher \( b \) value because they occupy more space.
• Correction in Gas Laws: Inaccuracies from ignoring molecular volume are rectified by the inclusion of \( b \) in the van der Waals equation.

Because of this, the concept of molecular volume is significant when working with gases at high pressures and low temperatures, where deviations from ideal behavior become more pronounced.

Gas Laws

Gas laws are mathematical relationships and equations that describe the behavior of gases. These laws include both ideal and non-ideal conditions. The importance of understanding these laws lies in their ability to predict how gases will respond to changes in pressure, volume, and temperature.From Ideal to Real Gases
Ideal gas laws, such as Charles’s Law, Boyle's Law, and Avogadro's Law, assume gases do not interact and occupy no space. These assumptions are accurate for ideal gases but do not hold for real gases as interactions and volume becomes apparent at extreme conditions.

• Ideal Gas Law: This formula \( PV = nRT \) sets a basis for behavior under ideal conditions, where \( P \) is pressure, \( V \) is volume, \( n \) is the number of moles, \( R \) is the gas constant, and \( T \) is temperature.
• Van der Waals Equation: Adjusts the ideal gas law by introducing constants \( a \) and \( b \), accounting for intermolecular forces and molecular volume, respectively.
• Significance of Van der Waals Equation: It provides greater accuracy for real gas behavior, especially under high-pressure and low-temperature conditions.

Studying gas laws, including corrections from the van der Waals equation, is crucial for applications in chemistry and engineering that involve real gases.