Problem 85 - Chemistry: The Central Science

Question

Based on their respective van der Waals constants (Table 10.3), is Ar or \(\mathrm{CO}_{2}\) expected to behave more nearly like an ideal gas at high pressures? Explain.

Step-by-Step Solution

Answer

Based on the van der Waals constants, Ar has weaker intermolecular forces (a = 1.355 L² atm/mol²) and a smaller particle size (b = 0.0321 L/mol) than CO₂ (a = 3.640 L² atm/mol², b = 0.0427 L/mol). Therefore, Ar is expected to behave more nearly like an ideal gas at high pressures compared to CO₂.

Step 1: Identify the van der Waals constants for Ar and CO₂

Refer to Table 10.3 and look for the van der Waals constants for Ar and CO₂. Note down the values for both gases. For Ar: a = 1.355 L² atm/mol² b = 0.0321 L/mol For CO₂: a = 3.640 L² atm/mol² b = 0.0427 L/mol

Step 2: Compare the "a" constants

Compare the "a" constants of Ar and CO₂ to determine which one has weaker intermolecular forces. Ar: a = 1.355 L² atm/mol² CO₂: a = 3.640 L² atm/mol² Since the van der Waals constant "a" for Ar (1.355 L² atm/mol²) is smaller than the "a" constant for CO₂ (3.640 L² atm/mol²), Ar has weaker intermolecular forces.

Step 3: Compare the "b" constants

Compare the "b" constants of Ar and CO₂ to determine which one has smaller particle size. Ar: b = 0.0321 L/mol CO₂: b = 0.0427 L/mol Since the van der Waals constant "b" for Ar (0.0321 L/mol) is smaller than the "b" constant for CO₂ (0.0427 L/mol), Ar has a smaller particle size.

Step 4: Conclusion

Based on the comparison of van der Waals constants "a" and "b", Ar has weaker intermolecular forces and a smaller particle size than CO₂. Thus, Ar is expected to behave more nearly like an ideal gas at high pressures compared to CO₂.

Key Concepts

Ideal Gas

An ideal gas is a theoretical gas composed of many randomly moving point particles that interact only through elastic collisions. The concept helps scientists model gas behavior because ideal gases follow the ideal gas law given by the equation:\[ PV = nRT \]where:

• \(P\) is the pressure
• \(V\) is the volume
• \(n\) is the amount of substance in moles
• \(R\) is the ideal gas constant
• \(T\) is the temperature in Kelvin

Ideal gases do not account for intermolecular forces or molecular volume, which means they cannot compress entirely or expand infinitely.
In reality, no gas perfectly behaves as an ideal gas but under certain conditions like high temperatures and low pressures, real gases approximate ideal gas behavior. This is because the particles are moving faster and have more space to do so without interacting with each other.

Intermolecular Forces

Intermolecular forces are the forces of attraction or repulsion between neighboring particles (atoms, molecules, or ions). They are distinct from chemical bonds.
These forces play a significant role in determining the physical properties of substances, including their boiling and melting points.
Several types of intermolecular forces exist:

• London dispersion forces: These are weak, temporary forces that occur due to the movement of electrons creating temporary dipoles in molecules.
• Dipole-dipole interactions: These occur when polar molecules align so that the positive end of one molecule is near the negative end of another.
• Hydrogen bonding: A strong type of dipole-dipole interaction occurring when hydrogen is bonded to highly electronegative atoms like oxygen or nitrogen.

The van der Waals constant "a" accounts for these intermolecular forces in the van der Waals equation, correcting the ideal gas law by considering the attractive forces between particles.

Van der Waals Constants

The van der Waals constants "a" and "b" are used in the van der Waals equation to adjust the ideal gas law for real gas behavior. The equation is presented as:\[ \left( P + \frac{an^2}{V^2} \right) (V - nb) = nRT \]Here:

• \(a\) corrects for the attractive intermolecular forces between particles, with larger values indicating stronger forces.
• \(b\) corrects for the volume occupied by the gas particles themselves, with larger values indicating larger particle sizes.

The van der Waals constants vary for different gases and help explain deviations from ideal behavior.
In the exercise example, Argon (Ar) and Carbon Dioxide (CO₂) have different values for "a" and "b", indicative of the differences in their intermolecular forces and molecular sizes.
This leads to the conclusion that Argon, having weaker attractions and smaller particle size, behaves more closely to an ideal gas under high pressure conditions compared to CO₂.