Problem 98 - Advanced Problems in Physical Chemistry for Competitive Examinations

Question

The behaviour of a real gas is usually depicted by plotting compressibility factor \(Z\) versus \(P\) at a constant temperature. At high temperature and high pressure, \(Z\) is usually more than \(1 .\) This fact can be explained by van der Waal's equation when (a) the constant \(a\) is negligible and not \(b\) (b) the constant \(b\) is negligible and not \(a\) (c) both constants \(a\) and \(b\) are negligible (d) both constants \(a\) and \(b\) arenot negligible

Step-by-Step Solution

Answer

The Van der Waals constant \(b\) is significant at high pressures, accounting for the finite size of gas molecules, which explains why \(Z > 1\) at high temperatures and pressures. Therefore, the correct answer is (b) the constant \(b\) is negligible and not \(a\).

Step 1: Understanding the Van der Waals Equation

The Van der Waals equation for a real gas is given by \((P + \frac{a}{V_m^2})(V_m - b) = RT\), where \(V_m\) is the molar volume, \(R\) is the gas constant, \(T\) is the temperature, \(P\) is the pressure, and \(a\) and \(b\) are the Van der Waals constants that correct for the intermolecular forces and the finite size of molecules, respectively.

Step 2: Considering High Pressure and Temperature

At high temperatures, the kinetic energy of the gas molecules is high, which tends to overcome the interactive forces between them. At high pressures, the volume \(V_m\) tends to become small. In such conditions, we analyze the significance of constants \(a\) and \(b\) in modifying the ideal gas behaviour as predicted by the Van der Waals equation.

Step 3: Analysing Constant \(a\) at High Temperatures

The constant \(a\) accounts for the attractive intermolecular forces. At high temperatures, these forces are less important compared to the kinetic energy of the gas molecules. Therefore, the effect of \(a\) is diminished, making it negligible in the equation, simplifying to \((P+\frac{a}{V_m^2})(V_m-b)\) becoming closer to \(PV_m=RT\).

Step 4: Analysing Constant \(b\) at High Pressures

The constant \(b\) corrects for the volume occupied by the gas molecules themselves. At high pressures, the available volume for the gas molecules is significantly reduced, and the correction due to \(b\) cannot be ignored because the finite size of the molecules becomes relatively more significant.

Step 5: Correlating Compressibility Factor \(Z\)

The compressibility factor \(Z\) is defined as \(Z=\frac{PV_m}{RT}\). If \(Z > 1\), this implies that the real gas occupies more volume than predicted by the ideal gas law under the given conditions. Since \(b\) cannot be neglected, but \(a\) can, the positive deviation of \(Z\) from 1 is explained by the non-negligible value of \(b\).

Step 6: Conclusion

The condition where the Van der Waals constant \(b\) is significant and not negligible while \(a\) is negligible explains why \(Z\) is more than 1 at high temperatures and high pressures. Therefore, the correct answer is (b) the constant \(b\) is negligible and not \(a\).

Key Concepts

Compressibility Factor

The compressibility factor (Z) is a useful dimensionless quantity in thermodynamics, representing how much a particular gas deviates from ideal gas behavior. It is defined as the ratio of the product of pressure (P) and molar volume (V_m) to the product of the gas constant (R) and temperature (T), given by the formula Z = \(\frac{PV_m}{RT}\). For an ideal gas, Z is exactly 1, as the gases strictly conform to the ideal gas law (PV_m = RT).

• Interpreting Z Values: When Z is above 1, it indicates that a gas is less compressible than predicted by the ideal gas law, likely due to repulsive intermolecular interactions which dominate at high pressures. Conversely, a Z below 1 suggests increased compressibility, often due to attractive forces between the molecules at lower pressures.
• Significance in Real Gas Behavior: Understanding Z aids in designing and analyzing systems where precise control of gas behavior is critical, such as in chemical reactors and refrigeration cycles.

In the context of the Van der Waals equation, when high temperatures and pressures make Z greater than 1, it aligns with the presence of finite molecular sizes represented by constant b, which accounts for the volume occupied by molecules themselves. Meanwhile, the intermolecular force constant a becomes less significant under these conditions and tends to be negligible.

Real Gas Behavior

Real gases exhibit behavior that deviates from the ideal model due to the effects of intermolecular forces and the finite volume occupied by the gas particles. The ideal gas law falls short in accurately predicting the properties of real gases, especially under conditions of high pressure and low temperature, where attractions between molecules and the volume of the molecules cannot be ignored.

• Van der Waals Equation: A significant improvement to model real gases is the Van der Waals equation, which adjusts for these non-ideal characteristics through two correction factors: a, which corrects for the attractive forces, and b, which accounts for the volume of the molecules.
• Adjustment for Non-Ideality: As discussed in the step-by-step solution, at high temperatures the kinetic energy of gas molecules allows them to overcome attractive forces, diminishing the impact of constant a. However, constant b remains significant at high pressures due to reduced space for movement, leading to a deviation in behavior from the ideal gas law that can be observed through an increased compressibility factor.

The interplay of these factors culminates in the phenomena where Z exceeds 1 at higher temperatures and pressures, emphasizing the gas molecules' physical presence and their inability to compress as an ideal gas would.

Intermolecular Forces

Intermolecular forces are the forces of attraction or repulsion that act between neighboring particles (atoms, molecules or ions). They are significantly weaker than the intramolecular forces that hold a substance together but are crucial in determining the physical and chemical properties of compounds.

• Types of Intermolecular Forces: These include London dispersion forces, dipole-dipole interactions, and hydrogen bonding. The strength and type of intermolecular forces present in a substance dictate its boiling and melting points, viscosity, and other physical properties.
• Impact on Gas Behavior: For gases, these intermolecular forces start becoming significant under high pressures and low temperatures. The Van der Waals constant a addresses the attractive part of these forces, leading to real gases condensing more readily than ideal gases under such conditions.

Given high temperature scenarios, where kinetic energy trumps these attractive forces rendering them unimportant, the Van der Waals equation simplifies by considering a negligible. This conceptual understanding is vital in predicting and calculating gas behavior in real-world applications, where intermolecular forces can markedly affect outcomes.