1.8
The Van der Waals Equation
The ideal gas law is based on two simplifying assumptions: first, that there are no intermolecular attractions between gas molecules, and second, that…
The ideal gas law, based on the assumptions of negligible intermolecular attractions and negligible volume of gas molecules, fails at high pressures and low temperatures.
Here, the van der Waals equation, a modified version of the ideal gas law, compensates for these deviations by introducing corrections.
The first correction in the pressure term adjusts for the difference between real gas pressure and ideal gas pressure. As the gas molecules attract one another, the real gas pressure is lower than the ideal value.
These attractive forces reduce both the frequency and force of collisions with container walls. As a result, reducing pressure is directly proportional to the square of the molar concentration of molecules.
The second correction is in the volume term, calculating the actual volume available for gas molecules as total volume minus the volume excluded by intermolecular repulsive interactions.
The constants 'a' and 'b', known as the van der Waals coefficients, represent the strength of attractive and repulsive interactions between gas molecules, respectively. Note that both coefficients are empirical constants characteristic of each gas and remain unaffected by temperature.
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Q1: Why does the ideal gas law fail at high pressures and low temperatures?
The ideal gas law assumes negligible intermolecular attractions and negligible molecular volume, assumptions that break down under extreme conditions. At high pressures and low temperatures, gas molecules are forced closer together, making intermolecular forces and molecular size significant. This causes deviation from ideal behavior, requiring corrections to predict real gas behavior accurately.
Q2: What does the pressure correction term in the van der Waals equation account for?
The pressure correction adjusts for attractive forces between gas molecules that reduce measured pressure below the ideal value. These intermolecular attractions decrease both collision frequency and force with container walls. The pressure reduction is directly proportional to the square of molar concentration, reflecting how attraction strength increases with molecular density.
Q3: How does the volume correction term modify the van der Waals equation?
The volume correction calculates actual available volume for molecular motion by subtracting the volume excluded by molecules themselves. For n moles where each molecule occupies volume b, the excluded volume is nb. The actual free volume becomes total volume minus nb, accounting for the physical space occupied by gas molecules.
Q4: What do the van der Waals coefficients 'a' and 'b' represent?
Coefficient 'a' represents intermolecular attraction strength; larger values indicate stronger cohesion and greater pressure correction. Coefficient 'b' represents excluded volume from repulsive interactions; larger values mean less free space available. Both are empirical constants unique to each gas and remain temperature-independent within the van der Waals model.
Q5: How do intermolecular attractions affect real gas pressure?
Intermolecular attractions pull molecules together, reducing the force and frequency of collisions with container walls. This causes real gas pressure to be lower than predicted by ideal gas behavior. The magnitude of this pressure reduction depends on molecular concentration squared, making it more significant at higher densities.
Q6: Why are van der Waals coefficients treated as temperature-independent?
Within the van der Waals model, coefficients 'a' and 'b' are treated as empirical constants characteristic of each gas that remain unaffected by temperature changes. This simplification allows the equation to provide reliable corrections across a range of conditions, though real substances may show some temperature dependence in practice.
Q7: When is the virial equation of state preferred over the van der Waals equation?
The virial equation of state is preferred for higher precision, particularly over wide ranges of temperatures and pressures. While the van der Waals equation offers valuable insights into real gas behavior, it is not universal for all substances. Virial coefficients, commonly tabulated at various temperatures, capture deviations from ideal behavior more accurately.