If Z is a compressibility factor, van der Waals equation at...

If $Z$ is a compressibility factor, van der Waals equation at low pressure can be written as:
[Main 2014]
(a) $Z=1+\frac{R T}{P b}$
(b) $Z=1-\frac{a}{V R T}$
(c) $Z=1-\frac{P b}{R T}$
(d) $Z=1+\frac{P b}{R T}$

Key Concepts

Compressibility Factor

The compressibility factor, typically denoted as Z, is a dimensionless quantity that reflects how much a real gas deviates from ideal gas behavior. It is defined as the ratio of the product of pressure and volume to the product of the number of moles, the gas constant, and temperature (PV/nRT). Values of Z different from one indicate non-ideal behavior due to molecular interactions and finite molecular volumes.

van der Waals Equation

The van der Waals equation is a modified form of the ideal gas law that accounts for non-ideal gas behavior by incorporating two correction factors: one that considers the finite volume occupied by gas molecules (repulsive forces) and another that accounts for the attractive forces between molecules. This equation introduces constants specific to each gas, making it a more accurate model under conditions where the ideal gas law fails.

Low Pressure Approximation

The low pressure approximation is a simplification used in gas equations where the pressure is sufficiently low that the corrections due to molecular volume and intermolecular attractions become small. Under these circumstances, terms in the equation that depend on pressure and its higher powers can be linearized or neglected, thereby simplifying the analysis and allowing an easier understanding of deviations from ideal behavior.

Perturbative Corrections

Perturbative corrections refer to the small modifications made to the ideal gas law to account for non-ideal effects. In the context of the van der Waals equation, the correction terms (associated with repulsive and attractive interactions) are expanded and approximated for conditions such as low pressure, leading to simplified expressions for the compressibility factor. This approach helps in identifying the leading contributions that cause deviations from ideality.

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