Deviation from Ideality

In the study of gases, an ideal gas is a theoretical gas composed of a set of randomly-moving, non-interacting point particles. The behavior of an ideal gas is described by the ideal gas law, which is a good approximation for many gases under moderate pressure and temperature. However, real gases often deviate from ideal behavior due to interactions between molecules and the finite volume occupied by the molecules. This deviation is most prominent under conditions of high pressure and low temperature.

Ideal Gas Law

The ideal gas law is a fundamental equation that describes the state of an ideal gas. It is given by:

$$ PV = nRT $$

where:

$P$ is the pressure of the gas,
$V$ is the volume of the gas,
$n$ is the number of moles of the gas,
$R$ is the universal gas constant, and
$T$ is the temperature of the gas in Kelvin.

Real Gases and Deviation from Ideality

Real gases deviate from the ideal gas law due to two main factors:

Intermolecular Forces: Attractive and repulsive forces between molecules affect the pressure of a gas. At high pressures (or low temperatures), these forces become significant.
Finite Molecular Volume: The ideal gas law assumes that the volume of the gas molecules is negligible compared to the volume of the container. However, at high pressures, the volume occupied by the gas molecules themselves cannot be ignored.

Van der Waals Equation

To account for these deviations, the Van der Waals equation modifies the ideal gas law to include terms for intermolecular forces and molecular volume:

$$ \left( P + \frac{a(n/V)^2} \right) (V - nb) = nRT $$

where:

$a$ is a measure of the magnitude of the intermolecular forces,
$b$ is the volume occupied by one mole of the gas molecules (excluded volume),
$n$ is the number of moles,
$V$ is the volume,
$P$ is the pressure,
$R$ is the gas constant, and
$T$ is the temperature.

Differences and Important Points

Aspect	Ideal Gas	Real Gas
Intermolecular Forces	Neglected	Considered (attraction and repulsion)
Molecular Volume	Neglected (molecules are point particles)	Considered (molecules have volume)
Equation of State	$PV = nRT$	$\left( P + \frac{a(n/V)^2} \right) (V - nb) = nRT$
Applicability	Low pressure, high temperature	High pressure, low temperature
Compressibility Factor	$Z = 1$ (always)	$Z \neq 1$ (varies with P and T)

Compressibility Factor (Z)

The compressibility factor $Z$ is a measure of the deviation from ideal gas behavior. It is defined as:

$$ Z = \frac{PV}{nRT} $$

For an ideal gas, $Z$ is always equal to 1. For real gases, $Z$ can be greater or less than 1, depending on the conditions.

Examples of Deviation from Ideality

High Pressure: As pressure increases, the volume of the gas decreases, and the molecules are forced closer together. This increases the effect of intermolecular forces, leading to a decrease in pressure compared to an ideal gas. This results in $Z < 1$.
Low Temperature: At low temperatures, the kinetic energy of the gas molecules decreases, and the effect of attractive forces becomes more significant. This can lead to liquefaction of the gas and a significant deviation from ideal behavior.
Noble Gases: At moderate conditions, noble gases (e.g., helium, neon) behave almost ideally because they have weak intermolecular forces.
Polar Gases: Gases like ammonia (NH₃) or water vapor (H₂O) show significant deviation from ideality even at moderate conditions due to strong dipole-dipole interactions.

Understanding the deviation from ideality is crucial in chemical engineering and physical chemistry, where accurate predictions of gas behavior are necessary for designing equipment and processes.