Behaviour of Real Gases — Explained

Detailed Explanation

The study of gases often begins with the concept of an ideal gas, a theoretical model based on a set of simplifying assumptions. These assumptions, derived from the Kinetic Molecular Theory of Gases, include: (1) gas molecules are point masses with negligible volume; (2) there are no intermolecular forces (attractive or repulsive) between gas molecules; (3) collisions between molecules and with container walls are perfectly elastic; and (4) the average kinetic energy of gas molecules is directly proportional to the absolute temperature.

Based on these assumptions, the ideal gas law, $PV = nRT$, was formulated, providing a simple and powerful tool for describing gas behavior under a wide range of conditions.

However, real gases, the actual gases found in nature, do not perfectly adhere to these ideal assumptions. While the ideal gas model serves as an excellent approximation under certain conditions (typically high temperatures and low pressures), its limitations become apparent when gases are subjected to high pressures or low temperatures. Understanding the 'behaviour of real gases' involves examining these deviations and developing more accurate models to describe them.

Conceptual Foundation: Why Real Gases Deviate

Real gases deviate from ideal behavior primarily due to the breakdown of two key assumptions of the Kinetic Molecular Theory:

1
Finite Volume of Gas Molecules: — The ideal gas model assumes that gas molecules are point masses, meaning they occupy no volume. In reality, gas molecules, though small, do possess a finite, non-zero volume. At low pressures, the volume occupied by the molecules themselves is negligible compared to the total volume of the container. However, as pressure increases, the molecules are forced closer together, and the actual volume available for their movement (the 'free volume') becomes significantly less than the container volume. The ideal gas law overestimates the available volume because it doesn't account for the space taken up by the molecules themselves.

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Intermolecular Forces: — The ideal gas model assumes no attractive or repulsive forces between gas molecules. In reality, gas molecules experience weak attractive forces (van der Waals forces, dipole-dipole interactions, hydrogen bonding) and strong repulsive forces when they come into very close contact. These attractive forces become significant at low temperatures (when molecules move slower and can 'linger' near each other) and high pressures (when molecules are closer together). These attractive forces tend to pull molecules towards each other, reducing the frequency and force of their collisions with the container walls. Consequently, the observed pressure exerted by a real gas is often less than what would be predicted by the ideal gas law, as the ideal gas law assumes no such 'internal' pressure reduction.

Key Principles: Quantifying Deviations - The Compressibility Factor (Z)

To quantify the deviation of real gases from ideal behavior, the compressibility factor (Z) is introduced:

$$Z = \frac{PV_{real}}{nRT}$$

For an ideal gas, $PV = nRT$, so $Z = 1$. For real gases, $Z$ can be greater than or less than 1, depending on the conditions and the specific gas:

If $Z < 1$ — This indicates that the real gas is more compressible than an ideal gas. This typically occurs at moderate pressures and low temperatures, where attractive intermolecular forces dominate. The attractive forces pull molecules closer, reducing the effective pressure and making the gas occupy a smaller volume than predicted by the ideal gas law. $PV_{real} < nRT$.
If $Z > 1$ — This indicates that the real gas is less compressible than an ideal gas. This typically occurs at very high pressures, where the finite volume of the molecules becomes the dominant factor. The molecules themselves occupy a significant portion of the container volume, reducing the free space available for movement and making the gas effectively 'harder' to compress. $PV_{real} > nRT$.
If $Z = 1$ — The gas behaves ideally. This occurs at very high temperatures and very low pressures.

The plot of Z versus pressure for various gases at a constant temperature is a crucial tool for visualizing these deviations. Typically, for most gases like $N_2$, $O_2$, $CO_2$, etc., Z first decreases below 1 (due to attractive forces) and then increases above 1 at higher pressures (due to molecular volume).

Hydrogen ($H_2$) and Helium ($He$) are exceptions; they generally show $Z > 1$ at all practical pressures at room temperature, as their molecules are very small and have extremely weak attractive forces, making the molecular volume effect dominant even at moderate pressures.

Derivations: The Van der Waals Equation of State

Johannes Diderik van der Waals proposed an equation in 1873 that attempts to correct the ideal gas law for the finite volume of molecules and intermolecular forces. The van der Waals equation for $n$ moles of gas is:

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

Let's break down the correction terms:

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Pressure Correction ($P + \frac{an^2}{V^2}$): — The term $\frac{an^2}{V^2}$ is added to the observed pressure ($P_{observed}$) to account for the attractive intermolecular forces. These forces reduce the impact of molecules on the container walls, leading to a lower observed pressure. The 'ideal' pressure (the pressure if there were no attractive forces) would be higher. The constant 'a' is a measure of the magnitude of attractive forces between gas molecules. A larger 'a' value indicates stronger attractive forces. The term is proportional to $n^2/V^2$ because the attractive forces depend on the number of interacting pairs of molecules, which is proportional to the square of the concentration ($n/V$).

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Volume Correction ($V - nb$): — The term $nb$ is subtracted from the observed volume ($V_{container}$) to account for the finite volume occupied by the gas molecules themselves. The constant 'b' represents the effective volume occupied by one mole of gas molecules. It's often called the 'excluded volume' or 'co-volume'. The actual volume available for the movement of molecules is $V - nb$. A larger 'b' value indicates larger molecular size.

Significance of 'a' and 'b' constants:

'a' (attraction parameter): — Reflects the strength of intermolecular attractive forces. Higher 'a' means stronger attractions. Units: $ext{L}^2 \text{atm mol}^{-2}$ or $ext{m}^6 \text{Pa mol}^{-2}$.
'b' (volume parameter): — Reflects the effective volume of the gas molecules. Higher 'b' means larger molecules. Units: $ext{L mol}^{-1}$ or $ext{m}^3 \text{mol}^{-1}$.

Critical Phenomena and Liquefaction of Gases

The behavior of real gases is crucial for understanding critical phenomena and the liquefaction of gases. As a gas is cooled and compressed, it eventually transitions into a liquid state. However, this transition is not always possible. For every gas, there exists a **critical temperature ($T_c$)** above which it cannot be liquefied, no matter how high the pressure applied. At or below $T_c$, a gas can be liquefied by applying sufficient pressure.

Critical Temperature ($T_c$): — The maximum temperature at which a gas can be liquefied by pressure alone. Above $T_c$, the kinetic energy of molecules is too high for intermolecular attractive forces to hold them together in a liquid state.
Critical Pressure ($P_c$): — The minimum pressure required to liquefy a gas at its critical temperature.
Critical Volume ($V_c$): — The volume occupied by one mole of the gas at its critical temperature and critical pressure.

These critical constants ($T_c$, $P_c$, $V_c$) can be related to the van der Waals constants 'a' and 'b':

$$T_c = \frac{8a}{27Rb}$$

$$P_c = \frac{a}{27b^2}$$

$$V_c = 3b$$

These relationships highlight how the molecular properties (intermolecular forces 'a' and molecular size 'b') directly influence the conditions required for liquefaction. Gases with higher 'a' values (stronger attractions) and lower 'b' values (smaller molecules) tend to have higher critical temperatures, making them easier to liquefy.

Real-World Applications

Understanding real gas behavior is vital in various fields:

Industrial Gas Production: — Processes like the liquefaction of air to separate nitrogen, oxygen, and argon rely heavily on the principles of real gas behavior and critical phenomena.
Refrigeration and Air Conditioning: — The working fluids (refrigerants) undergo phase changes (gas to liquid and back) under specific temperature and pressure conditions, which are governed by their real gas properties.
Chemical Engineering: — Designing reactors, pipelines, and storage tanks for gases requires accurate equations of state that account for real gas deviations, especially at high pressures and extreme temperatures.
Atmospheric Science: — Modeling the behavior of atmospheric gases, particularly at high altitudes or during weather phenomena, often requires real gas considerations.

Common Misconceptions

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All gases behave ideally at high temperature and low pressure: — While this is generally true, it's an approximation. No real gas is truly ideal. The statement should be: 'Real gases *approach* ideal behavior at high temperatures and low pressures.'
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Compressibility factor Z is always less than 1 for real gases: — This is incorrect. Z can be less than 1 (due to attractive forces) or greater than 1 (due to molecular volume), depending on the specific gas, temperature, and pressure. At very high pressures, Z is almost always greater than 1.
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Van der Waals equation is universally accurate: — While a significant improvement over the ideal gas law, the van der Waals equation is still an approximation. More complex equations of state (e.g., Redlich-Kwong, Berthelot, virial equations) exist for even greater accuracy, especially over wider ranges of temperature and pressure.
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'a' and 'b' are universal constants: — 'a' and 'b' are specific to each gas and reflect its unique molecular properties. They are not universal constants like R (the ideal gas constant).

NEET-Specific Angle

For NEET aspirants, the focus on real gases typically revolves around:

Conditions for ideal behavior: — Understanding when real gases approximate ideal behavior (high T, low P).
Interpretation of Z vs. P graphs: — Being able to explain why Z deviates from 1 and the significance of $Z < 1$ vs. $Z > 1$.
Van der Waals equation: — Knowing the form of the equation and the physical significance of the 'a' and 'b' constants. Qualitative comparison of 'a' and 'b' values for different gases (e.g., $CO_2$ has higher 'a' than $CH_4$ due to stronger attractions, $C_2H_6$ has higher 'b' than $CH_4$ due to larger size).
Critical constants: — Definitions of $T_c$, $P_c$, $V_c$ and their qualitative relationship to 'a' and 'b' (e.g., higher 'a' means higher $T_c$, easier liquefaction). Direct calculation using the formulas relating $T_c, P_c, V_c$ to 'a' and 'b' is also important.