The ideal gas is a theoretical substance whose behavior is perfectly predicted by simple equations. This model is useful for many everyday situations involving gases, such as air at room temperature and atmospheric pressure. However, the ideal gas is a simplification, and every real gas, including carbon dioxide (\(\text{CO}_2\)), deviates from this theoretical perfection. Understanding when \(\text{CO}_2\) stops behaving ideally is fundamental for industrial processes like carbon capture, refrigeration, and supercritical fluid extraction, where the gas is handled under extreme conditions. \(\text{CO}_2\) molecules possess both size and interactive forces, meaning they are inherently “real” and only approximate ideal behavior under specific circumstances.
Defining the Ideal Gas Model
The ideal gas model is based on the kinetic molecular theory, which relies on specific assumptions about gas particles. The first assumption is that gas particles are so small compared to the space they occupy that their individual volume is considered negligible. This allows the entire volume of the container to be treated as free space for the particles to move.
The second core assumption is that there are no attractive or repulsive forces acting between the gas particles or the container walls. Particles move randomly and independently, interacting only through perfectly elastic collisions. These two foundational ideas allow for the derivation of the simple Ideal Gas Law, PV = nRT, which connects pressure (P), volume (V), moles (n), and temperature (T) without accounting for molecular interactions or particle size.
Why CO₂ Deviates from Ideal Behavior
Carbon dioxide is a “real” gas because its molecules possess both a measurable size and intermolecular forces. The \(\text{CO}_2\) molecule occupies a finite volume that cannot be disregarded, especially when the gas is compressed. Although the molecule is linear and nonpolar overall, it possesses quadrupole moments, allowing for weak, short-range attractive forces, primarily London dispersion forces, to exist between neighboring \(\text{CO}_2\) molecules.
These forces are stronger in \(\text{CO}_2\) than in smaller gases like helium or hydrogen due to its larger size and greater number of electrons. The degree of deviation from ideal behavior is often quantified using the compressibility factor, Z, where Z is equal to the ratio PV/nRT. For an ideal gas, Z always equals 1 under all conditions. Because \(\text{CO}_2\) has both volume and attractive forces, its Z value is rarely precisely 1, confirming its non-ideal nature. At low pressures, attractive forces cause Z to dip below 1, while at very high pressures, the finite molecular volume causes Z to rise above 1.
External Conditions that Cause Model Breakdown
The approximations of the Ideal Gas Law become completely inaccurate for \(\text{CO}_2\) under conditions of high pressure and low temperature. These factors amplify the significance of molecular volume and intermolecular attraction. When the gas is subjected to very high pressure, the volume of the container is drastically reduced, forcing the molecules much closer together. The actual volume occupied by the \(\text{CO}_2\) molecules themselves becomes a significant fraction of the total container volume, making the assumption of negligible particle volume invalid.
Lowering the temperature causes the average kinetic energy of the \(\text{CO}_2\) molecules to decrease, making them move more slowly. This slower movement allows the weak intermolecular attractive forces to have a greater influence on molecular motion. These forces pull molecules toward one another, reducing the force of their impact on the container walls. The model’s failure is most dramatic near the critical point of \(\text{CO}_2\), which occurs at about 304 Kelvin (31 degrees Celsius) and 73 atmospheres of pressure. Near this point, \(\text{CO}_2\) transitions into a supercritical fluid, and the ideal gas equation is entirely inadequate for predicting its complex behavior.
Using Corrective Equations for Real Gases
When the Ideal Gas Law fails to predict the behavior of \(\text{CO}_2}\) under non-ideal conditions, a modified equation of state must be used. The most well-known corrective model is the Van der Waals equation, which introduces two specific correction terms to the original ideal gas formula. One term, represented by the constant ‘b’, is subtracted from the total volume and accounts for the finite volume occupied by the gas molecules themselves. This correction term addresses the failure of the model at high pressure.
The second term, represented by the constant ‘a’, is added to the measured pressure to account for the attractive intermolecular forces that reduce the effective pressure on the container walls. The ‘a’ term, which corrects for the effect of attraction, is particularly important at low temperatures where molecular movement is slow. Though the Van der Waals equation provides a substantial conceptual improvement, more complex equations of state, such as the Redlich-Kwong or Peng-Robinson equations, are often employed in engineering for higher-accuracy calculations involving \(\text{CO}_2\) across a wide range of industrial pressures and temperatures. These later equations build upon the Van der Waals framework by using more refined mathematical expressions for the attractive and repulsive forces between real gas molecules.