3.6.A.1 Deviations from Ideal Gas Behavior:

1.Interparticle Attractions:

These are the forces that act between molecules, as opposed to the intramolecular forces that hold atoms together within a molecule. The main types of intermolecular forces are:

a. London Dispersion Forces: Temporary attractions due to instantaneous dipoles.

b. Dipole-Dipole Interactions: Between polar molecules, where the positive end of one molecule attracts the negative end of another.

c. Hydrogen Bonding: A special, stronger case of dipole-dipole interaction, typically between hydrogen and electronegative atoms like oxygen, nitrogen, or fluorine.

d. Ion-Dipole Interactions: Occur when ionic compounds interact with polar molecules.

i. Effect of Intermolecular Forces on Deviations:

As temperature decreases (and pressure increases), molecules slow down, and intermolecular forces become more significant. This is especially true near condensation, where gases begin transitioning into liquids. In an ideal gas, intermolecular forces are considered negligible, meaning particles do not attract or repel each other.

However, in real gases, intermolecular forces cause deviations from the ideal gas behavior, particularly near the condensation point. Here’s how:

a. Attractive Forces and Deviations at Low Temperatures:

1. • When a gas is cooled, the molecules move more slowly, and the attractive intermolecular forces become more significant.
   • As these attractive forces act between molecules, the actual pressure of the gas becomes lower than predicted by the ideal gas law because the molecules are being pulled together (reducing the number of collisions with the walls of the container).
   • This is especially true near the condensation point when the gas is transitioning to liquid form.

b. Repulsive Forces at High Densities:

• • When the gas is compressed (at high pressure or low temperature), the molecules are packed closely together.
  • This leads to repulsive interactions because molecules don’t want to be compressed too much (they have finite sizes).
  • At this stage, the real gas behaves differently from an ideal gas, and the pressure can be higher than what the ideal gas law would predict.

iii. Near Condensation:

When gas is near its condensation point (for example, near the critical temperature), these intermolecular forces cause a real gas to deviate from ideal gas behavior. As gas molecules are attracted to each other more strongly, they tend to cluster, reducing the volume and pressure relative to what would be expected from the ideal gas equation. When the gas cools enough or the pressure increases sufficiently, these attractive forces dominate, causing the gas to condense into a liquid.

2. Particle Volume:

i. Ideal Gas Assumption:

The ideal gas law assumes that gas molecules have no volume and don’t interact with each other (except for elastic collisions). This assumption works well at low pressures and high temperatures, where the gas molecules are far apart and interact minimally.

However, at high pressures (or low volumes), the volume occupied by the individual gas molecules themselves cannot be ignored. As the pressure increases, gas molecules are pushed closer together, and their finite size starts to matter.

ii. How Finite Size Affects Gas Behavior:

a. Volume of Molecules Becomes Significant:

1. • In reality, gas molecules occupy a small but finite amount of space. The more tightly packed the molecules are (which happens at high pressures), the more this excluded volume (the space the molecules themselves occupy) affects the behavior of the gas.
   • As a result, the available volume for gas molecules to move is reduced, leading to a departure from ideal gas behavior.

b. Deviation from Ideal Gas Law:

1. 1. The ideal gas law assumes that the gas particles are point masses with no volume. However, at high pressures, this assumption breaks down because the molecules are closer together and occupy a non-negligible volume.
   2. The real gas law accounts for this by introducing a correction factor to the volume. In the Van der Waals equation, for example, the volume term is adjusted to account for the finite volume of the gas molecules:

\( \left(P + \frac{a}{V^2}\right)(V – b) = nRT \)

Where:

$b$ represents the volume occupied by the gas molecules themselves (the excluded volume).

$a$ accounts for intermolecular attractions, but here we’re focused on the $b$ term that corrects for the finite size of molecules.

c. Effect at High Pressure:

• • • • High pressure means the gas molecules are packed tightly together, and the space between them becomes very small. The ideal gas law doesn’t consider the fact that the molecules cannot be compressed indefinitely because they have finite size.
      • As pressure increases, the molecules’ physical volume becomes a more significant factor in determining the total volume of the gas, making the gas behave differently than predicted by the ideal gas law. The gas volume will be smaller than predicted by the ideal gas equation, since the volume occupied by the molecules themselves takes up some of the total volume.

d. Compression Limitations:

• • At extremely high pressures, the gas is essentially compressed to the point where the intermolecular forces and the volume of individual molecules dominate. In such conditions, the gas behaves more like a liquid or solid, and the compression is limited by the molecules’ inability to occupy less space.

3. Van der Waals Equation:

The Van der Waals equation is a modified version of the ideal gas law that accounts for both intermolecular forces and the finite size of gas molecules. It provides a more accurate description of real gases, particularly under conditions where the ideal gas law fails (like at high pressures or low temperatures).

The Van der Waals equation is:

$$\left(P + \frac{a}{V^2}\right)(V – b) = nRT$$

Where:

• P is the pressure of the gas.
• V is the volume of the gas.
• T is the temperature of the gas.
• n is the number of moles of the gas.
• R is the universal gas constant.
• a is a constant that accounts for the attractive intermolecular forces between gas molecules.
• b is a constant that accounts for the finite volume of gas molecules (the “excluded volume”).

i. How the Van der Waals Equation Modifies the Ideal Gas Law?

The Van der Waals equation adjusts the ideal gas law by modifying two aspects of gas behavior:

a. Intermolecular Attractions (Term with $a$):

1. • In real gases, molecules attract each other, especially at low temperatures or high pressures, which affects how they move and interact.
   • The term
     $\frac{a}{V^2}$

     represents these attractive forces. As the volume of the gas decreases (i.e., the gas is compressed), these attractions become more significant.

   • The
     $a$ constant is specific to each gas and reflects how strongly the molecules of that gas attract each other. The higher the $a$

     value, the stronger the attractive forces.

b. Finite Volume of Gas Molecules (Term with $b$):

• • The ideal gas law assumes that gas molecules have no volume. However, real molecules do have finite sizes, and at high pressures, the volume they occupy becomes significant.
  • The term ($(V – b)$

    accounts for the excluded volume — the space taken up by the molecules themselves. This means that the gas particles cannot be compressed into an arbitrarily small volume.

  • The b $\mathrm{<span\; style="font-family:\; Georgia,\; \text{'}Times\; New\; Roman\text{'},\; \text{'}Bitstream\; Charter\text{'},\; Times,\; serif;\; font-size:\; 16px;">constant\; represents\; the\; volume\; occupied\; by\; one\; mole\; of\; gas\; molecules.\; It\; is\; specific\; to\; the\; gas\; and\; depends\; on\; the\; size\; and\; shape\; of\; the\; molecules.</span>}$

ii. Key Concepts:

• Intermolecular Forces: The attractive forces between molecules cause the pressure of a real gas to be lower than predicted by the ideal gas law at high densities (low volumes), because molecules are attracted to each other and are less likely to collide with the walls of the container.
• Finite Size of Molecules: The finite size of the molecules reduces the available space for movement, causing a deviation from the ideal gas law, especially at high pressures.

iii. Practical Application of the Van der Waals Equation:

• The a and b constants are determined experimentally for each gas. These constants help describe the behavior of specific gases more accurately than the ideal gas law.
• For example, gases like oxygen (O₂) or carbon dioxide (CO₂) can be described by the Van der Waals equation, which will show how they behave differently from an ideal gas, especially under conditions where they are near their condensation point or compressed to high pressures.