Ideal Gas Law (PV = nRT)

Introduction

Key Concepts

Understanding the Ideal Gas Law

The Ideal Gas Law is an equation of state for a hypothetical ideal gas. It combines several gas laws to provide a single comprehensive equation:

Where:

• P = Pressure of the gas
• V = Volume of the gas
• n = Number of moles of the gas
• R = Universal gas constant ($8.314 \, \text{J/mol.K}$)
• T = Temperature in Kelvin

Assumptions of the Ideal Gas Law

The Ideal Gas Law is based on several key assumptions:

1. No intermolecular forces: Gas particles do not attract or repel each other.
2. Point particles: Gas molecules occupy no volume themselves.
3. Elastic collisions: Collisions between gas particles and with the container walls are perfectly elastic.
4. Random motion: Gas particles move in all directions with uniform speed.

Derivation from Combined Gas Laws

The Ideal Gas Law is derived by combining Boyle's Law, Charles's Law, and Avogadro's Law:

• Boyle's Law: $P \propto \frac{1}{V}$ (at constant T and n)
• Charles's Law: $V \propto T$ (at constant P and n)
• Avogadro's Law: $V \propto n$ (at constant P and T)

Combining these proportionalities leads to the Ideal Gas Law:

Applications of the Ideal Gas Law

The Ideal Gas Law has wide-ranging applications in both theoretical and practical chemistry:

• Predicting Gas Behavior: Calculating the change in one property when others are altered.
• Stoichiometric Calculations: Determining the amounts of reactants and products in gas-phase reactions.
• Engineering: Designing equipment that handles gases, such as compressors and expanders.
• Environmental Science: Estimating the concentrations of pollutants in the atmosphere.

Limitations of the Ideal Gas Law

While the Ideal Gas Law is widely used, it has limitations due to its underlying assumptions:

• High Pressure and Low Temperature: Under these conditions, real gases deviate significantly from ideal behavior as intermolecular forces become significant.
• Non-Point Molecules: Larger molecules occupy more space, violating the assumption that gas particles have no volume.
• Intermolecular Forces: Attraction or repulsion between molecules affects the pressure and volume, leading to deviations from the ideal equation.

Real Gases vs. Ideal Gases

Real gases exhibit behaviors that differ from those predicted by the Ideal Gas Law, especially under extreme conditions. The deviations are accounted for by the Van der Waals equation, which introduces correction factors for intermolecular forces and the finite volume of gas particles:

Where:

• a = Measure of the attraction between particles
• b = Volume occupied by one mole of particles

This modification provides a more accurate description of gas behavior under non-ideal conditions.

Graphical Representation of the Ideal Gas Law

The relationship between pressure, volume, and temperature can be visualized using various graphs:

• Pressure vs. Volume (P-V Diagram): Shows an inverse relationship; as volume increases, pressure decreases, demonstrating Boyle's Law.
• Volume vs. Temperature (V-T Diagram): Illustrates a direct relationship; as temperature increases, volume increases if pressure is constant, aligning with Charles's Law.
• Pressure vs. Temperature (P-T Diagram): Depicts a direct relationship; as temperature increases, pressure increases if volume is constant, consistent with Gay-Lussac's Law.

Calculations Involving the Ideal Gas Law

To apply the Ideal Gas Law in calculations, it's essential to ensure that all units are consistent:

• Pressure (P): Typically measured in atmospheres (atm), pascals (Pa), or torr.
• Volume (V): Measured in liters (L) or cubic meters (m³).
• Temperature (T): Must be in Kelvin (K). Convert Celsius to Kelvin by adding 273.15.
• Number of Moles (n): Amount of substance in moles.
• Gas Constant (R): Value of 0.0821 L.atm/mol.K is commonly used when pressure is in atm and volume in liters.

Example Calculation:

If 2 moles of an ideal gas are kept at a temperature of 300 K and a volume of 10 liters, what is the pressure exerted by the gas?

Using the Ideal Gas Law:

Partial Pressure and Dalton's Law

In a mixture of non-reacting gases, each gas exerts pressure independently. The total pressure is the sum of the partial pressures of individual gases, as described by Dalton's Law:

Where $P_1, P_2, P_3, \dots, P_n$ are the partial pressures of each gas in the mixture.

Mole Fraction and Its Role in Gas Mixtures

The mole fraction ($\chi$) represents the ratio of the number of moles of a particular gas to the total number of moles in the mixture:

The Partial Pressure of gas $i$ can be calculated using its mole fraction:

Applications in Real-World Scenarios

The Ideal Gas Law is applied in numerous real-world scenarios, including:

• Respiration: Calculating the volumes of air inhaled and exhaled under different conditions.
• Automobile Engineering: Designing engines and predicting the behavior of gases under varying temperatures and pressures.
• Meteorology: Understanding atmospheric pressure changes and weather patterns.
• Medical Technology: Managing gas mixtures used in anesthesia and respiratory therapies.

Temperature Dependence and Absolute Zero

Temperature plays a critical role in gas behavior. As temperature approaches absolute zero ($0 \, \text{K}$), the volume of an ideal gas would theoretically reach zero, and the pressure would become negligible. However, real gases exhibit liquefaction before reaching absolute zero due to intermolecular attractions.

Kinetic Molecular Theory and the Ideal Gas Law

The Kinetic Molecular Theory explains the behavior of gases and underpins the Ideal Gas Law:

• Particle Motion: Gas particles are in constant, random motion.
• Elastic Collisions: Collisions between gas particles and with container walls do not result in energy loss.
• Pressure Origin: Pressure is a result of collisions of gas particles with the container walls.
• Temperature Relationship: Temperature is directly proportional to the average kinetic energy of gas particles.

Comparison Table

Summary and Key Takeaways