Ideal Gas Notes: Assumptions and Equations

• Ideal Gas SL S1.5 Study Notes

Learning Outcomes

• An ideal gas comprises constantly moving particles with negligible volume and no intermolecular forces; collisions between particles are elastic.
• There is a need to recognize the assumptions in the ideal gas model.
• Real gases deviate from the ideal gas model, specifically at high pressure and/or low temperatures.
• The limits of the ideal gas model should be explained.
• The molar volume of an ideal gas is constant at a specific temperature and pressure.
• Investigate the relation between temperature, pressure, and volume for a fixed mass of an ideal gas, and analyze graphs showing these variables.
• The relationship between pressure, volume, amount, and temperature of an ideal gas is shown in the ideal gas equation PV = nRT and the combined gas law.
• Solve ideal gas equation problems.

Overview

• The ideal gas model assumes that gas particles do not attract or repel each other and occupy negligible volume. Additionally, it presumes that collisions between gas particles are perfectly elastic, meaning that there is no loss of kinetic energy during these collisions. This model is useful for simplifying calculations and understanding gas behavior, especially at low pressures and high temperatures where real gases behave more like ideal gases.
• Compare real gases vs. ideal gases
• The molar volume of an ideal gas
• Pressure, volume, amount, and temperature relation for an ideal gas

Assumptions of the Ideal Gas Model

• Gas molecules are in constant, random motion, moving in straight lines until colliding with another molecule or the container's side.
• These collisions are perfectly elastic; therefore, no energy is lost from the system.
• There are no intermolecular forces between gas particles.
• Due to absent intermolecular forces, an ideal gas will not condense.
• The kinetic energy of molecules is proportional to Kelvin temperature.
• Gas molecules' volume is insignificant compared to the container's volume they occupy.
• Vaporized water occupies 1600 times the volume of liquid water at standard temperature and pressure (STP), 273.15 K and 100 kPa, which equals >99.9% empty space so gas molecules can move freely.
• An ideal gas consists of elastically colliding particles, no intermolecular forces, and negligible volume compared to the gas' volume.

Pressure-Volume Relationships

• Boyle's Law: at constant temperature, the pressure of a gas is inversely proportional to its volume, expressed as p ∝ 1/V.
• Boyle's Law can also be expressed as pV = k (a constant) or P₁V₁=P₂V₂
• Gas molecules constantly strike and bounce off container walls to create measurable pressure.
• When volume is halved, pressure doubles.
• Graphs show an inverse relationship between pressure and volume.
• The SI unit of pressure is Pascal (Pa), where 1 Pa = 1 N m⁻² = 1 J m⁻³.
• Other pressure units include atmosphere (atm), millimeters of mercury (mm Hg), bar, and pounds per square inch (psi).
• Standard Temperature and Pressure (STP) for gases is 273.15 K and 100 kPa.

Worked Example Using Boyle's Law

• A weather balloon with 32.0 dm³ helium at 100.0 kPa pressure reaches 4500 m altitude, where the pressure is 57.7 kPa.
• To calculate the volume at that altitude, it's assumed that the amount of helium and temperature remain constant.
• According to Boyle's Law: V₂ = (V₁ × P₁) / P₂
• Using the values = ( 32.0 dm³ × 100 kPa)/ 57.7 kPa ≈ 55.5 dm³

Real Gases vs. Ideal Gases

• A gas deviates from ideal behavior when detectable intermolecular forces exist and/or a significant gas volume is occupied by the molecules.
• Gases commonly deviate at low temperature and/or high pressure.
• At low temperature, lower kinetic energy reduces the elastic rebound of gas molecules as intermolecular forces of attraction form.
• At high pressure, the volume of molecules is a significant portion of the total volume, and molecules are not compressible.
• Ideal gas conditions include low pressure and high temperature.
• At low pressure, space occupied by molecules is negligible due to very few unit volume molecules within the container.
• At high temperature, high kinetic energy prevents intermolecular attraction between gas particles.

The Molar Volume of an Ideal Gas

• Same volume of any two gases at the same temperature and pressure has an equal number of particles.
• The molar volume of an ideal gas, Vm, at STP, equals 22.7 dm³ mol⁻¹.

Worked Example

• With a 2.47 g mass, a 2.00 dm³ unknown gas has a molar mass of 28.0 g mol-1 at STP.
• n = V/Vm = (2.00 dm³)/(22.7 dm³ mol⁻¹) = 0.0881 mol
• M = m/n = (2.47 g) / (0.0881 mol) = 28.0 g mol⁻¹

Combined Gas Law

• Pressure is inversely proportional to volume and directly proportional to absolute temperature: p ∝ 1/V; p ∝ T
• The combined relationships can be shown as: p ∝ T/V
• (pV)/T =k (a constant) or (P₁V₁) / T₁= (P₂V₂)/ T₂
• This formula is known as the combined gas law.

Ideal Gas Equation

• pV = nRT
• R is known as universal gas constant, and it equals 8.31 J K⁻¹ mol⁻¹
• p is pressure in Pascals (Pa)
• V is volume in cubic meters (m³)
• T is temperature in Kelvin (K)
• n is the amount in moles (mol)

Explore the ideal gas model, its assumptions, and limitations. Compare ideal gases with real gases, focusing on molar volume and the relationship between pressure, volume, and temperature. Solve problems using the ideal gas equation.