Lecture Notes: Ideal Gas Law and Density of Gases

Overview

• Exploration of the Ideal Gas Law, relationship between properties of gases.
• Key focus: Understanding pressure, volume, moles, and temperature interrelations.
• Application in deriving other gas laws and determining gas density.

Key Concepts

Relationships Among Gas Properties

• Pressure (P) and Volume (V): Inversely proportional.
  • Relationship: [ P \times V = \text{constant} ]
• Volume (V) and Temperature (T): Directly proportional.
  • Relationship: [ \frac{V}{T} = \text{constant} ]
• Volume (V) and Moles (n): Directly proportional.
  • Relationship: [ \frac{V}{n} = \text{constant} ]

Combining the Relationships

• When none of the variables are constant, combine all relationships:
  • [ P \times \frac{V}{T} \times \frac{V}{n} = R ] (( R ): Ideal Gas Constant)
• Rearranged to the commonly used form:
  • [ PV = nRT ]

Ideal Gas Constant (R)

• ( R = 0.08206 ) L·atm/mol·K
• Units dictated by equation: liters, atmospheres, moles, Kelvin.

Application

• Used for scenarios with one set of conditions: ( P, V, n, T ).
• Units: Volume in liters, Pressure in atmospheres, Temperature in Kelvin.

Deriving Other Gas Laws

Using Ideal Gas Law

• From ( PV = nRT ), derive other gas laws by holding certain variables constant.
• Example:
  • Constant Moles and Temperature: ( PV_1 = PV_2 )
  • Constant Temperature and Pressure: ( \frac{V}{n} = \frac{V_1}{n_1} )

Combined Gas Law

• Holding moles constant derives:
  • [ \frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2} ]

Density of Gases

Deriving Density from Ideal Gas Law

• Start with: [ PV = nRT ]
• Rearrange: [ \frac{n}{V} = \frac{P}{RT} ]
• Multiply by molar mass for density: [ \text{Density} = \frac{P \times \text{Molar Mass}}{R \times T} ]

Example Calculation

• Problem: Calculate the density of CO2 at STP.
• Given:
  • Temperature ( T = 273 ) K
  • Pressure ( P = 1 ) atm
  • Molar Mass CO2 = 44 g/mol
• Formula:
  • [ \text{Density} = \frac{P \times \text{Molar Mass}}{R \times T} ]
  • Solution shows correct unit cancellation and results in grams per liter.

Using Molar Volume at STP

• Alternative method using standard molar volume (22.4 L/mol):
  • [ \text{Density} = \frac{\text{Molar Mass}}{\text{Molar Volume}} ]
  • Provides consistent results with less computation.

Conclusion

• The Ideal Gas Law is a versatile tool for deriving other relationships and calculating properties like density, especially under standard conditions (STP).