Ideal Gas Equation

This lesson covers: 

  1. How the combined gas law relates pressure, volume, temperature, and moles of gas.
  2. Understanding the equation of state for an ideal gas.
  3. Defining the molar gas constant R and its use in the equation.
  4. Introducing the Boltzmann constant k for relating pressure, volume, and temperature to the number of gas particles.

Deriving the combined gas law

The combined gas law is an amalgamation of three fundamental gas laws:

  • Boyle's Law - This law states that for a given amount of gas at a constant temperature, the pressure (P) is inversely proportional to its volume (V).
  • Charles's Law - For a fixed amount of gas at a constant pressure, the volume (V) directly varies with the temperature (T).
  • Avogadro's Law - This law states that at constant pressure and temperature, the volume (V) of a gas is directly proportional to the number of moles (n) of the gas.


By combining these proportionalities, we derive the combined gas law equation:


p V = n R T


Where:

  • P = Pressure (Pa)
  • V = Volume (m3)
  • n = Number of moles (mol)
  • R = Ideal gas constant (8.31 J K-1 mol-1)
  • T = Absolute temperature (K)

Worked Example - Calculating the volume of an ideal gas

Given a 2.00 mol sample of an ideal gas at a pressure of 101.3 kPa and a temperature of 300 K, find the volume of the gas.


Step 1: Rearranged formula

V = Pn R T​


Step 2: Substitution and correct evaluation

V=101.3×1032.00×8.31×300​ = 0.049 m3

The Boltzmann Constant k

The Boltzmann constant k, having the value 1.38 x 10-23 J K-1, modifies the ideal gas equation to focus on the number of particles (N) rather than the number of moles (n):


P V = N k T


Where: 

  • P = pressure (Pa)
  • V = volume (m3)
  • N = number of particles
  • k = Boltzmann constant (J K-1 )


This version of the equation connects pressure, volume, and temperature on a microscopic scale, with k representing the gas constant per particle.


The relationship between the constants R and k is:


k = NA​R​


Where:

  • R = Ideal gas constant (8.31 J K-1)
  • NA = Avogadro's number  (6.02 x 1023 mol-1)