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

$\text{V = }\frac{\text{n R T}}{\text{P}}$

Step 2: Substitution and correct evaluation

$V=\frac{2.00\times 8.31\times 300}{101.3\times {10}^3}{\text{ = 0.049 m}}^3$

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:

$\text{k = }\frac{\text{R}}{{\text{N}}_{\text{A}}}$

Where:

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