Ideal Gas Equation

$P V = n R T$

$P V = N k_{B} T$

The variables will be outlined below
The empirical gas laws can be combined to give a single constant, known as the ideal gas constant, R

A gas which obeys the ideal gas equation at all pressures, volumes and temperatures

$P V = n R T$

Constants

The ideal gas constant R is the macroscopic equivalent of the Boltzmann constant $k_{B}$
- The ideal gas constant is associated with macroscopic quantities such as volume and temperature
- The Boltzmann constant is associated with the thermal energy of microscopic particles

The Boltzmann constant is defined as the ratio of the ideal gas constant R and Avogadro's constant N_A:

$k_{B} = \frac{R}{N_{A}}$

$P V = N k_{B} T$

A gas has a temperature of –55°C and a pressure of 0.5 MPa. It occupies a volume of 0.02 m³.

Calculate the number of gas particles.

Answer:

Step 1: Write down the known quantities

Step 2: Write down the equation of state of ideal gases

$P V = n R T$

Step 3: Rearrange the above equation to calculate the number of moles n

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

Step 4: Substitute numbers into the equation

$n = \frac{(0.5 \times 10^{6}) \times 0.02}{8.31 \times 218} = 5.5 moles$

Step 5: Calculate the number of particles N

$n = \frac{N}{N_{A}} \Rightarrow N = n N_{A}$

$N = 5.5 \times (6.02 \times 10^{23}) = 3.3 \times 10^{24}$

The values for the gas constant, Avogadro constant and Boltzmann constant are all given in your data booklet.

Always make sure that temperature T is in kelvin for this topic. You must convert this from °C if not using the following conversion

$θ / ° C = T / K - 273 T / K = θ / ° C + 273$

Ideal Gas Equation (HL IB Physics)