Gas Law Relationships (DP IB Chemistry: SL)

• Gases in a container exert a pressure as the gas molecules are constantly colliding with the walls of the container

Gas particles exert a pressure by constantly colliding with the walls of the container

Changing gas volume

• Decreasing the volume (at constant temperature) of the container causes the molecules to be squashed together which results in more frequent collisions with the container wall
• The pressure of the gas increases

Decreasing the volume of a gas causes an increased collision frequency of the gas particles with the container wall

• The pressure is therefore inversely proportional to the volume (at constant temperature)
• This is known as Boyle's Law
• Mathematically, we say P ∝ 1/V or PV = a constant
• We can show a graphical representation of Boyle's Law in three different ways:
  • A graph of pressure of gas plotted against 1/ volume gives a straight line
  • A graph of pressure against volume gives a curve
  • A graph of PV versus P gives a straight line

Three graphs that show Boyle’s Law

Changing gas temperature

• When a gas is heated (at constant pressure) the particles gain more kinetic energy and undergo more frequent collisions with the container walls
• To keep the pressure constant, the molecules must get further apart and therefore the volume increases
• The volume is therefore directly proportional to the temperature in Kelvin (at constant pressure)
• This is known as Charles' Law
• Mathematically, V ∝ T or V/T = a constant
• A graph of volume against temperature in Kelvin gives a straight line

Increasing the temperature of a gas causes an increased collision frequency of the gas particles with the container wall (a); volume is directly proportional to the temperature in Kelvin (b)

Changing gas pressure

• Increasing the temperature (at constant volume) of the gas causes the molecules to gain more kinetic energy
• This means that the particles will move faster and collide with the container walls more frequently
• The pressure of the gas increases
• The temperature is therefore directly proportional to the pressure (at constant volume)
• Mathematically, we say that P ∝ T or P/T = a constant 
• A graph of temperature in Kelvin of a gas plotted against pressure gives a straight line

Increasing the temperature of a gas causes an increased collision frequency of the gas particles with the container wall (a); temperature is directly proportional to the pressure (b)

Pressure, volume and temperature

• Combining these three relationships together:
  • P/V = a constant
  • V/T = a constant
  • P/T = a constant

• We can see how the ideal gas equation is constructed
  • PV/T = a constant
  • PV = a constant x T

• This constant is made from two components, the number of moles, n, and the gas constant, R, resulting in the overall equation:
  • PV = nRT

Changing the conditions of a fixed amount of gas

• For a fixed amount of gas, n and R will be constant, so if you change the conditions of a gas we can ignore n and R in the ideal gas equation
• This leads to a very useful expression for problem solving
• Where P₁, V₁ and T₁ are the initial conditions of the gas and P₂, V₂ and T₂ are the final conditions

Answer:

Step 1: Rearrange the formula to change the conditions of a fixed amount of gas. Pressure is constant so it is left out of the formula

Step 2: Convert the temperature to Kelvin. There is no need to convert the volume to m³  because the formula is using a ratio of the two volumes

 V₁ = 20 dm³ 

V₂ = 10 dm³ 

T₁ = 25 + 273 = 298 K

Step 3: Calculate the new temperature

Answer:

Step 1: Rearrange the formula to change the conditions of a fixed amount of gas

Step 2: Substitute in the values and calculate the final pressure

P₁ = 80 kPa

V₁ = 2.00 dm³ 

V₂ = 2.25 dm³ 

T₁ = 20 + 273 = 293 K

T₂ = 70 + 273 = 343 K