# IB DP Chemistry: SL复习笔记1.2.4 The Ideal Gas Equation

### Ideal Gas Equation

#### Kinetic theory of gases

• The kinetic theory of gases states that molecules in gases are constantly moving
• The theory makes the following assumptions:
• The gas molecules are moving very fast and randomly
• The molecules hardly have any volume
• The gas molecules do not attract or repel each other (no intermolecular forces)
• No kinetic energy is lost when the gas molecules collide with each other (elastic collisions)
• The temperature of the gas is directly proportional to the average kinetic energy of the molecules
• Gases that follow the kinetic theory of gases are called ideal gases
• However, in reality gases do not fit this description exactly but may come very close and are called real gases
• The volume that a gas occupies depends on:
• Its pressure
• Its temperature

#### Ideal gas equation

• The ideal gas equation shows the relationship between pressure, volume, temperature and number of moles of gas of an ideal gas:

#### PV = nRT

P = pressure (pascals, Pa)

V = volume (m3)

n = number of moles of gas (mol)

R = gas constant (8.31 J K-1 mol-1)

T = temperature (Kelvin, K)

• The ideal gas equation can also be used to calculate the molar mass (M) of a gas

#### Worked Example

Calculate the volume, in dm3, occupied by 0.781 mol of oxygen at a pressure of 220 kPa and a temperature of 21 °C.

Step 1: Rearrange the ideal gas equation to find volume of the gas Step 2: Convert into the correct units and calculate the volume the oxygen gas occupies

p = 220 kPa = 220 000 Pa

n = 0.781 mol

R = 8.31 J K-1 mol-1

T = 21 oC = 294 K = 0.00867 m3

8.67 dm3

#### Exam Tip

A word about units...Students often mess up gas calculations by getting their unit conversions wrong, particularly from cm3 to m3. Think about what a cubic metre actually is - a cube with sides 1 m or 100 cm long.The volume of this cube is 100 x 100 x 100 = 1 000 000 or 10cm3So when you convert from m3 to cmyou MULTIPLY by 10and when you convert from cm3 to myou DIVIDE by 106  (or multiply by 10-6 which is the same thing)

#### Worked Example

Calculate the pressure of a gas, in kPa, given that 0.20 moles of the gas occupy 10.1 dm3 at a temperature of 25 oC.

Step 1: Rearrange the ideal gas equation to find the pressure of the gas Step 2: Convert to the correct units and calculate the pressure

n = 0.20 mol

V = 10.1 dm3 = 0.0101 m= 10.1 x 10-3 m

R = 8.31 J K-1 mol-1

T = 25 oC = 298 K P = 49 037 Pa = 49 kPa (2 sig figs)

#### Worked Example

Calculate the temperature of a gas, in oC, if 0.047 moles of the gas occupy 1.2 dm3 at a pressure of 100 kPa.

Step 1: Rearrange the ideal gas equation to find the temperature of the gas Step 2: Convert to the correct units and calculate the pressure

n = 0.047 mol

V = 1.2 dm3 = 0.0012 m= 1.2 x 10-3 m

R = 8.31 J K-1 mol-1

P = 100 kPa = 100 000 Pa T = 307.24 K = 34.24 oC= 34 oC  (2 sig figs)

#### Worked Example

A flask of volume 1000 cm3 contains 6.39 g of a gas. The pressure in the flask was 300 kPa and the temperature was 23 °C.Calculate the molar mass of the gas.

Step 1: Rearrange the ideal gas equation to find the number of moles of gas Step 2: Convert to the correct units and calculate the number of moles of gas

P = 300 kPa = 300 000 Pa

V = 1000 cm3 = 0.001 m= 1.0 x 10-3 m3

R = 8.31 J K-1 mol-1

T = 23 oC = 296 K n = 0.12 mol

Step 3: Calculate the molar mass using the number of moles of gas #### Exam Tip

To calculate the temperature in Kelvin, add 273 to the Celsius temperature, eg. 100 oC is 373 Kelvin. 