The Ideal Gas Equation Revision Notes for Grade 11 NSC Matric Physical Sciences

The Ideal Gas Equation

Introduction to the ideal gas equation

The ideal gas equation is one of the most important equations in chemistry, allowing us to calculate relationships between pressure, volume, temperature, and the amount of gas present. This equation combines several gas laws into one comprehensive formula that can solve a wide variety of problems involving gases.

Avogadro's law

In the early 1800s, scientist Amedeo Avogadro made an important observation about gases. He discovered that when you compare samples of different gases under identical conditions, they contain the same number of particles.

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Definition: Avogadro's law Equal volumes of gases, at the same temperature and pressure, contain the same number of molecules.

This law helps us understand that the volume of a gas is directly related to the number of particles (or moles) present, regardless of what type of gas it is.

Development of the ideal gas equation

Scientists combined different gas law equations to create one equation that includes temperature, volume, and pressure. This combined equation can be written as:

$\frac{pV}{T} = k$

where k is a constant value. However, the value of k changes depending on how much gas you have (the mass of gas).

When scientists calculated k for exactly 1 mol of any gas, they discovered something remarkable - the value was always the same:

$\frac{pV}{T} = 8,314$

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This special value is called the universal gas constant and is given the symbol R. The universal gas constant has units of J·K⁻¹·mol⁻¹, which means it's measured in joules per kelvin per mole.

The ideal gas equation formula

When we extend this relationship to any number of moles of gas (not just 1 mol), we get:

$\frac{pV}{T} = nR$

Rearranging this equation gives us the ideal gas equation:

$\boxed{pV = nRT}$

Where:

p = pressure (in Pascals, Pa)
V = volume (in cubic metres, m³)
n = number of moles of gas (in mol)
R = universal gas constant (8,314 J·K⁻¹·mol⁻¹)
T = temperature (in Kelvin, K)

Important points about units

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When working with the ideal gas equation, you must use SI units throughout your calculations. This is crucial for getting the correct answer.

Key unit conversions you need to know:

Temperature: Convert °C to K by adding 273
Pressure: Convert kPa to Pa by multiplying by 1000
Volume: Convert dm³ to m³ by dividing by 1000

Exam tip: A joule can be defined as Pa·m³, so when you use the ideal gas equation with the correct SI units, you'll get the right answer.

Worked example 1: Finding pressure

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Worked Example: Finding Pressure

Question: Two moles of oxygen (O₂) gas occupy a volume of 25 dm³ at a temperature of 40°C. Calculate the pressure of the gas under these conditions.

Solution:

Step 1: Write down all the information you know about the gas.

p = ?
V = 25 dm³
n = 2 mol
T = 40°C
R = 8,314 J·K⁻¹·mol⁻¹

Step 2: Convert the known values to SI units if necessary.

Convert volume to m³ and temperature to Kelvin:

V = 25/1000 = 0.025 m³
T = 40 + 273 = 313 K

Step 3: Choose the appropriate gas law equation.

Since we're dealing with all variables (temperature, pressure, volume, and amount of gas), we must use the ideal gas equation: pV = nRT

Step 4: Substitute the known values and calculate.

(0.025 m³)(p) = (2 mol)(8,314 J·K⁻¹·mol⁻¹)(313 K) (0.025 m³)(p) = 5204.564 Pa·m³ p = 208,182.56 Pa

The pressure will be 208,182.56 Pa or 208.2 kPa.

Worked example 2: Finding volume

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Worked Example: Finding Volume

Question: Carbon dioxide (CO₂) gas is produced from a chemical reaction. The gas is collected in a container at a pressure of 105 kPa and temperature of 20°C. If the number of moles of gas collected is 0.86 mol, what is the volume?

Solution:

Step 1: Write down all the information you know.

p = 105 kPa
V = ?
n = 0.86 mol
T = 20°C
R = 8,314 J·K⁻¹·mol⁻¹

Step 2: Convert to SI units if necessary.

Convert temperature to Kelvin and pressure to Pa:

p = 105 × 1000 = 105,000 Pa
T = 20 + 273 = 293 K

Step 3: Choose the appropriate equation.

Since all variables are changing, we use the ideal gas equation: pV = nRT

Step 4: Substitute and calculate.

(105,000 Pa)V = (8,314 J·K⁻¹·mol⁻¹)(293 K)(0.86 mol) (105,000 Pa)V = 2094.96 Pa·m³ V = 0.020 m³ = 20 dm³

The volume is 20 dm³.

Worked example 3: Finding temperature

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Worked Example: Finding Temperature

Question: Nitrogen (N₂) reacts with hydrogen (H₂) to produce ammonia (NH₃). 2 mol of ammonia gas is collected in a gas cylinder with volume 25 dm³ at pressure 195.89 kPa. Calculate the temperature of the gas inside the cylinder.

Solution:

Step 1: Write down the known information.

p = 195.89 kPa
V = 25 dm³
n = 2 mol
R = 8,314 J·K⁻¹·mol⁻¹
T = ?

Step 2: Convert to SI units.

V = 25/1000 = 0.025 m³
p = 195.89 × 1000 = 195,890 Pa

Step 3: Choose the appropriate equation.

Use the ideal gas equation: pV = nRT

Step 4: Substitute and calculate.

(195,890)(0.025) = (2)(8,314)T 4897.25 = 16,628(T) T = 294.52 K

The temperature is 294.52 K.

Worked example 4: Finding number of moles

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Worked Example: Finding Number of Moles

Question: Calculate the number of moles of air particles in a classroom of length 10 m, width 7 m, and height 2 m when the temperature is 23°C and the air pressure is 98 kPa.

Solution:

Step 1: Calculate the volume of air in the classroom.

The classroom is a rectangular prism, so: V = length × width × height = (10)(7)(2) = 140 m³

Step 2: Write down the known information.

p = 98 kPa
V = 140 m³
n = ?
R = 8,314 J·K⁻¹·mol⁻¹
T = 23°C

Step 3: Convert to SI units.

T = 23 + 273 = 296 K
p = 98 × 1000 = 98,000 Pa

Step 4: Use the ideal gas equation and solve for n.

pV = nRT (98,000)(140) = n(8,314)(296) 13,720,000 = 2477.572(n) n = 5537.7 mol

The number of moles in the classroom is 5537.7 mol.

Problem-solving strategy

When tackling ideal gas equation problems, follow these steps:

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Problem-Solving Strategy:

Write down all known information - list what you know and what you need to find
Convert all values to SI units - this is essential for getting the correct answer
Choose the appropriate equation - use pV = nRT when multiple variables are changing
Substitute values and solve - be careful with your arithmetic and units
Check your answer - does the result make sense?

Remember!

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Key Points to Remember:

The ideal gas equation is pV = nRT where R = 8,314 J·K⁻¹·mol⁻¹
Always use SI units: pressure in Pa, volume in m³, temperature in K
Temperature conversions: K = °C + 273
Pressure conversions: Pa = kPa × 1000
Volume conversions: m³ = dm³ ÷ 1000
The universal gas constant R is the same for all gases - that's why it's called "universal"
When all variables are changing, you must use the full ideal gas equation

The Ideal Gas Equation (Grade 11 NSC Matric Physical Sciences): Revision Notes