The Ideal Gas Law (HSC SSCE Chemistry): Revision Notes

The Ideal Gas Law

Introduction: Combining gas laws

In many real-world situations, such as scuba diving and the use of anaesthetics, all four properties of a gas can change simultaneously. These properties are:

• Pressure ($P$)
• Volume ($V$)
• Temperature ($T$)
• Amount of gas ($n$, measured in moles)

When all these conditions vary together, we need a single equation that accounts for all of them at once. This equation comes from combining three fundamental gas laws:

• Boyle's law (relating pressure and volume)
• Charles's law (relating volume and temperature)
• Avogadro's law (relating volume and amount)

When we combine these three laws mathematically, we obtain a relationship where the quantity $\frac{PV}{nT}$ remains constant:

$\frac{PV}{nT} = \text{constant}$

This combined expression is the foundation for understanding the behaviour of gases under varying conditions.

The ideal gas equation

Since the value of $\frac{PV}{nT}$ is constant for all gases, we can give this constant a specific symbol. Scientists use the letter $R$ to represent this universal value. Therefore, we can write:

$\frac{PV}{nT} = R$

Rearranging this equation by multiplying both sides by $nT$ gives us the more familiar form:

$PV = nRT$

This equation is known by several names:

• The ideal gas law
• The ideal gas equation
• The general gas equation

The constant $R$ is called the universal gas constant because its value is the same for all gases, regardless of their chemical identity.

Why is it called the "ideal" gas law?

The equation is described as "ideal" because it assumes perfect gas behaviour. In reality, real gases deviate slightly from this equation, particularly under extreme conditions:

Calculating the universal gas constant

We can determine the value of $R$ by using known information about gases. At standard conditions of $0°\text{C}$ (which equals $273.15 \text{ K}$) and $100 \text{ kPa}$ pressure, one mole of any ideal gas occupies $22.71 \text{ L}$.

Using the equation $R = \frac{PV}{nT}$ and substituting these values:

$R = \frac{PV}{nT} = \frac{100 \text{ kPa} \times 22.71 \text{ L}}{1 \text{ mol} \times 273.15 \text{ K}}$

$R = 8.314 \text{ kPa} \cdot \text{L} \cdot \text{K}^{-1} \cdot \text{mol}^{-1}$

Understanding the units of $R$

To properly express $R$ in its standard units, we need to understand the relationship between different units of measurement.

The pascal (Pa) is the SI unit of pressure, defined as one newton per square metre: $1 \text{ Pa} = 1 \text{ N} \cdot \text{m}^{-2}$

A litre equals $10^{-3} \text{ m}^3$ (one-thousandth of a cubic metre).

Therefore, when we multiply kilopascals by litres: $1 \text{ kPa} \times \text{ L} = 10^3 \text{ N} \cdot \text{m}^{-2} \times 10^{-3} \text{ m}^3 = 1 \text{ N} \cdot \text{m}$

One newton metre ($\text{N} \cdot \text{m}$) is the definition of a joule (J), which is the SI unit for work or energy.

Using this relationship, we can express $R$ in its standard units:

$R = 8.314 \text{ J} \cdot \text{K}^{-1} \cdot \text{mol}^{-1}$

This is the value of the universal gas constant you should remember and use in calculations.

Using the ideal gas law correctly

The key challenge when using $PV = nRT$ is ensuring all units are correct and consistent. Incorrect units are the most common source of errors when working with the ideal gas law.

Exam tip: Always include units in your calculations

When substituting values into the equation, write the units alongside the numbers. This helps you:

• Check that units cancel properly
• Verify your answer has the correct units
• Identify calculation errors more easily

Solving problems with the ideal gas law

The ideal gas equation $PV = nRT$ can be rearranged to solve for any one variable when the other three are known:

To find volume: $V = \frac{nRT}{P}$

To find pressure: $P = \frac{nRT}{V}$

To find temperature: $T = \frac{PV}{nR}$

To find moles: $n = \frac{PV}{RT}$

Worked examples

Practical application: Investigation

The ideal gas law can be applied to determine the amount of gas produced during chemical reactions. By measuring the volume of gas collected, along with the temperature and pressure, you can calculate the number of moles produced.

Investigation aim: To determine the number of moles of hydrogen gas produced when magnesium reacts with hydrochloric acid.

Basic principle: When magnesium metal reacts with hydrochloric acid solution, hydrogen gas is produced. The gas is collected in a burette, and its volume is measured. The gas pushes the acid solution out of the bottom of the burette. When the reaction is complete, the volume of solution released equals the volume of gas produced.

Chemical equation: $\text{Mg}(s) + 2\text{HCl}(aq) \rightarrow \text{MgCl}_2(aq) + \text{H}_2(g)$

Key measurements needed:

• Temperature of the acid solution
• Mass of magnesium used
• Volume of hydrogen gas collected
• Air pressure

Analysis: Use the ideal gas law ($PV = nRT$) to calculate the number of moles of hydrogen produced from the measured volume. Compare this with the number of moles calculated from the mass of magnesium that reacted, using stoichiometry.

This investigation demonstrates how the ideal gas law connects theoretical calculations with practical measurements in chemistry.

Remember!