S1.4.6 Avogadro's law

Avogadro's Law: Equal Volumes, Equal Molecules

You are holding two balloons of the same size—one filled with helium and the other with oxygen. Despite the difference in the types of gases, if the balloons are kept at the same temperature and pressure, they contain the same number of gas molecules.

This surprising fact is explained by Avogadro's Law, a fundamental principle that helps us understand the behavior of gases and their role in chemical reactions.

What is Avogadro's Law?

Definition

Avogadro's law

Avogadro's law states that equal volumes of all gases, measured under the same conditions of temperature and pressure, contain equal numbers of molecules.

This principle is grounded in the idea that the volume of a gas depends on the number of particles it contains, regardless of their type or size.

Whether the gas consists of single atoms (e.g., helium, He), diatomic molecules (e.g., oxygen, $O_{2}$ , or larger molecules (e.g., carbon dioxide, $C O_{2}$ , Avogadro's Law holds true.

Mathematical Expression of Avogadro's Law

The relationship described by Avogadro's Law can be expressed mathematically as:

$\frac{V_{1}}{n_{1}} = \frac{V_{2}}{n_{2}}$

Where:

$V_{1}$ and $V_{2}$ are the volumes of two gas samples.
$n_{1}$ and $n_{2}$ are the amounts of gas (in moles) in the respective samples.

This equation shows that the volume of a gas is directly proportional to the number of moles, provided the temperature and pressure remain constant.

Tip

When solving gas stoichiometry problems, you can use Avogadro's Law to relate volumes directly, without needing to calculate the number of moles.

Applications of Avogadro's Law in Stoichiometry

Avogadro's Law is especially useful in gas-phase reactions, where the volume ratios of reacting gases mirror their mole ratios in a balanced chemical equation. This simplifies calculations and eliminates the need for intermediate mole conversions.

Example question

Combustion of Hydrogen Sulfide

The combustion of hydrogen sulfide $H_{2}$ is represented by the following equation:

$2 H_{2} S (g) + 3 O_{2} (g) \to 2 H_{2} O (l) + 2 {SO}_{2} (g)$

If $0.908 d m^{3}$ of $H_{2} S$ gas is combusted, calculate:

The volume of oxygen $O_{2}$ gas consumed.
The volume of sulfur dioxide $S O_{2}$ gas produced.

Solution

Volume of $O_{2}$ : From the balanced equation, the stoichiometric ratio of $H_{2} S$ to $O_{2}$ is (2:3). Using Avogadro's Law: $V (O_{2}) = \frac{3}{2} \times V (H_{2} S) = \frac{3}{2} \times 0.908 {dm}^{3} = 1.36 {dm}^{3}$
Volume of $S O_{2}$ :The stoichiometric ratio of $H_{2} S$ to $S O_{2}$ is (1:1). Therefore, the volume of $S O_{2}$ produced is equal to the volume of $H_{2} S$ combusted: $V ({SO}_{2}) = V (H_{2} S) = 0.908 {dm}^{3}$

Common Mistake

Avogadro's Law applies only to gases. Do not use it to calculate the volume of liquid water $H_{2} O$ produced in this reaction.

Gas Volume Ratios and Reaction Stoichiometry

Avogadro's Law simplifies gas stoichiometry by allowing you to work directly with gas volumes instead of converting to moles. This is particularly helpful when all reactants and products are gases.

Example question

Incomplete Combustion of Hydrogen Sulfide

The incomplete combustion of hydrogen sulfide produces elemental sulfur instead of sulfur dioxide:

$2 H_{2} S (g) + O_{2} (g) \to 2 H_{2} O (l) + 2 S (s)$

If $1.25 d m^{3}$ of oxygen gas is consumed, calculate the volume of hydrogen sulfide gas combusted.

Solution

From the balanced equation, the stoichiometric ratio of $H_{2} S$ to $O_{2}$ is (2:1). Using Avogadro's Law:

$V (H_{2} S) = 2 \times V (O_{2}) = 2 \times 1.25 {dm}^{3} = 2.50 {dm}^{3}$

Practical Applications of Avogadro's Law

Chemical Industry: Avogadro's Law is essential for calculating gas volumes in industrial processes, such as the synthesis of ammonia (Haber process) or the production of sulfuric acid.
Environmental Science: The law helps estimate the volumes of greenhouse gases (e.g., $C O_{2}$ ) released during combustion reactions, aiding in environmental monitoring and climate modeling.
Breathing and Respiration: The principle that equal volumes of gases contain equal numbers of molecules explains why the oxygen you inhale and the carbon dioxide you exhale occupy roughly the same volume under normal conditions.

Theory of Knowledge

To what extent does the assumption of ideal gas behavior limit the accuracy of Avogadro's Law in real-world applications?
How do scientists account for deviations in the behavior of real gases?

Key Takeaways

Avogadro's Law establishes that the volume of a gas is directly proportional to the number of moles it contains, under constant temperature and pressure.
The volume ratios of reacting gases correspond to their mole ratios, simplifying stoichiometric calculations for gas-phase reactions.
Avogadro's Law assumes ideal gas behavior. Deviations may occur under high pressure or low temperature.

Self review

What volume of nitrogen gas $N_{2}$ is required to produce 3.00 $d m^{3}$ of ammonia $N H_{3}$ in the Haber process? The balanced equation is:
$N_{2} (g) + 3 H_{2} (g) \to 2 N H_{3} (g)$

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