S1.5.3 Molar volume of an ideal gas

Understanding Molar Volume and Graphical Relationships in Gases

You’re inflating a balloon for a party.
With each breath, it grows larger, and you might start to wonder: how much space does the air inside actually take up?
What if you could measure the volume of the gas and count the number of molecules inside?

This curiosity leads us to an important concept in chemistry: molar volume.

What is Molar Volume?

Definition

Molar volume

The molar volume of a gas is the volume occupied by one mole of an ideal gas under specific conditions of temperature and pressure.

This property is a cornerstone of gas behavior, as described by the ideal gas law.

At STP (Standard Temperature and Pressure):

Temperature = 273.15 K (0°C)
Pressure = 100 kPa
Molar Volume = 22.7 dm³ mol⁻¹

In simple terms, this means that one mole of any ideal gas will occupy 22.7 dm³ of space under these conditions, regardless of its chemical identity.

Example question

Calculating the Volume of a Gas at STP

Suppose you have 2 moles of oxygen gas (O₂) at STP. What is the total volume of the gas?

Solution

Using the molar volume at STP: $Volume = Moles \times Molar Volume$ $Volume = 2 mol \times 22.7 dm³/mol = 45.4 dm³$

Thus, 2 moles of oxygen gas occupy 45.4 dm³ at STP.

Tip

Always confirm that the temperature and pressure match STP conditions when using the molar volume of 22.7 dm³ mol⁻¹. If the conditions differ, use the ideal gas equation to calculate the volume.

Graphical Relationships Between Temperature, Pressure, and Volume

The behavior of gases can be visualized through graphs that illustrate the relationships between key variables: pressure (p), volume (V), and temperature (T).
These relationships are governed by the gas laws, which are derived from the ideal gas equation:

$p V = n R T$

Let’s explore these relationships one by one.

1. Pressure and Volume (Boyle’s Law)

Imagine squeezing a balloon. As you reduce its volume, you feel the pressure inside increase: this illustrates Boyle’s Law.

Definition

Boyle's law

Boyle's law states that at constant temperature and for a fixed amount of gas, pressure is inversely proportional to volume.

Mathematically:
$p \propto \frac{1}{V} or p V = constant$

Graphical Representation:

A graph of p vs. V forms a downward-sloping curve.
A graph of p vs. 1/V forms a straight line.

Example question

Boyle’s Law in Action

A gas occupies 4.0 dm³ at a pressure of 100 kPa. If the volume is reduced to 2.0 dm³, what will the new pressure be (assuming constant temperature)?

Solution

Using Boyle’s Law:

$p_{1} V_{1} = p_{2} V_{2}$

$100, kPa \times 4.0 dm³ = p_{2} \times 2.0 dm³$

$p_{2} = \frac{100 \times 4.0}{2.0} = 200 kPa$

The pressure doubles to 200 kPa when the volume is halved.

Common Mistake

Many students forget to keep the temperature constant when applying Boyle’s Law. Always ensure that no temperature change occurs during the process.

2. Volume and Temperature (Charles’s Law)

Now imagine heating a balloon. As the temperature rises, the balloon expands: this demonstrates Charles’s Law.

Definition

Charles's law

Charles's law states that at constant pressure and for a fixed amount of gas, volume is directly proportional to absolute temperature (in Kelvin).

Mathematically:
$V \propto T or \frac{V}{T} = constant$

Graphical Representation:

A graph of V vs. T (in Kelvin)produces a straight line passing through the origin.

Example question

Charles’s Law in Action

A gas occupies 3.0 dm³ at 273 K. What will its volume be at 546 K (assuming constant pressure)?

Solution

Using Charles’s Law:

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

$\frac{3.0, dm³}{273 K} = \frac{V_{2}}{546 K}$

$V_{2} = \frac{3.0 \times 546}{273} = 6.0 dm³$

The volume doubles as the temperature doubles (in Kelvin).

Note

Always convert temperature to Kelvin when using Charles’s Law. The Kelvin scale starts at absolute zero, where the volume of a gas theoretically becomes zero.

3. Pressure and Temperature (Gay-Lussac’s Law)

Finally, consider a sealed can of soda left in the sun. As the temperature rises, the pressure inside increases: this is explained by Gay-Lussac’s Law.

Definition

Gay-Lussac's law

Gay-Lussac's law states that at constant volume and for a fixed amount of gas, pressure is directly proportional to absolute temperature (in Kelvin).

Mathematically:
$p \propto T or \frac{p}{T} = constant$

Graphical Representation:

A graph of p vs. T (in Kelvin) produces a straight line passing through the origin.

Example question

Gay-Lussac’s Law in Action

A gas has a pressure of 150 kPa at 300 K. What will the pressure be at 450 K (assuming constant volume)?

Solution

Using Gay-Lussac’s Law:

$\frac{p_{1}}{T_{1}} = \frac{p_{2}}{T_{2}}$

$\frac{150 kPa}{300 K} = \frac{p_{2}}{450 K}$

$p_{2} = \frac{150 \times 450}{300} = 225 kPa$

The pressure increases as the temperature increases.

Reflection Questions

Self review

Why is it important to use Kelvin instead of Celsius when studying gas laws?
How would the graphical relationships change for a real gas under high pressure or low temperature?
Can you think of any situations where deviations from ideal gas behavior might have significant consequences?

Theory of Knowledge

How do the assumptions of the ideal gas model influence its applicability to real gases?
Consider the limitations of the model when analyzing gas behavior in extreme conditions, such as high pressures or low temperatures.

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S1.5.3 Molar volume of an ideal gas

Understanding Molar Volume and Graphical Relationships in Gases

What is Molar Volume?

Calculating the Volume of a Gas at STP

Graphical Relationships Between Temperature, Pressure, and Volume

1. Pressure and Volume (Boyle’s Law)

Graphical Representation:

Boyle’s Law in Action

2. Volume and Temperature (Charles’s Law)

Graphical Representation:

Charles’s Law in Action

3. Pressure and Temperature (Gay-Lussac’s Law)

Graphical Representation:

Gay-Lussac’s Law in Action

Reflection Questions

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Introduction to Molar Volume