Ideal Gas Law : PV = nRT Explained with Examples and Derivation

The ideal gas law is an equation that relates a gas’s pressure, volume, temperature, and amount in moles. This law is one of those equations that seems simple at first glance but explains an enormous part of the physical world. It helps describe why a balloon expands in warm air, why tyre pressure rises on a hot day, and why gas cylinders behave differently when compressed.

The formula is:

It connects pressure, volume, temperature, and the amount of gas in one clean relationship. Once you understand what each symbol means and how the units work, many gas-law questions become straightforward.

What Is the Ideal Gas Law?

The ideal gas law is an equation that relates a gas’s pressure, volume, temperature, and amount in moles:

It is used when a gas behaves approximately like an ideal gas, meaning its particles have negligible volume and do not strongly attract or repel one another. The law is most accurate at relatively low pressure and high temperature.

The Ideal Gas Law Formula: PV = nRT

The symbols in the equation mean:

The standard SI value of the gas constant is:

For most school, college, and introductory physics problems, using (R = 8.314) is enough.

An ideal gas is a simplified model of a gas. It assumes that gas molecules:

No real gas behaves perfectly this way, but many gases come close under ordinary conditions.

What Does PV = nRT Mean in Simple Words?

This law says that the behavior of a gas depends on four things:

For example, if you heat a gas in a sealed container, its pressure usually rises because the molecules move faster and strike the container walls more forcefully.

Which Value of R Should You Use?

The gas constant must match the units used in your calculation. This is one of the most common places where students lose marks.

The safest option is usually to convert everything into SI units and use:

Remember:

Why Must Temperature Be in Kelvin?

You must always use kelvin in gas-law calculations.

The reason is simple: gas laws depend on absolute temperature. Kelvin starts at absolute zero, the theoretical point where molecular motion reaches its minimum possible level.

Celsius does not start at absolute zero. A temperature of 0°C does not mean molecules have stopped moving. Therefore, using Celsius directly in gas-law ratios produces meaningless answers.

For example:

20°C = 293.15K

100°C = 373.15K

If you use 20 and 100 directly, you will get the wrong change in volume or pressure.

Derivation of the Ideal Gas Law

It can be derived from several experimental laws that describe gas behavior.

Boyle’s Law

Boyle’s law applies when temperature and the amount of gas stay constant.

This means pressure and volume are inversely related. If volume decreases, pressure increases.

Charles’s Law

Charles’s law applies when pressure and the amount of gas stay constant.

A gas expands when heated because its molecules move faster.

Avogadro’s Law

Avogadro’s law applies when pressure and temperature stay constant.

This means more moles of gas occupy more volume.

Combining the Laws

From Boyle’s law:

V ∝ 1/P

From Charles’s law:

V ∝ T

From Avogadro’s law:

V ∝ n

Combining all three relationships gives:

V ∝ nT/P

Multiplying by pressure:

PV ∝ nT

Replacing the proportionality constant with (R):

PV = nRT

That is the ideal gas law.

Kinetic Theory Derivation of PV = nRT

The ideal gas law can also be derived from kinetic theory.

Kinetic theory treats gas as a huge number of tiny particles moving randomly inside a container. When these particles collide with the container walls, they produce pressure.

For a gas with (N) molecules:

P = ⅓ Nm⟨c²⟩ / V

Multiplying both sides by volume gives:

PV = ⅓ Nm⟨c²⟩

The average translational kinetic energy of one molecule is:

Therefore:

m⟨c²⟩ = 3kBT

Substituting this into the pressure equation gives:

PV = NkBT

Since:

N = nNA

and:

R = NAkB

we get:

PV = nRT

This shows that temperature is directly connected to the average translational kinetic energy of gas molecules.

How to Solve Ideal Gas Law Questions

Use this five-step method every time.

Step 1: Write the given values

List pressure, volume, temperature, and moles clearly.

Step 2: Convert the units

Convert Celsius into kelvin. Convert litres into cubic metres if you are using SI units. Convert pressure into pascals when needed.

Step 3: Select the correct value of R

Match the gas constant to your pressure and volume units.

Step 4: Rearrange the formula

Start with:

PV = nRT

Then solve for the unknown.

For pressure:

P = nRT / V

For volume:

V = nRT / P

For moles:

n = PV / RT

For temperature:

Step 5: Check your answer

Ask yourself whether the answer makes physical sense. A hotter gas should usually have higher pressure or larger volume. A compressed gas should have higher pressure.

Practical Examples

Solved Example 1: Finding Volume Using PV = nRT

Question:
A container holds 2.0 mol of ideal gas at 300 K and a pressure of 1.5 × 10⁵ Pa. Find the volume.

Given

n = 2.0 mol

T = 300 K

P = 1.5 × 10⁵ Pa

R = 8.314 J mol⁻¹ K⁻¹

Formula

Solution

V = (2.0 × 8.314 × 300) / (1.5 × 10⁵)

V = 4988.4 / 150000

V = 0.0333 m³

Convert cubic metres to litres:

0.0333 m³ = 33.3 L

Answer:

V = 33.3 L

Solved Example 2: Boyle’s Law and Gas Compression

Question:
A gas occupies 0.50 m³ at a pressure of 2.0 × 10⁵ Pa. It is compressed at constant temperature to 0.20 m³. Find the new pressure.

Since temperature and moles remain constant, use Boyle’s law:

Rearrange:

Substitute the values:

P₂ = (2.0 × 10⁵ × 0.50) / 0.20

P₂ = 5.0 × 10⁵ Pa

Answer:

P₂ = 5.0 × 10⁵ Pa

The volume became smaller, so pressure increased.

Solved Example 3: Charles’s Law and a Hot-Air Balloon

Question:
Air in a balloon is heated from 15°C to 100°C at constant pressure. By what factor does the volume increase?

Convert temperatures to kelvin

T₁ = 15 + 273.15 = 288.15 K

T₂ = 100 + 273.15 = 373.15 K

At constant pressure:

V₂/V₁ = 373.15 / 288.15

V₂/V₁ = 1.295

Answer:
The volume increases by a factor of approximately:

V₂/V₁ ≈ 1.30

That is an increase of about 29.5%.

Hot-air balloons rise because heating the air reduces its density compared with the cooler air outside.

Solved Example 4: Combined Gas Law

Question:
A gas is initially at 1.0 × 10⁵ Pa, 0.010 m³, and 300 K. It is compressed to 0.004 m³ and heated to 400 K. Find the final pressure.

Use the combined gas law:

Rearrange:

Substitute the values:

P₂ = (1.0 × 10⁵ × 0.010 × 400) / (300 × 0.004)

P₂ = 4.00 × 10⁵ / 1.20

P₂ = 3.33 × 10⁵ Pa

Answer:

P₂ = 3.33 × 10⁵ Pa

The pressure rises because the gas is both compressed and heated.

Solved Example 5: Finding the Number of Moles

Question:
A 5.0 L container holds gas at 101.3 kPa and 298 K. How many moles of gas are present?

Use:

Since pressure is in kPa and volume is in litres, use:

R = 8.314 L kPa mol⁻¹ K⁻¹

Substitute:

n = (101.3 × 5.0) / (8.314 × 298)

n = 506.5 / 2477.6

n = 0.204 mol

Answer:

n = 0.204 mol

Boyle’s, Charles’s, Avogadro’s and Gay-Lussac’s Laws

The ideal gas law combines several simpler gas laws.

Ideal Gas Law vs Combined Gas Law

Students often confuse these two equations.

Use the Ideal Gas Law When:

You need to find pressure, volume, temperature, or moles for one gas state.

PV = nRT

Use the Combined Gas Law When:

The amount of gas remains constant, but pressure, volume, and temperature change between two states.

P₁V₁/T₁ = P₂V₂/T₂

The combined gas law is really a special form of the ideal gas law where (n) does not change.

When Can You Use the Ideal Gas Law?

This law works best when:

• Pressure is low.
• Temperature is high.
• The gas is far from condensing into a liquid.
• Molecular attractions are relatively weak.

Gases such as helium, hydrogen, nitrogen, oxygen, and argon can behave approximately ideally under many ordinary laboratory conditions.

However, the law becomes less accurate when gas particles are crowded together or strongly attracted to one another.

When Does the Ideal Gas Law Fail?

The law becomes less reliable at very high pressures and very low temperatures.

At high pressure, gas molecules are forced close together. Their own volume is no longer negligible, and repulsive effects become important.

At low temperature, molecules move more slowly. Attractive forces become more important, and the gas may eventually condense into a liquid.

A useful way to measure real-gas behavior is with the compressibility factor:

Z = PV / nRT

When:

Z = 1

the gas behaves ideally.

When:

Z < 1

attractive forces are more important.

When:

Z > 1

repulsive effects and molecular size become more important.

The van der Waals Equation

For gases that do not behave ideally, scientists often use the van der Waals equation:

(P + an²/V²)(V − nb) = nRT

Where:

a = correction for attractive forces between molecules

b = correction for the finite volume of gas molecules

The ideal gas law is still useful because it is simple, accurate enough in many conditions, and provides the foundation for more advanced gas models.

Common Ideal Gas Law Mistakes

Using Celsius Instead of Kelvin

Wrong:

T = 25

Correct:

T = 25 + 273.15 = 298.15K

Mixing Litres With Pascals

Do not use litres with R = 8.314 J mol⁻¹ K⁻¹. Convert litres to cubic metres first.

Using Gauge Pressure Instead of Absolute Pressure

The ideal gas law uses absolute pressure. Gauge pressure measures pressure above atmospheric pressure.

Forgetting to Convert Millilitres

1000 mL = 1 L

1000 L = 1 m³

Losing the Power of Ten

Scientific notation matters. A missing exponent can completely change the answer.

Using the Wrong Value of R

Always match your gas constant to your pressure and volume units.

Real-World Applications of the Ideal Gas Law

Final Takeaway

The ideal gas law is more than a formula to memorise. It is a practical way to understand how gases respond when pressure, temperature, volume, or amount changes.

Start with:

Convert temperature to kelvin. Choose the correct value of (R). Keep units consistent. Then check whether your answer makes physical sense.

Once you can do that confidently, questions involving balloons, tyres, weather, engines, gas cylinders, and laboratory experiments all become easier to understand.

Frequently Asked Questions