Ideal gas law

Core idea

Overview

The ideal gas law represents the equation of state for a hypothetical ideal gas, combining Boyle's, Charles's, and Avogadro's laws into one relationship. It establishes a mathematical connection between the pressure, volume, absolute temperature, and the molar amount of gas present in a system.

When to use: Use this equation when analyzing the behavior of gases at relatively low pressures and high temperatures where molecules act independently. It is the primary tool for determining a missing physical property of a gas sample when the other state variables are defined.

Why it matters: This relationship is essential for chemical engineering, meteorology, and the design of pneumatic systems. It allows for the calculation of gas density and molar mass, which are critical for industrial safety and atmospheric research.

Symbols

Variables

p = Pressure, V = Volume, n = Amount of Gas, T = Temperature, R = Gas Constant

p

Pressure

Pa

V

Volume

m^{3}

n

Amount of Gas

mol

T

Temperature

K

R

Gas Constant

J/molK

Walkthrough

Derivation

Understanding the Ideal Gas Law

The ideal gas law links pressure, volume, temperature, and moles for gases behaving ideally.

The gas behaves ideally (particles have negligible volume and no intermolecular forces).
Temperature is measured in kelvin (K).

1

State the Relationship:

Pressure P times volume V equals moles n times gas constant R times temperature T.

P V = n R T

2

Temperature Conversion:

Convert °C to K before substituting into PV = nRT.

T (K) = T (^{\circ} C) + 273

Note: In GCSE chemistry, you’ll often use this to find n, V, or P when conditions aren’t RTP/STP.

Result

T (K) = T (^{\circ} C) + 273

Source: AQA GCSE Chemistry — Quantitative Chemistry (Higher Tier)

Free formulas

Rearrangements

Solve for $p$

Ideal Gas Law: Make p the subject

p = \frac{n R T}{V}

Rearrange the Ideal Gas Law to solve for pressure (p).

Difficulty: 2/5

Solve for $V$

Ideal Gas Law: Make V the subject

V = \frac{n R T}{p}

Rearrange the Ideal Gas Law (pV=nRT) to make V (volume) the subject of the equation.

Difficulty: 2/5

Solve for $n$

Ideal Gas Law: Make n the subject

n = \frac{p V}{R T}

Rearrange the Ideal Gas Law to solve for 'n', the amount of gas, by isolating it on one side of the equation.

Difficulty: 2/5

Solve for $T$

Make T the subject

T = \frac{p V}{n R}

To make Temperature (T) the subject of the Ideal Gas Law, divide both sides of the equation by the product of the amount of gas (n) and the Gas Constant (R).

Difficulty: 2/5

Solve for $R$

Make R the subject of the Ideal Gas Law

R = \frac{p V}{n T}

Rearrange the Ideal Gas Law to solve for the Gas Constant, R.

Difficulty: 2/5

The static page shows the finished rearrangements. The app keeps the full worked algebra walkthrough.

Visual intuition

Graph

The graph forms a hyperbola because Volume appears in the denominator of the pressure formula, meaning that as Volume increases, Pressure decreases toward zero and as Volume approaches zero, Pressure increases toward infinity. For a chemistry student, this shape illustrates that at a constant amount of gas and temperature, a large Volume corresponds to a low Pressure while a small Volume forces a high Pressure. The most important feature of this curve is that it never reaches zero, which signifies that a gas can never be compressed into a state of zero Volume or expanded to reach zero Pressure.

Graph type: hyperbolic

Why it behaves this way

Intuition

Visualize a vast number of infinitesimally small, non-interacting particles (gas molecules) moving randomly and rapidly within a container, constantly colliding elastically with its walls and each other.

p

Macroscopic force per unit area exerted by gas molecules on container walls.

More molecules hitting the walls, or hitting them harder and faster, increases the pressure.

V

The total space available for the gas molecules to move within.

A larger volume means molecules travel further between wall collisions, reducing the frequency of impacts and thus pressure.

n

The total number of gas molecules expressed in moles.

More molecules in the same volume means more frequent collisions with the walls, increasing pressure.

R

A universal proportionality constant that relates the energy scale of the gas to its temperature and molar amount.

It's a fixed value that ensures the units are consistent and scales the relationship between the energy-related term (pV) and the temperature-related term (nT).

T

A measure of the average translational kinetic energy of the gas molecules, expressed in Kelvin.

Higher temperature means molecules move faster on average, leading to more energetic and frequent collisions with the walls.

Free study cues

Insight

Canonical usage

All quantities must be expressed in a consistent set of units, typically SI, or a set where the chosen value of the ideal gas constant (R) matches the units of pressure, volume, and temperature.

Common confusion

The most common mistake is using temperature in Celsius or Fahrenheit instead of Kelvin, or mixing unit systems for pressure, volume, and the gas constant (R) without ensuring consistency.

Unit systems

$p$ Pa (SI), atm, Torr - Pressure must be absolute pressure.

$V$ m^3 (SI), L - Volume of the gas.

$n$ mol - Amount of substance (moles) of the gas.

$T$ K - Absolute temperature (must be in Kelvin, not Celsius or Fahrenheit).

One free problem

Practice Problem

A 2.50 mole sample of oxygen gas is placed in a 5.00 L container at a temperature of 300 K. Calculate the pressure in atmospheres using R = 0.0821 L·atm/mol·K.

Amount of Gas2.5 mol

Volume5 m^3

Temperature300 K

Gas Constant0.0821 J/molK

Solve for: $p$

Hint: Rearrange the formula to p = nRT / V.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

In car tyre pressure in winter, Ideal gas law is used to calculate Pressure from Volume, Amount of Gas, and Temperature. The result matters because it helps turn a changing quantity into a total amount such as area, distance, volume, work, or cost.

Study smarter

Tips

Always convert temperatures to Kelvin by adding 273.15 to the Celsius value.
Ensure the units for pressure and volume match the units of the universal gas constant R being used.
Remember that this law assumes gas particles have no volume and no attractive forces, which is an approximation of real behavior.

Avoid these traps

Common Mistakes

Using Celsius.
Using dm³ without checking R units.
Forgetting that temperature must be in Kelvin (add 273).
Using the wrong value of R for the units being used.

Keep going

Related Formulas

Common questions

Frequently Asked Questions

The ideal gas law links pressure, volume, temperature, and moles for gases behaving ideally.

Use this equation when analyzing the behavior of gases at relatively low pressures and high temperatures where molecules act independently. It is the primary tool for determining a missing physical property of a gas sample when the other state variables are defined.

This relationship is essential for chemical engineering, meteorology, and the design of pneumatic systems. It allows for the calculation of gas density and molar mass, which are critical for industrial safety and atmospheric research.

Using Celsius. Using dm³ without checking R units. Forgetting that temperature must be in Kelvin (add 273). Using the wrong value of R for the units being used.

In car tyre pressure in winter, Ideal gas law is used to calculate Pressure from Volume, Amount of Gas, and Temperature. The result matters because it helps turn a changing quantity into a total amount such as area, distance, volume, work, or cost.

Always convert temperatures to Kelvin by adding 273.15 to the Celsius value. Ensure the units for pressure and volume match the units of the universal gas constant R being used. Remember that this law assumes gas particles have no volume and no attractive forces, which is an approximation of real behavior.

References

Sources

Atkins' Physical Chemistry
Halliday, Resnick, and Walker, Fundamentals of Physics
Wikipedia: Ideal gas law
IUPAC Gold Book: Ideal gas
NIST CODATA 2018
Atkins' Physical Chemistry, 11th ed.
IUPAC Gold Book
Atkins' Physical Chemistry (e.g., Peter Atkins, Julio de Paula, James Keeler)

Overview

Variables

Derivation

State the Relationship:

Temperature Conversion:

Rearrangements

Graph

Intuition

Insight

Practice Problem

Real-World Context

Tips

Common Mistakes

Related Formulas

Gas Density

Osmotic pressure

Mole fraction

Frequently Asked Questions

Sources