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Ideal Gas Law

Use the ideal gas law and partial pressures with fixed-variable labels and compatible units.

AP Chemistry Topic 3.4 · CED printed p. 63 / PDF p. 70. This teaching, practice and feedback were added for the AP edition; original author material remains separately traceable.

Before you start
  • Moles and unit cancellation
  • Ratios and inverse dependence

Keep the units and fixed variables visible

The ideal-gas law PV=nRTPV=nRT connects pressure, gas volume, amount and absolute temperature. Kelvin temperature is essential: 0 °C is 273.15 K, not zero thermal motion. Choose an R consistent with pressure and volume units, such as 0.08206 L atm mol−1 K−1 or 8.314 J mol−1 K−1 with SI pressure and volume. Convert before substituting.

For a fixed amount, P ∝ T at fixed V, V ∝ T at fixed P, and P ∝ 1/V at fixed T. State those conditions in both a calculation and a graph. P versus V is curved for a fixed-temperature ideal sample; P versus 1/V is linear. P versus Kelvin T at fixed n,V extrapolates to the origin in the mathematical ideal model. It is not a guarantee that a real gas can remain gaseous all the way to 0 K.

Each ideal component contributes pressure

For an ideal mixture, every component experiences the same T and V and contributes Pi = niRT/V. Sum the contributions to get total pressure. The mole fraction Xi=nintotalX_i=\frac{n_i}{n_{\mathrm{total}}} therefore gives Pi=XiPtotalP_i=X_iP_{\mathrm{total}}. Fractions are based on moles, not grams. Changing another component’s amount at fixed T,V raises total pressure but does not change the first component’s partial pressure.

If gas is collected over water and the water-vapor pressure is given, subtract that partial pressure from total pressure to obtain the dry gas pressure. Do not subtract an unrelated atmospheric value or a temperature. Pressure measured from a gas mixture includes every gas-phase contributor under the stated model.

Let's work through it

Work it through

A 2.00 L container at 300. K holds 0.0800 mol N2 and 0.0200 mol O2. Use R = 0.08206 L atm mol−1 K−1.

  1. Total gas amount = 0.1000 mol. Ptotal = (0.1000)(0.08206)(300.)/2.00 = 1.2309 atm → 1.23 atm.
  2. XN2 = 0.800 and XO2 = 0.200. Thus partial pressures are 0.985 atm and 0.246 atm.
  3. If N2 is added at fixed T,V, O2 partial pressure remains nO2RT/V = 0.246 atm even though its mole fraction decreases.
  4. The fixed-volume P-versus-Kelvin-T graph is linear for this fixed total amount. Its slope is nR/V = 0.00410 atm K−1.

Make the picture move

Interactive concept model

Connect pressure to particles hitting the walls.

Predict first. Hold two quantities fixed, then change the third. Predict pressure before reading the output. Does doubling temperature double characteristic speed?

PV = nRT; at fixed identity, characteristic speed ∝ √T

Connect pressure to particles hitting the walls.An inverse pressure–volume curve at selected amount and temperature, with a dot marking the selected volume. At 0.5 mol, 300 K, and 15 L, the ideal pressure is 0.821 atm. At fixed n and T, pressure varies inversely with volume. The accessible data table and explanation give the same relationships.511.317.523.83000.6771.352.032.71Selected VPressure (atm)Volume (L)
  • P(V), selected n and T
  • Labeled selected or reference state
An inverse pressure–volume curve at selected amount and temperature, with a dot marking the selected volume. At 0.5 mol, 300 K, and 15 L, the ideal pressure is 0.821 atm. At fixed n and T, pressure varies inversely with volume.
Gas particles and pressure schematicA still schematic box contains representative particles and direction arrows. Changing volume changes the box width; changing amount changes the representative count; changing temperature changes arrow length.0.5 mol · 15 L · 300 KArrow length is schematic: speed scale = 1.00 relative to 300 K.
A still schematic box contains representative particles and direction arrows. Changing volume changes the box width; changing amount changes the representative count; changing temperature changes arrow length.

On a small screen, scroll inside the graph or diagram to read its labels. The values and explanation also appear below.

Selected pressure0.821 atm
PV/(nT)0.08206 L atm mol⁻¹ K⁻¹
Characteristic speed / speed at 300 K1.00

At 0.5 mol, 300 K, and 15 L, the ideal pressure is 0.821 atm. At fixed n and T, pressure varies inversely with volume.

Read the data and comparisons
Gas particles and pressure: model values and comparisons (numbers rounded for display)
Volume (L)Pressure (atm)n (mol)T (K)
52.460.5300
101.230.5300
150.8210.5300
200.6150.5300
300.410.5300

Macroscopic / data. Pressure 0.821 atm is the computed macroscopic outcome.

Particle view. Arrows show a characteristic speed scale, not identical speeds in a real gas. The diagram uses representative counts rather than individual molecules in a mole.

Symbolic view. P = (0.5 mol)(0.08206 L atm mol⁻¹ K⁻¹)(300 K)/(15 L).

Why it works. At the same temperature, reducing volume increases the rate of wall collisions. Increasing the amount increases particles per volume. Increasing temperature raises average translational kinetic energy and the typical momentum transferred in a collision. The gas law uses absolute temperature.

Conditions and limits. Ideal gas: negligible particle volume and intermolecular attractions, equilibrium temperature, fixed gas identity. R = 0.08206 L atm mol⁻¹ K⁻¹. Dots are representative particles; box width qualitatively tracks volume without a quantitative geometric scale. This still picture does not simulate motion, collisions, or a velocity distribution. Real gases depart from ideal behavior, especially at high pressure or low temperature.

Try it from memory

Pause & retrieve

Try it before you peek.

Write your answer and the reason from memory. Then compare with the worked answer. Your self-assessment is a study decision, not an automatically graded score.

Question 1 of 30 self-assessed

Question 1

Why does doubling temperature mean doubling Kelvin temperature rather than Celsius temperature?

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See every recall question with its answer

These references preserve each prompt and its answer for direct evidence links. Try the interactive questions above before opening the worked reasoning.

Retrieval 1

Why does doubling temperature mean doubling Kelvin temperature rather than Celsius temperature?

Worked answer and reason

The ideal law uses absolute thermal temperature; Celsius has an arbitrary offset.

Proportionality only applies to the absolute scale.

Retrieval 2

At fixed T,V, does adding He change the partial pressure of existing Ne in an ideal mixture?

Worked answer and reason

No, provided Ne amount stays fixed. Its partial pressure remains nNeRT/V.

Other components change total pressure and mole fraction but not that component’s independent contribution.

Retrieval 3

What plot makes fixed-n,T pressure behavior linear?

Worked answer and reason

P versus 1/V, with slope nRT.

The inverse dependence is linear in reciprocal volume, not volume.

Catch the tempting mistake

Use total solution-like liquid volume instead of gas container volume.
Gas V is the available sample/container volume under the model.
Mole fraction is mass of component divided by total mass.
It uses component moles divided by total moles.
Adding another gas lowers existing gas partial pressure because its fraction falls.
At fixed T,V and original amount, total pressure rises to compensate the smaller fraction.

Put it to work

Give the chemical reason as well as the calculation. Worked solutions are on a separate page.

Scientific calculator workspace

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Scientific calculator

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For the teacher
  • Require fixed-variable labels on graphs.
  • After calculating fractions, ask what happens when another gas is added at fixed volume.

Unit planning and formative guidance →

Source objectives and scope

Source edition: ap-chemistry-ced-fall2024-local-b5dfe867. See this topic in the course outline →

3.4.A
Explain the relationship between the macroscopic properties of a sample of gas or mixture of gases using the ideal gas law.
3.4.A.1
The macroscopic properties of ideal gases are related through the ideal gas law: EQN: PV = nRT.
3.4.A.2
In a sample containing a mixture of ideal gases, the pressure exerted by each component (the partial pressure) is independent of the other components. Therefore, the partial pressure of a gas within the mixture is proportional to its mole fraction (X), and the total pressure of the sample is the sum of the partial pressures. EQN: PA = Ptotal × XA, where XA = moles A/total moles; EQN: Ptotal = PA + PB + PC + …
3.4.A.3
Graphical representations of the relationships between P, V, T, and n are useful to describe gas behavior.
Science references for this teaching
Glossary and deeper reading · 5 refreshers

These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.

  • Mole

    A mole is an amount of substance containing exactly 6.02214076 × 10²³ specified entities. The entities can be atoms, molecules, ions, or formula units, so the counted entity must be named. For example, one mole of oxygen molecules contains two moles of oxygen atoms.

    Read more: Moles and Molar Mass →

  • Real-gas behavior

    Real-gas behavior includes effects of particle attractions and particle volume that the ideal gas law neglects. Those effects compete, so actual pressure can fall below or rise above the ideal prediction under different conditions. For example, attraction can lower pressure while excluded particle volume can raise it under strong compression.

    Read more: Deviation from Ideal Gas Law →

  • Mixture

    A mixture contains two or more substances whose proportions can vary. Its components retain their chemical identities until a chemical reaction changes them. For example, water and ethanol can form mixtures with many different compositions.

    Read more: Composition of Mixtures →

  • Mole fraction

    Mole fraction is a component's amount in moles divided by the mixture's total amount in moles. The fractions of all components sum to one. For example, one mole of one gas mixed with three moles of another gives the first gas a mole fraction of 0.25.

  • Partial pressure

    A gas's partial pressure is the pressure it would exert alone in the mixture's volume at the same temperature. In an ideal mixture, the component partial pressures add to the total pressure. For example, a component with mole fraction 0.20 contributes 0.20 of the total ideal pressure.

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