The Ideal Gas Law
Use a stated ideal-gas model and controlled variables to relate pressure, volume, amount, and temperature.
AP Physics 2 Topic 9.2 · College Board CED printed p. 29 / PDF p. 36 · See this topic in the course outline
The ideal gas law is an equation of state: it relates the pressure, volume, temperature, and amount of gas in one equilibrium state. Before using a ratio, decide which quantities really stay fixed.
Before you start
- Kelvin temperature and absolute pressure.
- Proportional and inverse graphs; slope and extrapolation.
- A mole counts particles: Avogadro’s constant N_A = 6.02 × 10²³ mol⁻¹ is the number of particles per mole; N = nN_A.
What the ideal model assumes
Treat gas atoms as tiny particles with random instantaneous velocities. Their own volumes occupy a negligible fraction of the container, they collide elastically, and appreciable forces act only during brief collisions. Between collisions the atoms move freely. Random means many different instantaneous directions, not zero speed.
These assumptions remove sustained intermolecular attractions and exclude internal structure. A dilute gas far from condensation often approximates this model well. Crowding or strong interactions can make it fail. Elastic collisions conserve the pair’s kinetic energy while redistributing energy among atoms; they do not force equal speeds. State the ideal model before using its equation rather than assuming every real gas behaves exactly this way.
Counting atoms or moles in the same equation
relates equilibrium state variables. P is absolute pressure in Pa, V is volume in m³, n is amount in mol, N is the number of atoms, and T is kelvin. Avogadro’s constant gives the number of particles per mole: one mole contains approximately 6.02 × 10²³ atoms in this monatomic model. R = 8.31 J/(mol·K), N = nN_A, and R = N_Ak_B, so the atom and mole forms say the same thing. The product Pa·m³ has units J. Use the whole volume occupied by the gas, not the sum of the atomic volumes.
For a sealed sample, n stays fixed and . At fixed V, heating raises P because normal impulses and collision frequency rise. At fixed T, compression increases number density and raises P. At fixed P, heating needs a larger V. If gas leaks or is injected, n must remain in the comparison; there is no fixed-amount shortcut.
Use graph shape and intercept to test the gas model
Graphs reveal which gas-law relationship is being tested. With fixed amount, the following comparisons follow directly by solving PV = nRT for the plotted variable.
| Controls | Plot | Prediction and meaning |
|---|---|---|
| V fixed | P versus T in K | Straight line through origin; slope nR/V |
| P fixed | V versus T in K | Straight line through origin; slope nR/P |
| T fixed | P versus V | Decreasing hyperbola; PV is constant |
| T fixed | P versus 1/V | Straight line through origin; slope nRT |
For a constant-volume gas thermometer, a plot of P against Celsius temperature is also linear, but its horizontal intercept is near −273.15 °C. Extend the best-fit line to P = 0 to estimate this limiting temperature, equivalent to 0 K. The measured points need not include a zero-pressure point. Real gases may condense before reaching that intercept, so this is an extrapolation of the ideal model, not a proposed procedure for cooling a real sample to zero pressure. A nonzero intercept on a P-versus-kelvin or P-versus-1/V graph can flag offsets, leaks, dead volume, or changing temperature.
Use consistent units to find the amount of gas
A sealed ideal gas occupies 2.00 L at 1.20 × 10⁵ Pa and 300 K. Find n and N.
Choose the equilibrium gas sample; V = 2.00 × 10⁻³ m³.
Why this step? The SI gas constant requires cubic meters and pascals.
n = PV/(RT) = (1.20 × 10⁵)(2.00 × 10⁻³)/(8.31 × 300) = 0.0963 mol.
Why this step? Solve the equation of state for the amount rather than treating the gas law as an energy transfer equation.
N = nN_A = 0.0963(6.02 × 10²³) = 5.80 × 10²² atoms.
Why this step? Avogadro’s constant counts 6.02 × 10²³ particles per mole, so multiplying amount in mol by N_A converts the same sample to an atom count.
Answer and check
n = 0.0963 mol and N = 5.80 × 10²² atoms. Check: PV = 240 J = nRT.
Extrapolate a gas thermometer line
At fixed volume a gas gives P = 100 kPa at 0 °C and 120 kPa at 55 °C. Estimate the Celsius temperature for zero pressure.
Slope = (120 − 100)/(55 − 0) = 0.364 kPa/°C. Write P = 100 + 0.364T_C.
Why this step? Two measurements define a linear model; Celsius is appropriate for locating its offset intercept.
Set P = 0: T_C = −100/0.364 ≈ −275 °C.
Why this step? The horizontal intercept represents the extrapolated zero-pressure limit.
Compare with −273.15 °C and discuss measurement uncertainty and ideal-gas assumptions.
Why this step? An approximate measured intercept does not imply a new value of absolute zero or guarantee a real gas stays gaseous there.
Answer and check
The extrapolated intercept is approximately −275 °C, close to the ideal limit. This estimate is based on a best-fit model extended outside the measured range.
Make the model respond
Focused Physics 2 model
Ideal gas: pressure and molecular speed
Predict first. Predict how doubling temperature or volume changes pressure. Which change affects molecular speed?
PV = nRT; vᵣₘₛ = √(3RT/M); n = 0.500 mol of helium
- Selected T = 300 K
- Reference T = 300 K
- Selected or reference point
On a narrow screen, scroll within a figure to read its labels. The table and explanation provide the same physics in text.
At 300 K and 12 L, P = 103.9 kPa and vᵣₘₛ = 1,367 m/s.
Read the model data and comparisons
| Quantity | Value | Unit | Meaning or limit |
|---|---|---|---|
| Amount n | 0.5 | mol | Fixed; helium molar mass 0.004002602 kg/mol |
| Temperature | 300 | K | |
| Volume | 0.012 | m³ | |
| Pressure | 1.039 × 10^5 | Pa | |
| Rms speed | 1,367 | m/s | |
| Monatomic internal energy | 1,871 | J |
At fixed amount and temperature, larger volume lowers the collision rate per area and pressure varies inversely with volume. At fixed volume, pressure is proportional to absolute temperature. The rms speed depends on temperature and molecular mass, not volume. A warmer equilibrium speed distribution has a higher characteristic speed and greater width; the arrows here only illustrate a speed scale.
Assumptions and limits. Dilute ideal monatomic helium in thermal equilibrium; pointlike particles with negligible interaction energy except elastic collisions. n = 0.500 mol and M = 0.004002602 kg/mol stay fixed. Container and arrow sizes are schematic. The model does not calculate a Maxwell–Boltzmann distribution or nonideal-gas corrections.
Try it from memory
Write your answer and reason before opening the worked answer. Your self-assessment is a study decision; it is not an automatically graded score.
Question 1
A sealed gas doubles its volume while its kelvin temperature triples. What happens to its pressure?
Worked answer and reason
P₂/P₁ = (T₂/T₁)(V₁/V₂) = 3/2.
The number of atoms stays fixed, so PV/T is constant. Neither Boyle’s nor a constant-volume law alone applies.
Question 2
For one isothermal gas sample, what graph would turn its curved P-versus-V relationship into a line? What does the slope mean?
Worked answer and reason
Plot P versus 1/V. Its slope is nRT and its ideal intercept is zero.
Solving PV = nRT for P gives P = (nRT)(1/V). The stated controls make the slope constant.
Question 3
A gas becomes very dense and close to condensation. Which ideal assumptions become questionable?
Worked answer and reason
The atomic volume may no longer be negligible and attractions between collisions may matter.
The ideal gas description neglects finite particle size and sustained interactions; those approximations are less suitable in a crowded gas.
JavaScript enables saved drafts and self-assessment. You can always work on paper.
Catch the tempting mistake
- If the temperature doubles in degrees Celsius, pressure doubles.
- Gas-law ratios require kelvin. For example, 20 °C to 40 °C changes 293 K to 313 K, a much smaller ratio.
- PV is always constant.
- PV is constant only when both amount and temperature are fixed.
- Extrapolating a gas graph proves a real gas remains gaseous to absolute zero.
- Extrapolation describes the ideal model; real gases often liquefy before that limiting state.
Put it to work
Give the physical reason as well as the calculation. The worked solutions are on a separate page.
- Problem 3 · numerical state comparison; model assumptions
- Problem 4 · data graph; extrapolation; graph translation
- Problem 13 · quantitative PV cycle; signed energy table; entropy evaluation
Scientific calculator workspace
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Scientific calculator
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For the teacher
- Have students annotate fixed N, V, P, or T directly on each graph.
- Check use of absolute pressure if a device reports gauge pressure. Add ambient pressure before applying PV = nRT.
- Accept a zero-pressure intercept close to −273 °C from uncertain data, with an explanation of the ideal-model extrapolation.
Source objectives and scope
- 9.2.A — Describe the properties of an ideal gas.
Suggested skills are teaching guidance. The course applies all science practices across its objectives.
Science references for this teaching
- College Board AP Physics 2 Course and Exam Description, Unit 9 (PDF pages 34–42; printed pages 27–35) — All Unit 9 learning objectives, essential knowledge clauses, and AP scope boundaries; source equations checked in the supplied pinned PDF.
- OpenStax University Physics Volume 2, Molecular Model of an Ideal Gas — Supporting open-textbook reading for ideal-gas assumptions, absolute pressure, and state-variable graphs; real-gas extensions in the source are beyond this unit’s AP requirements.
- OpenStax University Physics Volume 2, First Law of Thermodynamics — Supporting open-textbook reading for energy accounting; authoring translates any work-by convention into the AP work-on convention.
- OpenStax University Physics Volume 2, Entropy — Optional conceptual supporting reading for qualitative entropy and reversibility; quantitative entropy calculations in the source are outside this AP treatment.
Related source reading
These alternative explanations and examples retain their source context. Draft placement does not add accepted exam coverage.
- ideal gas definition · alternative source explanation
Glossary and deeper reading · 12 refreshers
These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.
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Pressure
Pressure is the normal force per unit area exerted on a surface. It is a scalar measured in pascals, with one pascal equal to one newton per square meter. For example, a gas exerts pressure throughout its volume through many microscopic collisions, even when its bulk motion is zero.
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Temperature
Temperature is a state quantity that determines the direction of net energy transfer through thermal contact. For a classical ideal gas it is proportional to the average translational kinetic energy per particle, rather than to the total energy of the sample. For example, a small hot metal sample can have a higher temperature but less total internal energy than a large cooler sample.
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Kelvin temperature scale
The kelvin scale measures thermodynamic temperature from absolute zero. A temperature interval of one kelvin has the same size as one degree Celsius, but their zero points differ. For example, temperature ratios in the ideal gas law must use kelvin values rather than Celsius values.
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Absolute pressure
Absolute pressure is pressure measured relative to a perfect vacuum. It includes the atmospheric contribution that gauge pressure omits and is required in the ideal gas law. For example, a tire’s absolute pressure is its gauge reading plus the surrounding atmospheric pressure.
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Mole
A mole is an amount of substance containing Avogadro’s number of specified entities. Those entities must be named, such as atoms, molecules, or ions, and a mole is not itself a mass unit. For example, one mole of helium atoms and one mole of oxygen molecules contain the same number of specified entities but have different masses.
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Avogadro’s constant
Avogadro’s constant is the number of specified entities per mole, approximately 6.02 × 10²³ per mole. It converts between an amount in moles and a count of particles. For example, doubling the number of moles in a sample doubles its particle count when the entity being counted stays the same.
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Instantaneous velocity
Instantaneous velocity is the rate of change of position at a particular moment. On a position-versus-time graph, its component is the slope of the tangent at that time. For example, the velocity just before a ball lands differs from the average velocity of its entire flight.
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Elastic collision
An elastic collision conserves the total kinetic energy of the colliding system. Total momentum is also conserved when the system has negligible net external impulse, although each object’s momentum and energy can change. For example, ideal gas atoms redistribute their speeds in elastic collisions without changing the pair’s total kinetic energy.
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Kinetic energy
Kinetic energy is energy associated with motion relative to a chosen reference frame. For a nonrelativistic translating object it is one half of its mass times speed squared. For example, doubling a cart’s speed quadruples its translational kinetic energy when its mass stays fixed.
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State variable
A state variable is determined by a system’s present equilibrium state rather than by the route used to reach it. Internal energy and entropy are state properties, whereas heat and work describe transfers along a process. For example, two gas processes with identical endpoints have the same internal-energy change but can involve different amounts of work.
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Heat
Heat is energy transferred between systems because of a temperature difference. It describes a transfer rather than a substance or energy store contained in an object. For example, when a warmer mug transfers energy to a cooler hand, the mug loses internal energy and the hand gains it.
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Impulse
Impulse is the change in linear momentum produced by a force over a time interval. It equals the signed vector area under a force-versus-time graph and is measured in newton-seconds. For example, a padded landing spreads the stopping impulse over more time, reducing the average force for the same momentum change.