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Chemistry 2e · Gases

Relating Pressure, Volume, Amount, and Temperature: The Ideal Gas Law

9 min read
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Standard constants used (R = 0.08206 L·atm·mol⁻¹·K⁻¹, 0 °C = 273.15 K, STP molar volume 22.4 L/mol) are widely published reference values (NIST/IUPAC conventions); verify against current primary sources before formal citation.
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On this page 9 sections
  1. In 30 seconds
  2. Why this matters
  3. The college version
  4. Eli explains
  5. Worked example
  6. Key takeaway
  7. Check yourself
  8. Study tools
  9. Sources & references

In 30 seconds

Four properties describe any gas sample: pressure P, volume V, temperature T, and amount of substance n. The gas laws of the 1600s–1800s were discovered one pair at a time — hold two properties constant, change a third, and watch the fourth respond. Boyle found that squeezing a gas raises its pressure and shrinks its volume at constant temperature; Charles found that heating a gas expands it at constant pressure; Avogadro found that more gas means more volume at constant pressure and temperature. The combines all of these into a single equation:

PV = nRT

where R is the ideal gas constant. This equation describes an ideal gas — a simplified model in which gas particles have negligible volume and do not attract one another. Real gases (air, oxygen, helium, carbon dioxide) follow it closely at ordinary temperatures and pressures, making the ideal gas law one of the most useful relationships in all of chemistry.

Why this matters

  • Predicting and controlling systems: Any closed container of gas — a scuba tank, weather balloon, anesthesia circuit, car airbag, or industrial reactor — can be analyzed with the ideal gas law. Given any three of P, V, n, T, the fourth is calculable.
  • Healthcare: Oxygen tanks for patients are filled to high pressure so a small tank holds many moles of O₂. The pressure–volume–temperature relationships here explain why oxygen cylinders must be stored away from heat.
  • Weather and aviation: Weather balloons expand as they rise because external pressure drops; hot-air balloons lift because heating expands the air inside. Both are in action.
  • Diving safety: How dissolved gases behave in divers' blood (decompression sickness risk) depends on how gas pressure and volume change with depth.
  • Exams: Boyle's, Charles's, Avogadro's, and the combined gas laws, plus PV = nRT with unit conversions, are among the most tested concepts in general chemistry.

The college version

Core Concepts

Temperature must be in kelvin

Every gas law in this chapter requires absolute temperature in kelvin (K). The starts at absolute zero (0 K = −273.15 °C), where particle motion is minimal. Convert with:

T(K) = T(°C) + 273.15

Why kelvin? The gas laws are proportionalities — double the kelvin temperature and the volume doubles. That only works when the scale starts at true zero. Using Celsius would make "doubling" meaningless, and dividing by a negative Celsius temperature would produce nonsense. Convert to kelvin before substituting into any gas law.

Boyle's law: pressure and volume at constant n and T

Robert Boyle (1662) showed that at constant temperature and amount, pressure and volume are inversely proportional:

P1V1 = P2V2

Squeeze a gas into half its volume and the pressure doubles. This is why a syringe with a blocked tip is hard to push and why a balloon shrinks when you sit on it.

Charles's law: volume and temperature at constant n and P

Jacques Charles (1787) showed that at constant pressure and amount, volume is directly proportional to kelvin temperature:

V1T1 = V2T2

Heat a gas and its particles move faster, striking walls harder and more often; to keep pressure constant, the container must expand. This is why hot-air balloons rise and why a balloon left in a hot car expands.

Gay-Lussac's (Amontons') law: pressure and temperature at constant n and V

At constant volume and amount, pressure is directly proportional to kelvin temperature:

P1T1 = P2T2

A sealed soda can left in a hot car heats up; the gas above the liquid gains pressure, which is why cans can bulge or burst. Aerosol cans carry "do not incinerate / store below 120 °F" warnings for exactly this reason.

Avogadro's law: volume and amount at constant P and T

Equal volumes of gases at the same temperature and pressure contain equal numbers of particles:

V1n1 = V2n2

Blow more air into a balloon and it grows; release gas and it shrinks. This law bridges gas volume and moles, making gas stoichiometry possible (Topic 3).

The combined gas law

When several conditions change at once (amount fixed), the individual laws merge into one expression:

P1V1T1 = P2V2T2

This is the workhorse equation for problems where a gas sample changes conditions — a balloon rising, a tire warming, a syringe being compressed. Each individual law is just the combined law with one variable held constant.

The ideal gas law

Combining all four proportionalities gives:

PV = nRT

with n in moles and R the ideal gas constant. The value of R depends on the pressure and volume units:

Units of P and VValue of R
atm and L0.08206 L·atm/(mol·K)
kPa and L8.314 L·kPa/(mol·K)
mmHg (torr) and L62.36 L·mmHg/(mol·K)

At (273.15 K and 1 atm), one mole of any ideal gas occupies 22.4 L — the standard molar volume. The ideal gas law works best at low pressure and high temperature, where real gases behave most ideally (see Topic 6 for deviations).

Common Confusions

Do Not ConfuseWithDifference
Kelvin temperatureCelsius temperature in gas lawsGas laws require kelvin; 0 °C is 273.15 K, not "zero"
Boyle's law (inverse)Charles's law (direct)Boyle: P vs V at constant T (one up, other down). Charles: V vs T at constant P (both up or down)
R = 0.08206R = 62.36 or 8.3140.08206 pairs with atm·L; 62.36 with mmHg·L; 8.314 with kPa·L. Match units or answers are off by 760×
Volume change with T (Charles)Volume change with P (Boyle)Same variable (V) but different partner and opposite proportionality direction
STPRoom temperatureSTP is 273.15 K and 1 atm (molar volume 22.4 L); room conditions are roughly 298 K and 1 atm
Ideal gasReal gasIdeal gas assumes point particles with no attractions; real gases deviate at high P / low T (Topic 6)
n (moles)Massn is amount in moles; convert mass to moles with molar mass before using PV = nRT
Eli, the EliExplains learning guide

Eli explains

The same idea, in plain words

Explain it like I’m 10

Imagine a box of bouncy balls. If you make the box smaller without losing balls, the balls hit the walls more often — that's Boyle's law (smaller box, higher pressure). If you warm the balls so they bounce faster, they push the walls harder and the box stretches — that's Charles's law (hotter, bigger). If you add more balls, the box must grow to fit them — that's Avogadro's law. The ideal gas law is the one big rule tying it all together: more balls, smaller box, or hotter balls all mean more wall-pushing.

Worked example

Example 1: Charles's law — balloon in a freezer

A balloon holds 3.00 L of air at 25.0 °C at constant pressure. It is placed in a freezer at −10.0 °C. What is the new volume?

Step 1 — write the formula (constant P, constant n):

V1T1 = V2T2

Step 2 — convert temperatures to kelvin:

T1 = 25.0 + 273.15 = 298.15 K  ; T2 = -10.0 + 273.15 = 263.15 K

Step 3 — solve for V2 and substitute:

V2 = V1 × T2T1 = 3.00 L × 263.15 K298.15 K = 2.65 L

The volume shrank from 3.00 L to 2.65 L — sensible, because cooling slows the particles so the balloon contracts.

Example 2: Combined gas law — a balloon rising

A weather balloon contains 25.0 L of helium at 1.00 atm and 20.0 °C. It rises to where the pressure is 0.500 atm and the temperature is −30.0 °C. Find the new volume.

Step 1 — write the formula (amount fixed):

P1V1T1 = P2V2T2

Step 2 — convert to kelvin:

T1 = 20.0 + 273.15 = 293.15 K  ; T2 = -30.0 + 273.15 = 243.15 K

Step 3 — solve for V2 and substitute:

V2 = V1 × P1P2 × T2T1 = 25.0 L × 1.00 atm0.500 atm × 243.15 K293.15 K = 41.5 L

Check the direction: pressure halved (should double the volume), temperature dropped (should shrink it). Doubling then shrinking ~17% gives a net increase — 41.5 L from 25.0 L is reasonable.

Example 3: Ideal gas law — moles of gas in a tank

How many moles of nitrogen gas are in a 10.0 L cylinder at 25.0 °C and 15.0 atm?

Step 1 — write the formula:

PV = nRT

Step 2 — convert T to kelvin: T = 25.0 + 273.15 = 298.15 K.

Step 3 — solve for n and substitute (using R = 0.08206 L·atm/(mol·K) since P is in atm and V in L):

n = PVRT = (15.0 atm)(10.0 L)(0.08206 L·atm/(mol·K))(298.15 K) = 6.13 mol

Check units: atm, L, and K cancel, leaving mol. The cylinder holds about 6.1 mol of N₂ — roughly 172 g of gas.

Key takeaways

  • The ideal gas law is PV = nRT; use R = 0.08206 for atm·L units.
  • ALWAYS convert Celsius to kelvin (TK = T°C + 273.15) before using any gas law.
  • Boyle: P1V1 = P2V2 (inverse). Charles: V1/T1 = V2/T2 (direct, K only). Gay-Lussac: P1/T1 = P2/T2 (direct, K only). Avogadro: V1/n1 = V2/n2.
  • Combined gas law P1V1T1 = P2V2T2 handles any fixed-amount change-of-conditions problem.
  • At STP (273.15 K, 1 atm), 1 mol ideal gas = 22.4 L.
  • Write the formula first, substitute numbers with units, then compute — units must cancel to give the correct final unit.
  • The ideal gas law is a model: real gases deviate at high pressure and low temperature.

Check yourself

5 review questions from the chapter. Try each one, then open the answer.

  1. Write the ideal gas law and state what each symbol means.

    Show answer

    PV = nRT: P = pressure, V = volume, n = moles, T = kelvin temperature, R = ideal gas constant.

  2. A gas occupies 5.00 L at 300 K. At constant pressure and amount, what volume does it occupy at 600 K?

    Show answer

    V2 = V1(T2/T1) = 5.00 L × (600/300) = 10.0 L. Doubling kelvin temperature doubles volume (Charles's law).

  3. Why must temperature be in kelvin in all gas law calculations?

    Show answer

    The gas laws are proportionalities that only hold on a scale starting at absolute zero; doubling kelvin temperature doubles particle kinetic energy and volume, but doubling Celsius temperature has no such meaning.

  4. A 2.00 L flask contains 0.500 mol of gas at 27.0 °C. What is the pressure in atm?

    Show answer

    T = 27.0 + 273.15 = 300.15 K; P = nRT/V = (0.500)(0.08206)(300.15)/(2.00) = 6.16 atm.

  5. What is the volume of 1.00 mol of an ideal gas at STP?

    Show answer

    22.4 L (standard molar volume at STP: 273.15 K, 1 atm).

Keep learning

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Study tools & related lessonsKey vocabulary · Related

Key vocabulary

Ideal gas law
Equation PV = nRT relating pressure, volume, amount, and temperature
Ideal gas constant (R)
The proportionality constant in PV = nRT; value depends on units
Boyle's law
At constant T and n, pressure and volume are inversely proportional
Charles's law
At constant P and n, volume is directly proportional to kelvin temperature
Gay-Lussac's law
At constant V and n, pressure is directly proportional to kelvin temperature
Avogadro's law
At constant P and T, volume is proportional to moles of gas
STP
Standard temperature and pressure: 273.15 K and 1 atm
Kelvin scale
Absolute temperature scale starting at 0 K = −273.15 °C

Sources & references

  1. openstax.org — Chemistry 2e

This lesson was adapted from the open educational references above; their licenses and attributions are preserved. See Copyright & Licensing.

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