Chemistry: Atoms First 2e · Gases

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

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In 30 seconds
Why this matters
The college version
Eli explains
Worked example
Key takeaway
Check yourself
Study tools
Sources & references

In 30 seconds

Four properties describe a gas sample: pressure P, volume V, temperature T, and amount n (in moles). Centuries of experiments showed that these four are not independent — changing one forces changes in the others. The empirical gas laws each hold two of them fixed and describe how the other two respond. Combined, they collapse into a single powerful equation, the :

$PV = nRT$

where R is the ideal gas constant. This topic builds each empirical law (Boyle's, Charles's, Avogadro's, and Gay-Lussac's), shows how they merge into the , and then introduces the ideal gas law with its constant and its standard reference conditions. The ideal gas law is the workhorse of this chapter: nearly every gas calculation — from weather balloons to scuba tanks to gas stoichiometry — flows from it.

Why this matters

Predicting behavior: the ideal gas law lets you compute any one of P, V, n, or T if you know the other three — no experiment required.
Real-world systems: aerosol cans, car airbags, weather balloons, scuba tanks, and anesthesia gas mixtures are all designed using these relationships.
Medical and safety context: knowing how pressure rises with temperature () explains why aerosol cans warn "do not incinerate," and why tires can over-pressure on hot roads.
Foundation for later topics: gas stoichiometry, effusion/diffusion, and kinetic-molecular theory all build on PV = nRT.
Exams: ideal-gas-law rearrangements and combined-gas-law problems are among the most frequently tested calculations in general chemistry.

The college version

Core Concepts

Boyle's law: pressure and volume (constant n, T)

Robert Boyle found that at fixed temperature and amount of gas, pressure and volume are inversely proportional: squeeze a gas into half its volume and its pressure doubles.

$P 1 V 1 = P 2 V 2$

This is why a sealed syringe is hard to push once the plunger is nearly home, and why a rising weather balloon expands as outside pressure falls.

Charles's law: volume and temperature (constant n, P)

At fixed pressure and amount, gas volume is directly proportional to its absolute temperature ():

$V 1 T 1 = V 2 T 2$

Doubling the kelvin temperature doubles the volume. Temperature must always be in kelvin (K = ^°C + 273.15) because the law describes proportionality to absolute temperature — dividing by a Celsius number gives nonsense.

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

At fixed volume and amount, gas pressure is directly proportional to absolute temperature:

$P 1 T 1 = P 2 T 2$

Heating a sealed rigid container raises the pressure — the mechanism behind the aerosol-can warning and the pressure buildup in a closed pressure cooker.

Avogadro's law: volume and amount (constant P, T)

At fixed pressure and temperature, gas volume is directly proportional to the number of moles:

$V 1 n 1 = V 2 n 2$

Equal volumes of gases at the same temperature and pressure contain equal numbers of molecules — the idea that lets chemists use gas volumes as mole counts in reactions.

The combined gas law

If n is constant but P, V, and T all change, the first three laws combine into:

$P 1 V 1 T 1 = P 2 V 2 T 2$

You can "derive" any of the individual laws by holding one quantity constant and canceling it.

The ideal gas law and the gas constant

Merging all four proportionalities (with n included) gives:

$PV = nRT$

The constant R has different numerical values depending on the pressure and volume units:

$R = 0.08206 L atm mol -1 K -1 = 8.314 J mol -1 K -1 = 62.36 L torr mol -1 K -1$

Choose the R whose units match your problem's pressure unit, and always use kelvin for temperature. Standard temperature and pressure () is defined as exactly 273.15 K (0 °C) and 1 atm; at STP, one mole of an ideal gas occupies 22.4 L (the ).

The "ideal" assumption

The ideal gas law treats gas molecules as point particles with no attractions and no volume. Real gases deviate at high pressure and low temperature; the law remains an excellent approximation for most gases near room temperature and 1 atm. (Non-ideal behavior is covered later in this chapter.)

Common Confusions

Do Not Confuse	With	Difference
Celsius temperature	Kelvin temperature	Gas laws require kelvin; K = ^°C + 273.15. Using °C directly in Charles's or Gay-Lussac's law gives wrong answers
Boyle's law	Charles's law	Boyle: P vs V at constant T (inverse); Charles: V vs T at constant P (direct)
Gay-Lussac's law	Charles's law	Gay-Lussac: P vs T at constant V; Charles: V vs T at constant P — both need kelvin
0 °C in kelvin	273 K	0 °C = 273.15 K, not 273; use 273.15 for calculations requiring three or more significant figures
R = 0.08206	R = 8.314	Same constant, different units: 0.08206 L atm mol⁻¹ K⁻¹ vs 8.314 J mol⁻¹ K⁻¹; pick the one matching your units
Volume at STP (22.4 L/mol)	Volume at any conditions	22.4 L/mol applies only at 273.15 K and 1 atm; otherwise compute with the ideal gas law
Combined gas law	Ideal gas law	Combined law fixes n and compares two states; ideal gas law relates all four variables in a single state

Eli explains

The same idea, in plain words

Explain it like I’m 10

Think of a bouncy castle full of kids. Squeeze the castle smaller and the kids bounce harder against the walls — that's Boyle's law. Warm the kids up so they run faster and they push harder — that's Gay-Lussac's law. Put more kids in and the castle stretches bigger — that's Avogadro's law. The ideal gas law is one big recipe that puts all three ideas together: more kids, faster kids, or a smaller castle all mean harder pushing.

Worked example

Example 1: Boyle's law — compressing a syringe

A syringe contains 40.0 mL of air at 1.00 atm. The plunger is pushed to compress the air to 25.0 mL while the temperature stays constant. What is the new pressure?

Step 1 — Write Boyle's law:

$P 1 V 1 = P 2 V 2$

Step 2 — Solve for P₂ and substitute:

$P 2 = P 1 V 1 V 2 = (1.00 atm)(40.0 mL) 25.0 mL = 1.60 atm$

Dimensional check: mL cancels, leaving atm. Sense check: volume decreased, so pressure increased — 1.60 atm > 1.00 atm. ✓

Example 2: Combined gas law — a rising weather balloon

A balloon holds 5.00 L of helium at ground level, where P = 1.00 atm and T = 27.0 ^°C. It rises to an altitude where P = 0.500 atm and T = -23.0 ^°C. What is its new volume? (Assume no gas escapes.)

Step 1 — Convert temperatures to kelvin:

$T 1 = 27.0 + 273.15 = 300.2 K, T 2 = -23.0 + 273.15 = 250.2 K$

Step 2 — Write the combined gas law and solve for V₂:

$P 1 V 1 T 1 = P 2 V 2 T 2 \Rightarrow V 2 = V 1 \times P 1 P 2 \times T 2 T 1$

Step 3 — Substitute:

$V 2 = 5.00 L \times 1.00 atm 0.500 atm \times 250.2 K 300.2 K = 8.34 L$

Sense check: lower pressure expands the balloon (factor of 2), but colder temperature contracts it (factor ~0.83); net result is growth to 8.34 L, consistent with balloons swelling at altitude.

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

A scuba tank has a volume of 11.0 L and is filled to a pressure of 204 atm at 25.0 °C. How many moles of air does it contain?

Step 1 — Convert temperature to kelvin:

$T = 25.0 + 273.15 = 298.2 K$

Step 2 — Write the ideal gas law and solve for n:

$PV = nRT \Rightarrow n = PV RT$

Step 3 — Substitute with R = 0.08206 L atm mol^-1K^-1 (P in atm):

$n = (204 atm)(11.0 L) (0.08206 L atm mol -1 K -1)(298.2 K) = 91.8 mol$

Dimensional check: atm and L cancel, K cancels, leaving mol. A tank of ~92 mol of air at 204 atm is consistent with the ~3,000 psi fill pressure of scuba tanks.

Example 4: Molar volume at STP

Confirm that one mole of an ideal gas occupies 22.4 L at STP.

Step 1 — Write the ideal gas law solved for V:

$V = nRT P$

Step 2 — Substitute STP values (T = 273.15 K, P = 1.00 atm, n = 1.00 mol):

$V = (1.00 mol)(0.08206 L atm mol -1 K -1)(273.15 K) 1.00 atm = 22.4 L$

This is the origin of the 22.4 L/mol molar volume used throughout gas stoichiometry.

Key takeaways

Ideal gas law: PV = nRT; R = 0.08206 L atm mol^-1K^-1 when P is in atm.
Temperature must be in kelvin in every gas-law equation: K = ^°C + 273.15.
Boyle's law (PV = const), Charles's law (V/T = const), Gay-Lussac's law (P/T = const), Avogadro's law (V/n = const).
Combined gas law: P₁V₁/T₁ = P₂V₂/T₂ — use it when n is fixed and two or more variables change.
STP = 273.15 K and 1 atm; molar volume of an ideal gas at STP = 22.4 L/mol.
To solve, rearrange PV = nRT for the unknown, substitute values with units, and check that units cancel.
Pick the R value matching the problem's pressure unit (0.08206 for atm, 62.36 for torr/mmHg).

Check yourself

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

Write the ideal gas law and give the value of R when pressure is in atm and volume in liters.
Show answer
PV = nRT; R = 0.08206 L atm mol^-1K^-1.
A gas occupies 3.00 L at 1.00 atm. If the temperature and moles are constant, what volume does it occupy at 3.00 atm?
Show answer
Boyle's law: V₂ = (1.00 × 3.00)/3.00 = 1.00 L. Pressure tripled, so volume fell to one third.
Convert 25.0 °C to kelvin. Why is this conversion mandatory in gas-law problems?
Show answer
25.0 + 273.15 = 298.2 K. Gas laws are proportional to absolute temperature; °C values are relative to an arbitrary zero and give incorrect ratios.
A rigid 2.00 L canister of gas at 300 K is heated to 450 K. If volume and moles are constant, by what factor does the pressure increase?
Show answer
Gay-Lussac's law: P₂/P₁ = T₂/T₁ = 450/300 = 1.50 — pressure increases by 50%.
How many moles of gas are in a 22.4 L container at STP? What gas law relationship makes this a quick answer?
Show answer
1.00 mol. At STP the molar volume is 22.4 L/mol, so 22.4 L holds exactly 1 mol (Avogadro's law / ideal gas law).
Which value of R should you use if a problem gives pressure in torr, and what is it?
Show answer
R = 62.36 L torr mol^-1K^-1, so torr cancels when pressure is in torr.

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Study tools & related lessonsKey vocabulary · Related

Key vocabulary

Ideal gas law: PV = nRT, linking pressure, volume, moles, and kelvin temperature
Ideal gas constant R: The proportionality constant whose value depends on pressure/volume units
Boyle's law: At constant n and T, P and V are inversely proportional
Charles's law: At constant n and P, V is proportional to kelvin T
Gay-Lussac's law: At constant n and V, P is proportional to kelvin T
Avogadro's law: At constant P and T, V is proportional to n
Combined gas law: P₁V₁/T₁ = P₂V₂/T₂ for fixed n
STP: Standard temperature and pressure: 273.15 K and 1 atm
Molar volume: Volume occupied by one mole of gas (22.4 L at STP)
Kelvin: Absolute temperature scale starting at -273.15 ^°C

Sources & references

openstax.org — Chemistry Atoms First 2e

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

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