How do you fix a dented ping-pong ball?
Drop a dented ping-pong ball into hot water and it slowly pops back out. Bike tires pumped up hard can burst in the summer sun. Block the tip of a syringe and push the plunger, and it gets harder and harder to push. Describing a fixed amount of gas takes three quantities: pressure P, volume V, and temperature T. Hold one of them constant, and the other two have a simple relationship. Those relationships are the three gas laws in this lesson.
Constant temperature: Boyle’s law
For a fixed amount of gas at constant temperature, pressure is inversely proportional to volume. This is Boyle’s law玻意耳定律:
Halve the volume, and twice as many particles are packed into each liter, so they hit the walls twice as often and the pressure doubles.
Constant pressure: Charles’s law
For a fixed amount of gas at constant pressure, volume is directly proportional to the absolute (kelvin) temperature. This is Charles’s law盖-吕萨克定律（人教版物理）:
Warmer particles move faster and hit harder. To keep the pressure the same, the gas must expand so the hits are spread out. A graph of V against Celsius temperature t is a straight line that, extended, crosses the axis at −273.15 °C, where the volume would be zero. That temperature is absolute zero绝对零度. The Kelvin scale热力学温标 starts there:
The 273.15 is exact. Celsius temperatures in problems are usually given to the ones place (such as 25 °C), and then adding 273 is enough: 25 + 273 = 298 K, the same as adding 273.15 and rounding to the ones place. (The AP equation sheet writes K = °C + 273.15.) Add 273.15 only when the temperature has decimals: 25.0 °C is 298.2 K. Every example in this chapter follows this rule.
(Real gases liquefy long before they get that cold, and absolute zero itself cannot be reached.)
Constant volume: Gay-Lussac’s law
For a fixed amount of gas at constant volume, pressure is directly proportional to the kelvin temperature. This is Gay-Lussac’s law查理定律（人教版物理）:
With the volume fixed, hotter particles hit the walls more often and harder, so the pressure rises. That is why tire pressure goes up in hot weather.
Two sets of names, swapped
In China these laws are taught in high-school physics, not chemistry. Boyle’s law has the same name in both countries, but the other two names are swapped:
| Held constant | Relationship | US chemistry | Chinese physics textbook |
|---|---|---|---|
| T | PV constant | Boyle’s law | 玻意耳定律 (Boyle) |
| P | V/T constant | Charles’s law | 盖-吕萨克定律 (Gay-Lussac) |
| V | P/T constant | Gay-Lussac’s law | 查理定律 (Charles) |
Both Charles and Gay-Lussac studied both relationships, and the two traditions settled on different names. Remembering the relationships is safer than remembering names: check which quantity is held constant, and you know which relationship applies. Chinese physics books also write pressure as a lowercase p.
Pick a mode, move the slider, record a few points, and look at the shape of the graph:
🔧 Math tool card · Slope & Linear RelationshipsSkip it if you already know this
Direct proportion: a straight line through the origin. The graph of y = kx is a line through the origin with slope k = y/x. At constant pressure, V = kT, so a V–T graph looks like this.
Inverse proportion: straighten the curve. The graph of y = k/x is a curve that is hard to judge by eye. Plot y against 1/x instead, and you get a straight line through the origin. For Boyle’s law, P against V is a curve, but P against 1/V is a line.
Extrapolation. Volume and Celsius temperature are related by V = k(t + 273.15), a straight line that doesn’t pass through the origin. Extend the line to V = 0, and it crosses the axis at t = −273.15 °C. Drawing a line through data and extending it beyond the measurements is called extrapolation.
A V–t graph (t in °C) for a gas at constant pressure is extended until it reaches the temperature axis. Where does it cross?
V = k(t + 273.15), so V = 0 at t = −273.15 °C, absolute zero.
Example 1: pushing a syringe
A syringe holds 50.0 mL of air at 1.00 atm. At constant temperature, it is compressed to 20.0 mL. What is the new pressure?
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What is constant? Temperature, and the amount of gas (the syringe is sealed), so use Boyle’s law, P₁V₁ = P₂V₂.
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List the knowns. P₁ = 1.00 atm, V₁ = 50.0 mL, V₂ = 20.0 mL; find P₂.
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Rearrange and substitute.
The volume units cancel, so there is no need to convert to liters.
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Check. The volume became 2/5 as large, so the pressure should become 5/2 as large: 1.00 × 2.5 = 2.50. It matches.
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A common mistake. Treating P and V as directly proportional: P₂ = 1.00 × 20.0 ÷ 50.0 = 0.400 atm. Squeezing a gas can’t lower its pressure; the plunger gets harder to push.
Example 2: a balloon in liquid nitrogen
A balloon has a volume of 2.00 L at 25 °C. It is placed in liquid nitrogen at −196 °C at constant pressure. What is its new volume?
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What is constant? Pressure and the amount of gas, so use Charles’s law, V₁/T₁ = V₂/T₂.
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Convert to kelvins.
The Celsius temperatures are given to the ones place, so adding 273 is enough (see above).
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Rearrange and substitute.
77 K has only two significant figures, so the answer has two.
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Check. The Kelvin temperature drops to 77/298 ≈ 1/4 of its value, so the volume shrinks to about 1/4: 2.00 L ÷ 4 = 0.50 L, close to 0.52 L.
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Why kelvins? Using Celsius gives 2.00 × (−196) ÷ 25 = −15.7 L, a negative volume, which is impossible.
- Constant T, volume halved: the pressure
P is inversely proportional to V, so halving V doubles P.
- Constant P, kelvin temperature doubled: the volume
V is directly proportional to T.
- Constant V, kelvin temperature halved: the pressure
P is directly proportional to T.
- Constant T, pressure quadrupled: the volume
V is inversely proportional to P.
- Constant P, temperature raised from 27 °C to 327 °C: the volume
300 K to 600 K doubles the kelvin temperature, so the volume doubles, not 327 ÷ 27 ≈ 12 times.
- Constant V, temperature raised from −73 °C to 127 °C: the pressure
200 K to 400 K, so the pressure doubles.
- Constant T, volume doubled: the pressure
P is inversely proportional to V.
- Constant T, volume cut to 1/4: the pressure
V becomes 1/4 as large, so P becomes 4 times as large.
Common mistakes
Mistake 1: “Going from 10 °C to 20 °C doubles the volume.” Gas laws need kelvin temperatures. 283 K to 293 K is an increase of only about 3.5%, so the volume grows by only about 3.5%.
Mistake 2: “P is inversely proportional to V, so the P–V graph is a straight line.” An inverse proportion graphs as a curve. P is directly proportional to 1/V, so the P–1/V graph is the straight line.
Mistake 3: “Charles’s law is P ∝ T.” In US textbooks, Charles’s law is V ∝ T at constant pressure, and P ∝ T at constant volume is Gay-Lussac’s law. (Chinese physics textbooks use the opposite names, so check which quantity is held constant.)
Quick check
A fixed amount of gas at constant pressure is heated from 27 °C to 127 °C. Its volume becomes how many times as large?
T₁ = 300 K and T₂ = 400 K, so V₂/V₁ = T₂/T₁ = 400/300 = 4/3.
A fixed amount of gas at constant temperature is compressed from 6.0 L to 2.0 L. Its pressure becomes how many times as large?
P₁V₁ = P₂V₂, so P₂/P₁ = V₁/V₂ = 6.0/2.0 = 3.
Summary
- For a fixed amount of gas: P₁V₁ = P₂V₂ at constant T (Boyle), V₁/T₁ = V₂/T₂ at constant P (Charles), and P₁/T₁ = P₂/T₂ at constant V (Gay-Lussac).
- Always use kelvins: T = t + 273.15 K. Extrapolating a V–t graph to V = 0 gives absolute zero, −273.15 °C.
- Chinese physics textbooks swap the names: V ∝ T is 盖-吕萨克定律 and P ∝ T is 查理定律.
On to AP (for later)
AP Chemistry Unit 3 treats these laws as special cases of the ideal gas law PV = nRT (2.8.4). You won’t need the names, but you will need to read relationships from graphs and explain them with particle collisions, for example: “At constant volume, raising the temperature increases the average kinetic energy of the particles, so they hit the walls more often and with more force, and the pressure rises.”