Charles's law: volume and temperature of gases

The balloon rises: volume follows temperature

A hot-air balloon is Charles's law made visible. The burner roars, the air inside the envelope warms, and the balloon swells and lifts. Nothing about the outside pressure changed; the heat alone did the work.

The rule is a direct proportion:

$$\frac{V_1}{T_1} = \frac{V_2}{T_2}$$

At constant pressure, volume divided by absolute temperature is constant. Raise the kelvin temperature by 10 percent and the volume grows by 10 percent. The temperature must be in kelvins, because the proportion only works on a scale that starts at zero energy.

A worked example: a balloon's envelope holds 2,200 m^3 of air at $15^\circ\text{C}$ (288 K). The burner heats the air to $100^\circ\text{C}$ (373 K) at constant pressure. The new volume the same air would occupy is $V_2 = V_1T_2/T_1 = 2200 \times 373/288 \approx 2,850$ m^3. The envelope cannot stretch that far, so the extra 650 m^3 worth of air escapes out the mouth, leaving fewer, lighter contents behind, which is exactly why the balloon becomes buoyant.

Why warming expands a gas: the particle view

At the particle level, Charles's law is a story about pushing back. Heat the gas and its particles move faster and strike the walls harder. In a flexible container like a balloon, the walls yield: they move outward until the collisions spread over enough new area that the pressure falls back to match the outside air.

Equilibrium returns when the collision rate per unit area is restored. Faster particles need more wall area to keep the same pressure, so the volume grows in step with the temperature. Cool the gas and the reverse happens: slower particles need less wall, and the outside atmosphere squeezes the container smaller.

A worked example: a flexible weather balloon holds 100 L of helium at $20^\circ\text{C}$ (293 K) at ground-level pressure. It rises into air at $-40^\circ\text{C}$ (233 K) where the outside pressure is much lower, but consider temperature alone first: $V_2 = 100 \times 233/293 \approx 79.5$ L. Cold alone would shrink it by a fifth. In reality the falling outside pressure expands it far more, which is why weather balloons swell enormously before they burst.

Solving V1/T1 = V2/T2 problems

Charles's-law problems are routine once the kelvin habit is automatic:

1. Convert every temperature to kelvins first: $K = ^\circ\text{C} + 273.15$.
2. Check units: the two volumes must match each other; the temperatures are both in K.
3. Solve for the unknown: $V_2 = V_1T_2/T_1$ or $T_2 = T_1V_2/V_1$.
4. Sanity-check the direction: warmer means bigger, cooler means smaller.

A worked example: a gas occupies 3.00 L at $25^\circ\text{C}$ and constant pressure. It is cooled until its volume is 2.40 L. Find the final temperature: $T_1 = 298$ K, so $T_2 = T_1V_2/V_1 = 298 \times 2.40/3.00 = 238$ K, which is $238 - 273 = -35^\circ\text{C}$. The volume fell to 80 percent, so the kelvin temperature fell to 80 percent too, a consistency check that catches most arithmetic slips.

The straight line that revealed absolute zero

Jacques Charles never published his 1780s measurements; Joseph Gay-Lussac did, and the straight line he plotted pointed somewhere remarkable. Volume versus temperature graphs as a straight line, and extending that line backward hits zero volume at $-273.15^\circ\text{C}$.

No real gas reaches zero volume, they all liquefy first, but the intercept revealed something profound: a temperature scale with a true zero, the point of minimum particle motion. Lord Kelvin built the absolute scale on this insight, and every gas calculation since has used it.

A worked example of extrapolation: a student measures 100 mL of air at $20^\circ\text{C}$ and 93 mL at $0^\circ\text{C}$. The volume fell 7 mL over 20 degrees, a slope of 0.35 mL/°C. Extending down from 93 mL at $0^\circ\text{C}$: $93 \div 0.35 \approx 266$ degrees below zero, landing near $-266^\circ\text{C}$, within a few percent of the true $-273.15^\circ\text{C}$. Two rough measurements point at absolute zero, which shows how tightly volume and temperature are linked.

Ballooning, engines, and heat limits

Charles's law governs anything where heated gas is free to expand. Hot-air balloons are the flagship: pilots control altitude by heating or venting, trading temperature for lift. Engine cooling systems include expansion tanks because hot coolant and trapped air need room to grow.

It also sets limits. A spray can left in a hot car does not just face rising pressure; any flexible seal or pouch inside expands too. And in the laboratory, gas thermometers exploit the law directly: measure a fixed-pressure gas volume and you have measured the temperature.

A worked example with a safety edge: a sealed flexible pouch holds 500 mL of air at $25^\circ\text{C}$ (298 K) and is rated to 600 mL before bursting. Left on a dashboard at $70^\circ\text{C}$ (343 K), the air tries to reach $V_2 = 500 \times 343/298 \approx 575$ mL. It survives, but only just; at $85^\circ\text{C}$ (358 K) it would reach 601 mL and fail. Thermal expansion leaves thin margins, which is why heat warnings print maximum temperatures, not just cautions.

Common mistakes

Real-world applications

• Operating hot-air balloons by heating the envelope air

• Calibrating thermometers from gas expansion

• Designing engine cooling systems that vent expanding gases

• Predicting weather balloon burst altitudes