Charles’s Law: Volume and Temperature
Charles’s law states that for a fixed amount of gas at constant pressure, the volume is directly proportional to the absolute temperature in kelvins: V1/T1 = V2/T2. In brief: a gas that occupies 2.00 L at 300 K will occupy 2.20 L at 330 K if the pressure and the amount of gas do not change.
Charles’s law explains why a hot-air balloon rises, why a basketball left in the cold looks deflated and why the kelvin scale has an absolute zero. This article gives the formula, the worked method, a table of data and the history of the law, together with the most common errors.
What does Charles’s law say?
When a gas is heated at constant pressure, its volume increases in direct proportion to the absolute temperature:
V / T = k, or V1/T1 = V2/T2
Doubling the kelvin temperature doubles the volume. The law is valid only for a fixed number of moles of gas at constant pressure. In molecular terms, hotter particles move faster and hit the walls harder, so the walls (for example, a balloon or a piston) must move outward to keep the pressure from changing.
Why must temperature be in kelvins?
The volume of a gas is proportional to the kelvin temperature, not to the Celsius temperature. If volume is plotted against temperature in °C, the result is a straight line that does not pass through the origin. Extrapolated to zero volume, the line crosses the temperature axis at about −273.15 °C. This point is absolute zero, the lowest temperature possible, and is defined as 0 K.
Real gases condense to liquids before reaching absolute zero, so the volume never actually becomes zero. The extrapolation still shows that the kelvin scale is the natural scale for gas calculations. To convert, use K = °C + 273.15; the value 273 is accurate enough for most school problems.
How do you solve Charles’s law problems?
- Write the known values and label them as initial (1) and final (2).
- Convert all temperatures to kelvins.
- Rearrange to find the unknown: V2 = V1 × T2 / T1.
- Substitute and calculate.
- Check: a higher temperature must give a larger volume.
Worked example 1: heating a gas
A balloon has a volume of 4.50 L at 20 °C. What is its volume at 80 °C if the pressure does not change?
T1 = 20 + 273 = 293 K and T2 = 80 + 273 = 353 K. V2 = 4.50 L × 353 K / 293 K = 5.42 L.
Worked example 2: cooling a gas
A 250 mL sample of nitrogen at 25 °C is cooled in an ice bath to 0 °C. Find the new volume.
T1 = 298 K and T2 = 273 K. V2 = 250 mL × 273 / 298 = 229 mL. The sample shrinks by about 8 %.
Worked example 3: finding the temperature
A gas occupies 1.20 L at 300 K. At what temperature will it occupy 1.50 L?
T2 = T1 × V2 / V1 = 300 K × 1.50 / 1.20 = 375 K, or 102 °C.
Charles’s law data table
| Temperature (°C) | Temperature (K) | Volume (L) | V/T (L/K) |
|---|---|---|---|
| −173 | 100 | 1.00 | 0.0100 |
| −73 | 200 | 2.00 | 0.0100 |
| 27 | 300 | 3.00 | 0.0100 |
| 127 | 400 | 4.00 | 0.0100 |
| 227 | 500 | 5.00 | 0.0100 |
| 327 | 600 | 6.00 | 0.0100 |
Each doubling of the kelvin temperature doubles the volume (100 K to 200 K, and 200 K to 400 K), but the same temperature steps in °C do not.
Who discovered Charles’s law?
The French physicist Jacques Charles studied the thermal expansion of gases around 1787 and found that oxygen, nitrogen, hydrogen and carbon dioxide expand by about the same amount when heated by the same number of degrees. He did not publish his results. Joseph Louis Gay-Lussac published detailed measurements in 1802 and credited Charles, which is why the law carries Charles’s name, although in France it is often called the Gay-Lussac law. Jacques Charles also made the first manned hydrogen balloon flight in 1783, an early application of the same physics.
Where does Charles’s law apply?
- Hot-air balloons: heating the air inside makes it expand, which lowers its density below that of the surrounding air.
- Tires and balls: in cold weather the air contracts, although in a closed rigid container the pressure effect (Gay-Lussac) dominates.
- Baking: gases in dough expand in the oven, helping bread to rise.
- Cryogenics: a balloon dipped in liquid nitrogen shrinks sharply, and recovers when it warms.
- Engines: hot combustion gases expand and push the piston.
Which mistakes are common?
- Using Celsius: this is the most common error. Always convert to kelvins before dividing.
- Forgetting the conditions: pressure and the amount of gas must be constant.
- Putting temperatures in the wrong order: T2 goes with V2, and T1 with V1.
- Expecting the volume to reach zero: real gases liquefy first, so the proportionality only holds above the condensation point.
- Mixing volume units: the ratio is unit-free only when both volumes use the same unit.