**Charles’s law** or the **law of volumes** is an ideal gas law that states that the volume and temperature of a fixed amount of gas are proportional at constant pressure. Doubling the temperature of a gas doubles its volume. Halving the temperature of a gas halves its volume. The law takes its name from French scientist Jacques Charles, who formulated the law in the 1780s.

Charles’s law states that increasing the temperature of a gas at constant pressure increases its volume.

### Charles’s Law Formula

There are a few ways to state Charles law as a formula:

V ∝ T

V/T = k

V = kT

V_{1}/T_{1} = V_{2}/T_{2}

V_{2}/V_{1} = T_{2}/T_{1}

V_{1}T_{2} = V_{2}T_{1}

Here, T is absolute temperature, V is volume, and k is a non-zero constant. Note that absolute temperature means Celsius and Fahrenheit temperature must be converted to Kelvin. The graph of volume versus pressure shows the linear relationship. Also, the line points toward the origin, although a gas could never reach it because it would change into a liquid or solid first.

### Examples of Charles’s Law in Everyday Life

It’s easy to find examples of Charles’s law in everyday life.

- Hot air balloons fly based on Charles’s law. Heating the air in the balloon increases the balloon’s volume. This decreases its density, so the balloon rises in the air. To come down, chilling the air (not-heating-it) allows the balloon to deflate. The gas becomes more dense and the balloon sinks.
- If you take a filled balloon outside on a hot day, it expands (and may pop!). If you take it outdoors on a winter day, it deflates but returns to its normal volume when you take it indoors again. You can even use a balloon as a poor sort of thermometer, using Charles’s law.

### Charles’s Law Example Calculation

#### Example #1

A gas occupies 221 cm^{3} at a temperature of 0 °C and pressure of 760 mm Hg. Find its volume at 100 °C.

First, don’t worry about the pressure. The number doesn’t enter into the calculation. All that matters is that it’s a constant.

Use the equation:

V_{1}/T_{1} = V_{2}/T_{2}

Convert 0 °C and 100 °C to Kelvin:

V_{1} = 221cm^{3}; T_{1} = 273K (0 + 273); T_{2} = 373K (100 + 273)

Plug the values into the equation and solve for V_{2}:

V_{1}/T_{1} = V_{2}/T_{2}

221cm^{3} / 273K = V_{2 }/ 373K

V_{2 } = (221 cm^{3})(373K) / 273K

V_{2 } = 302 cm^{3}

#### Example #2

Find the final temperature of a sample of nitrogen gas at constant pressure if it starts at 27 °C and changes volume from 600 mL to 700 mL.

First convert the temperature to Kelvin.

T_{1} = 273 + 27

T_{1} = 300 K

Next, plug in the numbers.

V_{1}/T_{1} = V_{2}/T_{2}

600 mL/300 K = 700 mL/T_{2}

(T_{2})(600 mL/300 K) = 700 mL

T_{2} = (700 mL)/(600 mL/300 K)

T_{2} = (700 mL)/(2mL/K)

T_{2} = 350 K

### Why Temperature Must Be in Kelvin

Charles’s law calculations require temperature on an absolute scale, such as the Kelvin scale. So, using the formula requires converting from Celsius or Fahrenheit to Kelvin. There are two reasons for this. First, the negative temperatures on the Celsius and Fahrenheit scales could lead to impossible negative volume calculations. Second, the energy doesn’t scale properly using relative scales. So, a gas at 20 K has twice the energy of a gas at 10K, but the same is not true of as gas at 20 °C compared to 10 °C or 20 °F compared to 10 °F.

### What Happens at Absolute Zero?

Like the other ideal gas laws, Charles’s law doesn’t apply under extreme conditions. It doesn’t make sense at absolute zero. First, matter can’t have zero volume. Second, a gas at constant pressure eventually changes into a liquid or solid as temperature drops.

### References

- Fullick, P. (1994).
*Physics*. Heinemann. ISBN 978-0-435-57078-1. - Gay-Lussac, J. L. (1802). “Recherches sur la dilatation des gaz et des vapeurs” [Research on the expansion of gases and vapors].
*Annales de Chimie*. 43: 137–75. - Krönig, A. (1856). “Grundzüge einer Theorie der Gase“.
*Annalen der Physik*. 99 (10): 315–22. doi:10.1002/andp.18561751008