Charles’ law is an empirical law, that is, based on experimental observations, which establishes the relationship between the volume and the temperature of a gas when the pressure and mass or number of moles are constant. The first to enunciate it was the French physicist Jacques Alexandre César Charles, at the end of the 18th century. According to this law, the volume of a fixed sample of a gas held at constant pressure is directly proportional to the absolute temperature . In other words:

This law states that if a gas is doubled in absolute temperature, its volume will also double. In fact, if the temperature is multiplied by any factor, the volume will also be multiplied by the same factor, as long as the amount of gas and its pressure are kept constant.

## Charles’s law in equation form

Like any law of proportionality, the above relationship can be rewritten in the form of an equation simply by introducing a suitable constant of proportionality. That is to say:

where K is a constant of proportionality that depends on the amount of gas and its pressure.

As can be seen, this equation has the form of an increasing linear function with slope K. It is observed experimentally that this slope increases with the number of moles of the gas and decreases with the pressure. In addition, all the lines that are built at different values of P and n, when extrapolated to a volume of zero, intersect the temperature axis at -273.15 °C, which corresponds to absolute zero. This behavior is shown below:

## Changes of state and Charles’s law

Charles’ law can be rearranged by dividing both sides of the equation by the temperature, in which case the right-hand side will just be the constant of proportionality:

In other words, Charles’s law predicts that if the pressure and the number of moles are held constant, the relationship between the volume and the absolute temperature will remain constant. This means that if we carry out a process in which a gas changes from an initial to a final state in an isobaric manner (at P = constant), the relationship between the initial volume and the temperature will be equal to the relationship between the volume and the final temperature, that is:

This equation can be used to determine both the volume and the initial or final temperature, when the other three variables are already known.

## Examples of the application of Charles’s law

Below are two examples of typical gas-related problems that can be solved using Charles’s law.

### Example 1: Doubling the volume

*Determine the final temperature of an ideal gas that is initially at 25°C and that is heated until its volume increases to twice its initial value.*

#### Solution

The data provided by the problem is:

T i = 25 °C

V f = 2. V i

The first thing we must do is transform the temperature to Kelvin, since Charles’s law relates volume to absolute temperature and the centigrade scale is a relative scale.

We can now apply Charles’s law to determine the final temperature. We don’t need to know the values of the volumes, just the relationship between them.

Therefore, the final temperature will be 596.30 K or 323.15 °C.

### Example 2: Lowering the temperature by half

*If a helium sample was originally at -130.15°C, cooled to -180.15°C at constant pressure, and its final volume turned out to be 10.0 L, what was the initial volume?*

#### Solution

In this case, we have the following data:

T i = -130.15 °C

T f = -180.15 °C

V f = 10.0L

As before, we must start by determining the absolute temperatures, and then apply Charles’s law.

Now we can apply Charles’s law:

The helium sample must have started from an initial volume of 15.38 L.

## Charles’ law constant of proportionality and the ideal gas law

The ideal gas law represents an equation of state that completely describes an ideal gas when we know three of four state functions, namely pressure, temperature, volume, or number of moles. The equation is given by:

where R is the universal ideal gas constant, P is the pressure of the gas, and all other variables are the same as in Charles’ law. This equation can be rewritten as:

This law applies to ideal gases under any set of conditions, including those in which Charles’s law applies. Therefore, in the case that the pressure and the number of moles are kept constant, the above expression must be equivalent to Charles’s law. By comparison, we can see that the Charles’ law constant of proportionality is then equal to the factor in parentheses:

As can be seen, this expression for the constant of proportionality agrees with the experimental observation that it remains constant when n and P are constant; increases as n increases and decreases as P increases.

## References

Britannica, The Editors of Encyclopaedia. (2020, February 18). *Charles’s law | Definition & Facts* . Encyclopedia Britannica. https://www.britannica.com/science/Charless-law

Britannica, The Editors of Encyclopaedia. (2021, November 8). *Jacques-Charles | French physicist* . Encyclopedia Britannica. https://www.britannica.com/biography/Jacques-Charles

Chang, R. (2021). *Chemistry *( 11th ed.). MCGRAW HILL EDUCATION.

*Gas Laws* . (nd). Chem.FSU. https://www.chem.fsu.edu/chemlab/chm1045/gas_laws.html

Libretexts. (2020, August 22). *Gas Laws: Overview* . Chemistry LibreTexts. https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Supplemental_Modules_(Physical_and_Theoretical_Chemistry)/Physical_Properties_of_Matter/States_of_Matter/Properties_of_Gases/Gas_Laws/Gas_Laws%3A_Overview

Libretexts. (2021, April 30). *14.4: Charles’s Law* . Chemistry LibreTexts. https://chem.libretexts.org/Bookshelves/Introductory_Chemistry/Book%3A_Introductory_Chemistry_(CK-12)/14%3A_The_Behavior_of_Gases/14.04%3A_Charles’s_Law