#### Learning Objective

- State Charles’ Law and its underlying assumptions

#### Key Points

- The lower the pressure of a gas, the greater its volume (Boyle’s Law); at low pressures, [latex]\frac{V}{273}[/latex] will have a larger value.
- Charles’ and Gay-Lussac’s Law can be expressed algebraically as [latex]\frac{\Delta V}{\Delta T} = constant[/latex] or [latex]\frac{V_1}{T_1} = \frac{V_2}{T_2}[/latex] .

#### Terms

- absolute zerothe theoretical lowest possible temperature; by international agreement, absolute zero is defined as 0 K on the Kelvin scale and as −273.15° on the Celsius scale
- Charles’ lawat constant pressure, the volume of a given mass of an ideal gas increases or decreases by the same factor as its temperature on the absolute temperature scale (i.e. gas expands as temperature increases)

## Charles’ and Guy-Lussac’s Law

Charles’ Law describes the relationship between the volume and temperature of a gas. The law was ﬁrst published by Joseph Louis Gay-Lussac in 1802, but he referenced unpublished work by Jacques Charles from around 1787. This law states that at constant pressure, the volume of a given mass of an ideal gas increases or decreases by the same factor as its temperature (in Kelvin); in other words, temperature and volume are directly proportional. Stated mathematically, this relationship is:

[latex]\frac {V_1}{T_1}=\frac{V_2}{T_2}[/latex]

## Example

- A car tire filled with air has a volume of 100 L at 10°C. What will the expanded volume of the tire be after driving the car has raised the temperature of the tire to 40°C?
- [latex]\frac {V_1}{T_1}=\frac{V_2}{T_2}[/latex]
- [latex]\frac {\text{100 L}}{\text{283 K}}=\frac{V_2}{\text{313 K}}[/latex]
- [latex]V_2=\text{110 L}[/latex]

## V vs. T Plot and Charles’ Law

A visual expression of Charles’ and Gay-Lussac’s Law is shown in a graph of the volume of one mole of an ideal gas as a function of its temperature at various constant pressures. The plots show that the ratio [latex]\frac{V}{T}[/latex] (and thus [latex]\frac{\Delta V}{\Delta T}[/latex]) is a constant at any given pressure. Therefore, the law can be expressed algebraically as [latex]\frac{\Delta V}{\Delta T} = \text{constant}[/latex] or [latex]\frac{V_1}{T_1} = \frac{V_2}{T_2}[/latex].

## Extrapolation to Zero Volume

If a gas contracts by 1/273 of its volume for each degree of cooling, it should contract to zero volume at a temperature of –273°C; this is the lowest possible temperature in the universe, known as absolute zero. This extrapolation of Charles’ Law was the first evidence of the significance of this temperature.

## Why Do the Plots for Different Pressures Have Different Slopes?

The lower a gas’ pressure, the greater its volume (Boyle’s Law), so at low pressures, the fraction \frac{V}{273} will have a larger value; therefore, the gas must “contract faster” to reach zero volume when its starting volume is larger.