# Relationship between Pressure and Temperature

When thinking of the effects of heat on aerosol cans and inflated tires, it is clear that pressure and temperature are somehow related. This relationship is fairly straightforward and is most well-known, in its simplest form, as Gay-Lussac’s Law. This simple formulation makes intuitive sense and provides a framework for understanding how temperature can affect everyday objects with pressurized air. Understanding this property of gas has many applications, especially in heating and cooling (like in the production of air conditioning units) and in meteorology (understanding the formation of clouds).

Simply put, the pressure of a given system is proportional to its temperature. If you increase the temperature, while holding the volume constant and not allowing any molecules to escape, the pressure will increase. This makes sense on a very simple level. By raising the temperature, the gas molecules in the system have greater kinetic energy (essentially, they move faster). Pressure is the result of collisions between molecules and their container, so with greater kinetic energy the system will see more collisions.

Gay-Lussac’s law is written mathematically below. P is the pressure, T is the temperature, and k is the gas constant that relates the units for temperature and pressure.

Gay-Lussac’s Law: P = k x T

This relationship can be further expanded by considering a comparison between two systems under different conditions. This is known as the Law of Pressure-Temperature. It holds true because pressure and temperature are proportional, so the ratio of pressure to temperature should be identical between two systems that have the same volume and number of molecules. This pressure law was discovered by Amontons and forms the basis for modern thermometers, which utilize changes in pressure caused by temperature to change their volume.

Law of Pressure-Temperature: P1/T1 = P2/T2

So, for a system with really high pressure, the temperature will also be high. An identical system, with low pressure will have a low temperature. The ratios for these two systems will be identical and the law will hold.

It is worth noting that for all of these systems, the volume must be held constant. This is because the volume of the system is linked to the pressure and temperature as well (this is demonstrated through the Ideal Gas Law and the Combined Gas Law). When the temperature of a system increases, the pressure increases. To accommodate the increased pressure, the system volume will increase. This is why heating aerosol cans can be so dangerous. They are held at a fixed pressure, so increasing the temperature without allowing the volume to increase causes an intense buildup of pressure until it bursts.

Resources

Gay-Lussac at Illinois at Urbana-Champaign

Pressure and Temperature: University of Wisconsin at Madison

Gas Laws: Georgia State University