Gas Pressure and Temperature

Meaning of gas pressure and temperature

We first need to know the meaning of the pressure and temperature of a gas. For this, we look at a gas as a bunch of particles flying around randomly inside a closed container of constant volume. See the animation below for the setup. Once in a while, the particles of the gas will hit the container wall and bounce back inwards, and this bounce exerts a force outwards on the container, perpendicular to the walls. This way, every unit area of the container experiences the same force from the bouncing gas particles.

We see that pressure is measured in units of $\mathrm{N}/{\mathrm{m}}^2$, which we call pascal. The symbol of this unit is $\mathrm{Pa}$.

The particles in a gas all have roughly the same speed (if they did not, bouncing off of each other would quickly result in a redistribution of the kinetic energy of the particles, and therefore distributed velocities), so there is a certain amount of average kinetic energy per gas particle. This energy per particle defines the temperature of the gas: the faster the particles move, the higher the temperature of the gas.

Relationship between gas pressure and temperature

From the definitions and the setup of the gas in a closed container, we see that an increase in gas temperature, and therefore an increase in the average speed of the gas particles, will lead to more frequent (if the volume of the closed gas container remains constant) and more violent bounces against the walls of the container. In other words, heating up a gas in a container of constant volume increases the number of collisions between particles and the walls of the container, and it increases the average force of the collisions with the container. This means that the total force per unit area of the container walls increases, which, by definition, means that this increases the pressure of the gas.

We can also visualise this increase in pressure using the animation above. If the speed of the particles increases, then we intuitively see that the particles exert a larger force on the container walls. Thus, the relationship between the temperature and the pressure of a gas - given a constant container volume - is positive: an increase in gas temperature means an increase in gas pressure. Inversely, given a constant container volume, an increase in gas pressure means an increase in gas temperature, because the only way for the pressure to increase is for the particles to move faster, which means that the temperature increases.

Gas pressure and temperature graph

As the relationship between gas pressure and temperature is positive, the pressure-temperature graph corresponding to a gas consists of an increasing curve. As it turns out, the graph will actually be a straight line (because of the linearity we mentioned in the deep dive above), but you do not need to know this. Below, we see the graph corresponding to a typical gas you may find on Earth.

Graph of the pressure as a function of the temperature of a typical gas - StudySmarter Originals

We see that around the freezing point of water $\left(273\mathrm{K}\right)$, the pressure of this gas is approximately $100\mathrm{kPa}$. In practice, this is the pressure of the air around us: atmospheric pressure is approximately $100\mathrm{kPa}$. We also see that at a temperature of $0\mathrm{K}$, the pressure is $0\mathrm{Pa}$: this is logical because particles don't move at a temperature of absolute zero, so they don't bounce off of any walls either.

Gas pressure and temperature experiments

Examples of using the gas pressure-temperature relationship

Suppose we have an empty closed container of constant volume: it only contains air, and this air has a certain pressure