## 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.

The (perpendicular) force per unit area that bouncing gas particles exert on a wall is called the **pressure** of the gas.

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

When we write out units in full, we always write them in all lower case, even if they are named after people. For example, the unit pascal is named after the physicist Pascal. Symbols of units *can* have capital letters, of which there are many examples, including $\mathrm{Pa}$.

If the particles in a gas exert a (perpendicular) force of $4\mathrm{N}$on a $0.1{\mathrm{m}}^{2}$area of a wall, then the pressure of that gas is $40\mathrm{Pa}$.

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.

For a closed container of constant volume (i.e. a constant volume and a constant number of particles in the gas), the gas pressure is a constant multiplied by the temperature. In other words, the relationship between gas pressure and temperature is linear. In practice, this means the following.

If, in a closed container of constant volume, the temperature doubles, then the pressure also doubles. If the temperature drops, then the pressure also drops, by the same factor.

## 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.

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

Take any empty closed container of constant volume and heat it up, for example by using fire. The temperature of the air inside the container will rise, so the pressure of the air will increase as well. At some air temperature, the force that the air exerts on the container through its pressure becomes too big for the container to stay in one piece: the container explodes. Obviously, this experiment is dangerous to perform in practice, but it is a nice and simple thought experiment.

This is an experiment you *can* do at home. Take a plastic water (or soft drink) bottle, empty it such that mainly air is inside, and close it off with the cap. Now put the bottle in your freezer and wait for about 5 minutes. The bottle probably has dents in it. These are caused by the decrease in pressure of the air inside the bottle because of the cooling! Taking the bottle out of the freezer will warm the air inside it up again, and the bottle will un-dent itself in front of you. This un-denting visualises the increase in air pressure inside the bottle, which is a direct result of the reheating of that air.

## 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

If we heat up the container, the air will be heated as well, so the pressure of the air inside the container will increase. On the other hand, if we put the container in the freezer, the gas temperature will decrease, so the gas pressure will decrease.

If we cannot measure the temperature of the container directly, but we are able to measure the pressure of the air inside the container, we can still say things about the temperature of the air in the container. Namely, if the pressure is increasing, we know that the temperature must be increasing, and if the pressure is decreasing, we know that the temperature must be decreasing.

## Gas Pressure and Temperature - Key takeaways

- When thinking about gases, visualise particles flying around randomly. If there is a wall, they bounce off it, and this bouncing exerts a force on the wall.
- The pressure of a gas is the force per unit area that it would exert on a (potentially purely hypothetical) wall.
- The temperature of a gas is dictated only by the average energy of the particles in the gas.
- For a gas in a closed container of constant volume, an increase in temperature always goes together with an increase in pressure, and vice versa.
- The graph of pressure versus temperature is an increasing line.
- Heating up a closed container of constant volume will make the gas pressure increase, which will at some point make the container explode.
- We can say things about the temperature of a gas by measuring its pressure, and vice versa.

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##### Frequently Asked Questions about Gas Pressure and Temperature

How does temperature affect gas pressure?

Temperature affects gas pressure as follows: for an ideal gas in a constant volume and particle number, the pressure is linear in temperature. This means that a percentage change in temperature causes the same percentage change in pressure.

How does temperature change with pressure?

Temperature changes with pressure as follows: for an ideal gas in a constant volume and particle number, the temperature is linear in pressure. This means that a percentage change in pressure causes the same percentage change in temperature.

What causes gas pressure and temperature to change?

Causes of gas pressure and temperature changes can be a change in the particle density of the gas (either by shrinking its volume or by adding particles), or manual heating of the gas.

What happens to gas when the temperature increases?

An increase in temperature of a gas will either lower its density (if the pressure is being held constant, e.g. in the atmosphere), or increase its pressure (if its density is being held constant, e.g. in a closed container).

What is the equation and formula for calculating gas pressure and temperature?

The equation/formula for calculating gas pressure from temperature and vice versa is the ideal gas law: *pV=NkT*. If you know the particle density of a gas, you have a direct (linear) relationship between gas pressure and temperature.

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