Pressure and Density

Pressure Definition

From now on, when we mention gases, we are referring to ideal gases.

So, how does the gas exert this pressure? The answer is collisions. Ideal gases are in constant motion and can move in every direction. Because of this, they are bound to collide with each other and the walls of their container. When a gas particle collides with the container, it exerts a force onto it, then bounces off. The greater the number of collisions, the greater the pressure.

Below is a diagram of this process:

Fig.1-Gas particles collide with the container to create pressure

The gas particle particles with the lines coming off them are colliding with the container and are about to bounce back off. The other particles in the container are also moving and can also collide with the container at a later time

Density Definition

Now, let's look at the definition of density.

Density answers the question, "How much of this substance is contained within this volume?"

Below is an example of what density looks like for gases:

Fig.2-When the volume is the same, more moles=higher density

For gases, the formula for density is:

$$\frac{n}{V}$$

Where n is the number of moles and V is volume

Since the volume for each sample is the same, the container with more gas particles (right) has a greater density.

Relationship Between Pressure and Density

Pressure and density have a direct relationship, meaning that if one increases, so does the other.

To put it mathematically:

$$P \propto \frac{n}{V}$$

Where P is pressure, n is the number of moles, V is volume, and ∝ is the symbol for "proportional to"

This also means that pressure is directly proportional to the number of moles, but inversely proportional (one goes up, the other goes down) to volume, since volume is in the denominator.

So, why is this? Well, let's think back to our definitions. Pressure is based on two things: the number of collisions (i.e. more collisions equals greater force) and the area of the container.

If the volume is stable, and the number of moles increases (net increase in density), the number of collisions is also going to increase.

Fig.3-An increase in density due to an increase in moles causes an increase in pressure

Basically, the particles have less room to move freely, which increases the likelihood of a collision.

Now let's talk about what happens when volume is changed (number of moles is stable). If the volume is decreases, not only will the area decrease, but the number of collisions will increase as well.

Fig.4-When volume decreases (density increase), the pressure increases

Pressure is force/area, so a decrease in volume leads to an increase in pressure (more collisions due to less space) and a decrease in the area.

In our introduction, I talked about popping balloons. The reason why the balloon pops is because of this relationship. When you step/stand on the balloon, you are decreasing the volume, so the pressure must increase. When the pressure gets too much for the balloon to withstand, it bursts.

Ideal Gas Law

The relationship between pressure and density is shown by the Ideal Gas Law.

Let's rearrange this formula, so it clearly shows the relationship between pressure and density

$$PV=nRT$$

$$P=\frac{nRT}{V}$$

$$P=\frac{n}{V}*\frac{RT}{V}=Density\cdot \frac{RT}{V}$$

As you can see, the ideal gas law shows that pressure and density (n/V) are directly proportional.

Pressure and Density Examples

Now that we understand the relationship between pressure and density, let's work on some examples!

Let's do one more, shall we?