Have you tried to pop a balloon? If you step or sit on one, it will eventually pop with a loud "bang", scaring anyone close enough.
But why does it pop? Balloons pop because of the relationship between pressure and density (or more specifically, pressure and volume). In this article, we will learn about the relationship between pressure and density, so keep reading to learn the answer to our query!
- This article covers the topic of pressure and density
- First, we will define pressure and density
- Then, we will look at the relationship between pressure and density
- Next, we will look at the ideal gas law and how it relates to pressure and density
- Lastly, we will work on some examples related to pressure and density
Pressure Definition
Pressure is the force exerted by one substance onto another, divided by the area of the receiving substance. For a gas, pressure is the force exerted by a gas onto the walls of its container, divided by the area of the container.
An ideal gas is a hypothetical gas that approximates the behavior of "real gases". The properties of an ideal gas are:
- Negligible volume
- Negligible mass
- No interactions between particles
- Fully elastic collisions (no loss in kinetic energy)
- Particles are in constant motion
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 is a substance's mass per volume (m/V). For gases, we often use the number density, which would be the number of moles (n) per volume (n/V).
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.
When volume increases, the total density decreases, which is why there is the inverse relationship between pressure and volume
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.
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.
The ideal gas law is used to show the behaviors of ideal gases, and therefore approximate the behavior of real gases.
The formula is:
$$PV=nRT$$
Where P is pressure, V is volume, n is the number of moles, R is the gas constant, and T is temperature
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!
A 0.56 L balloon contains 1.35 mol of helium. If the same amount of helium was pumped into a 0.76 L balloon, which balloon would have the greater pressure?
Let's look at our relationship:
$$P \propto \frac{n}{V}$$In this case, volume (V) is increasing. Since pressure and volume have an inverse relationship, the 0.56 L balloon would be the one with the greater pressure
At 1 atmosphere and 0 °C, helium has a density of 0.179 g/L. If the pressure is raised to two atmospheres, what will happen to the density?
Let's take another look at our formula:
$$P \propto \frac{n}{V}$$
Since pressure is directly proportional to density, an increase in pressure means there will also be an increase in density
Let's do one more, shall we?
A 2.5 L container (container A) of hydrogen has a pressure of 1.35 atm. Another container (container B) of hydrogen is 3.2 L, with a pressure of 1.14 atm. Which container has more moles of hydrogen?
Let's rearrange our equation, so we can see this relationship better:
$$P \propto \frac{n}{V}$$
$$PV \propto n$$
So the product of pressure and volume is directly proportional to moles, meaning that whichever box has the larger product will have more moles of gas
$$PV \propto n$$
$$(1.35\,atm)(2.5\,L) \propto n$$
$$3.375\,atm*L \propto n$$
$$PV \propto n$$
$$(1.14\,atm)(3.2\,L) \propto n$$
$$3.648\,atm*L \propto n$$
Since container B has a greater pressure-volume product, it will have more moles of hydrogen
Pressure and Density - Key takeaways
- Pressure is the force exerted by one substance onto another, divided by the area of the receiving substance. For a gas, pressure is the force exerted by a gas onto the walls of its container, divided by the area of the container.
- Density is a substance's mass per volume (m/V). For gases, we often use the number density, which would be the number of moles (n) per volume (n/V).
- 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.
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