You may know that you aren't supposed to put an aerosol can in a fire because it could explode, but do you know why? In this lesson, we will explain Gay-Lussac's law, which shows the relationship between the temperature and pressure of a gas.
Temperature and Pressure Relationship
Johnny Dalton is enjoying his day at the beach on Ideal Island, the place where all gases behave ideally. What makes them behave so ideally? Ideal gas particles move rapidly and randomly, they don't lose energy when they collide, and they do not have any intermolecular forces. This means that when two ideal gas particles get near each other, they are not attracted to each other like real gas particles may be.
As he lays his towel out and gets his sunblock out, he notices a warning on the back of the aerosol can: Contents under pressure; do not heat. Why would heating this aerosol can be a cause for warning? Johnny thinks for a minute and then recalls some of the ideas of kinetic molecular theory. He remembers that as the temperature of a gas increases, the gas particles move faster and faster. After all, temperature is really just a measure of the average kinetic energy of the particles. Also, when gas particles are moving quickly, they hit the insides of their container more frequently. Finally, because pressure is just a measure of the force of the particles hitting the inside of the container, the more times it hits the inside of the container, the higher the pressure will be!
As heat rises, particles in a gas move faster, increasing pressure.
Now, the container that is holding these particles may not be able to withstand those high amounts of pressure. If the pressure inside of the container exceeds the limits of the container, an explosion could occur.
If you live in cold climates, you may notice that the pressure in your car's tires decreases as it gets colder. This is because the air particles inside the tires are colder and moving slower. They don't hit the inside walls of the tires as frequently, and the pressure in your car's tires decreases.
This relationship between temperature and pressure is known as Gay-Lussac's law. It states that if the volume of a container is held constant as the temperature of a gas increases, the pressure inside the container will also increase. As with the other gas laws, this one can be represented in the form of an equation:
P1/T1 = P2/T2
Recall that we use 1s and 2s to indicate the quantities before (1s) and after (2s) a change has taken place. Also, note that the units for pressure do not matter, as long as they are the same throughout the entire equation. The units for temperature must be Kelvins or the equation will not work. This is because the Kelvin scale is an absolute scale - it doesn't go negative. Finally, this equation only works for an ideal gas. Most gases that surround you and me behave very much like ideal gases, so we can use this equation as an approximation for the gases we encounter.
Use this equation for calculating the relationship between pressure and temperature.
Practice Question 1
Let's say you are making dinner using a pressure cooker with an initial internal pressure of 1 atmosphere and an internal temperature of 100 degrees Celsius (or 373 K). Assuming the volume does not change and the gas behaves ideally, what will the pressure inside the container be if the temperature is raised to 200 degrees Celsius (or 473 K)?
To solve this, you will need to make sure you are using the Kelvin temperatures. Substituting the numbers into the correct locations in the equation given to us:
1 atm / 373 K = P2 / 473 K
Solving for the final pressure can be done by cross-multiplying and dividing. This gives us a final pressure of 1.27 atm.
Practice Question 2
Let's try a more complicated example. A container of an ideal gas has an initial pressure of 300 torr and the temperature is 25 degrees Celsius. If the pressure in the container is increased to 1 atm, what temperature change caused this increase in pressure?
Well, before we start to solve this problem, we must first convert one of the pressure units so they are both the same. I am going to convert atmospheres into torr, but you could easily convert torrs into atmospheres. I know that each atmosphere of pressure is equal to 760 torr, so I'm going to substitute in 760 torr for my 1 atmosphere.
Before I can solve this equation, I also need to convert 25 degrees Celsius to Kelvin. This can be done by adding 273 to the temperature in Celsius. This gives me 298 Kelvins. When I plug all these numbers into the formula, I get:
300 torr / 298 K = 760 torr / T2
In this sample problem, the units for pressure need to be the same.
I'm going to solve for T2 by cross-multiplying the 298 K and the 760 torr. I will then divide by the 300 torr to give me a final temperature of 755 K. This should make sense because the final pressure was a bit more than double the initial pressure, which would mean that the final temperature should be a bit more than double the initial temperature.
Every day you are surrounded by gases that behave in a very predictable manner. As the temperature of a gas increases, the particles move faster. If a container can expand, it will. If a container cannot expand, this increase in particle speed will result in an increase in pressure inside the container. This pressure and temperature relationship is an example of Gay-Lussac's law. It is the reason pressure cookers are able to cook food really fast, and it's the reason Johnny will make sure to keep the sunblock aerosol can away from the fire. Gay-Lussac's law can also be represented in equation form as P1/T1 = P2/T2. Using the equation helps out a great deal when solving numerical problems.
This lesson can help you gain the skills to do the following:
- Explain the relationship of temperature to pressure in gases using Gay-Lussac's law
- Identify Gay-Lussac's law in equation form
- Solve problems using Gay-Lussac's equation
- Describe real-world applications of Gay-Lussac's law