Gas Laws

In this section, we'll examine how the four properties we've learned about are related to each other mathematically, and the physical explanations for these relationships. We will learn a number of equations known as gas laws. These laws will form the basis for the Ideal Gas Law, which we'll study in the next section.

A Thought Experiment

Imagine a container of fixed volume and insulated so that the temperature inside stays constant. Imagine it contains 2 moles of gas and the pressure is 700 torr. What do you think would happen to the pressure if we added another 2 moles of gas?

Since pressure is due to collisions with the walls of the container, you should think about how doubling the number of gas particles from 2 to 4 moles will affect the number and force of the collisions in the container. How will doubling the number of gas particles change the number of collisions that occur each second? How will doubling the number of gas particles change the force with which the average particle collides with the wall?

The Kinetic Molecular Theory (KMT) says that the pressure is due to collisions with the walls of the container and depends on how many gas particles there are and how violent the collisions are. Since the energy depends only on the temperature, which doesn’t change, the collisions won’t be harder but adding gas will increase the number of collisions, thus increasing the pressure.

Doubling the number of gas particles, it turns out, exactly doubles the number of times each second a particle strikes the wall of the container. Thus the pressure doubles as well, going from 700 torr to 1400 torr.

This relationship - when one variable doubles, the other doubles - is known as direct proportionality. And it doesn't just apply to doubling. If you were to keep the size and temperature of a container constant and cut the amount of gas inside by a third, the pressure would also be cut by a third. Increase the gas by a factor of 3.845? The pressure increases by that factor as well. This type of relationship can be represented a few different ways.

P = k*n P/n = k P₁/n₁ = P₂/n₂

The version at left shows the relationship as function: at a given V & T, if you take the number of moles of gas you have and multiply by some constant k, you get the pressure. Thus, if n doubles, P doubles. Note: when we say k is a constant, it only holds for the specific values of V and T we have in a given setting. If they change, we will need a different value of k.

The version in the middle emphasizes precisely what k is, and what is staying constant here. In this arrangement, k is the quotient of P and n, so that quotient will always have the same value.

The third version is meant to be applied to a system going from some starting condition to some ending condition, where the moles of gas are changing, without changing V or T. We call the start point 1 and the end point 2. Because the quotient P/n is constant, it will have the same value at points 1 and 2, so we can set the two equal to each other. This will allow us to figure out how a particular change in n will affect P.

We will now learn about a few other mathematical relationships between gas variables. In each case, we will be holding two variables constant (like we did here with V and T) and seeing how the other two variables affect one another.

Boyle's Law

Now imagine a container whose volume can change, which is sealed so that no gas molecules can flow in or out. Suppose such a container has a volume of 1.0 L and, without changing the amount of gas it contains or its temperature, we decrease its volume to 0.5 L. What will happen to the pressure?

Again, consider how the number of collisions and the force of the average collision will be affected. Here’s a hint: although the number of gas particles has not changed, the total volume of their container has decreased, so each particle has less room to move around in. As before, since we have not changed the temperature, the force of the average collision will not change.

Again, because the temperature has not changed, the collisions will be no more or less violent, but the number of collisions will change. In one third the volume, each gas particle has, on the average, one half the distance to go before hitting the wall of the container, and will therefore do so two times as often, increasing the pressure by a factor of two. This prediction can be tested by experiment, as was done by Robert Boyle, who formulated this principle, which is named after him. His experiments showed that, for constant n and T, pressure and volume are inversely proportional: as one increases by a certain factor, the other decreases by the same factor.

As with the previous thought experiment, there are a few ways this relationship can be represented.

P = k*(1/V) PV = k P₁V₁ = P₂V₂

The main way Boyle's law will be applied in this class is the third version of this equation, shown on the right. It illustrates how pressure and volume vary over a particular change (if n and T are kept constant). We can see the math at work in the image above. The value of P₁V₁ can be calculated using the values on the left: 1.0 L*100 mm Hg. Doing the same on the right with P₂V₂ comes out to 0.5 L*200 mm Hg, with both these values equal to 100 L*mm Hg.

The way this equation is typically used is that, given three of the variables (two pressures and two volumes), a fourth can be solved for. For example, consider a container with a volume of 2.8 L containing gas with a pressure of 0.56 atm. If the container is compressed to 1.9 L, what would be the new pressure? As shown below, we can set up the equation with our three known variables, and some algebra allows us to find the new pressure P₂ to be 0.83 atm.

As part of this week's lab exercise, you will test Boyle's law yourself using a sealed syringe, which will allow you to change the volume of a gas and see how pressure is affected. Of course, your test will be qualitative: you will not be able to exactly measure the pressure and see if it changes by a particular amount. However, you should be able to observe that, when the volume of gas in the syringe is cut in half, its pressure feels about twice as great, roughly speaking.

Gay-Lussac's Law

Imagine, now, that you have a closed, rigid container – the number of moles of gas doesn’t change, nor can the volume. What would happen to the pressure if you increased the temperature from 300 K to 400 K? Let's think about it as we did the previous thought experiments.

When the temperature of a gas rises, under Kinetic Molecular Theory, it means the particles have more kinetic energy because they are travelling faster. So, while there are just as many of them and they are in the same space, they will hit the container walls more frequently (because they cross the container more quickly) and more forcefully (because something faster exerts more force).

Thus, for a fixed sample of gas at constant volume, increasing the temperature causes an increase in pressure. And, as before, if we do experiments to measure exactly how much the pressure increases, we can observe a direct proportionality, meaning if the temperature is halved, the pressure is halved, and so on. This is known as Gay-Lussac's law, or alternately as Amonton's law.

P = k*T P/T = k P₁/T₁ = P₂/T₂

As before, we can visualize the equation a few different ways, with the most functionally useful one for this class being shown at right. As before, if you know three of the values in this equation, you can solve for the fourth. Examples of this type of problem can be found in your lab workbook.

As with Boyle's Law, you will test Gay-Lussac's law in the lab exercises for this lesson. You will place a container equipped with a pressure gauge into hot and cold water and observe how the gauge changes. Using a thermometer, you will be able to measure both P and T in this setup, making this a quantitative observation of Gay-Lussac's law, meaning one done with numbers attached.

Charles' Law

Let's revisit our example of Boyle's Law, above. In that case, we had a fixed quantity of gas, and we adjusted the volume of a piston at constant temperature. Now, we will imagine the opposite process: we will heat the gas in the piston and allow the volume to change in response. Importantly, we will keep the pressure in the chamber constant. This can be accomplished by allowing the piston to move freely, as it will always move to the point at which the pressures inside and outside the chamber are equal.

What will happen in this case? As with Gay-Lussac's Law, the heated particles will speed up, hitting the container with increasing force. However, the instant they do so, because the piston is free to move, it will rise, expanding the container. Once it reaches the point where the increased particle speed is exactly balanced out by the reduced frequency of collisions (due to the larger volume), it will stop moving.

Thus, we can say that, with constant n and P, the volume of a gas sample is proportional to its temperature. Of course, don't forget that the temperature must be measured in Kelvin for this to work.

V = k*T V/T = k V₁/T₁ = V₂/T₂

Once again, this relationship turns out to be a direct proportionality, thus we can write and solve the same kinds of equations for Charles' law as we did with Boyle's and Gay-Lussac's.

Avogadro's Law

Two of the gas properties we have discussed - pressure and temperature - are intensive properties, meaning they are independent of how much gas you have (air at sea level has a pressure of 1 atm, and that isn't affected by whether you have a gallon of air or a milliliter). The other two properties - volume and moles - are extensive; they are measures of the quantity of gas, either in terms of space or molecules.

Let's let pressure and temperature be constant, and consider a few different samples of gas. If they are all at the same pressure and temperature, then logically their volume should be proportional to the number of moles in them. Put another way: if you have two samples of gas, and each one has the same number of particles, moving around with the same energy, exerting the same pressure on the container ... then they must have the same volume!

This principle is called Avogadro's Law. It is formulated similarly to the other gas laws (see below) but it isn't generally applied in quite the same way: it is generally used to compare two gas samples at the same conditions of temperature and pressure, rather than a single sample before and after a transformation. However, the algebra used to apply the equations is very similar to the other laws.

V = k*n V/n = k V₁/n₁ = V₂/n₂

STP and Molar Volume

One of the important features of Avogadro's law (and all the gas laws) is that it applies no matter what gas you have: methane behaves the same as helium, and the same as all other gases. Why is this useful? Well, if you look at the middle equation above, you can see that, for a given temperature and pressure, the value of V/n is equal to a constant k. If we plug a value of 1 mole in for n, we get a result that may surprise you: at a given temperature and pressure, one mole of any gas always occupies the same volume

Standard Temperature and Pressure

To make maximum use of this fact, scientists have settled on a common "Standard Temperature and Pressure," (STP) namely a pressure of 1 atm and a temperature of 0 °C (273 K). There is nothing inherently special about these values except that they are convenient to work at and remember. Thus, scientists have settled on them.

Molar Volume

At standard temperature and pressure, all gases have a molar volume of 22.4 L/mol. So 1 mol of neon occupies 22.4 L at STP, 0.5 mol of carbon dioxide occupies 11.2 L, 3.8 mol of krypton occupies 85.1 L, etc. At different temperatures and pressures, the molar volume will be different, but can be calculated.

How are these values calculated? By using the Ideal Gas Law, which synthesizes the individual gas laws we have discussed here into a single equation, which is the topic of our next section.

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