### Understanding the behavior of gases in containers

In the early days of the development of chemistry, a few key researchers performed enough experiments on gases to be able to determine some empirical patterns in bulk behavior. They derived a number of "gas laws," including the ideal gas law, which we use to model most gases in most situations.

Much later, researchers in statistical mechanics came at these laws from the microscopic point of view, that is, by focusing on the behavior of small ensembles of atoms and molecules and building up the properties of gases from there.

You can get some sense of that by working through the section on the kinetic molecular theory of gases.

While the gas laws discussed in this section are very useful, it's important to realize that they were developed under a number of assumptions that break down at high pressure, when gas atoms and molecules exert significant attractive forces on one another, and in a few other special circumstances. When that does happen, we have other ways to deal with predicting the behavior of those gases.

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### empirical

An empirical rule or law is one that is based on experiment and observation instead of pure mathematical logic. Empirical observations inform theoretical investigations, and theory invites experiment in order to confirm, disprove or improve the theory.

### 1. Boyle's law

Boyle's law says that the product of pressure and volume of an ideal gas, all other things (number of moles and temperature) being constant, is always constant

Another way to say that is that if a gas system in state 1 is changed to state 2 by adjusting the pressure or volume, then the product (PV) is the same in both:

You can picture it like this. Consider a gas in a cubic container with dimensions L × L × L. With a fixed number of gas particles, there will be a fixed average number of collisions with the walls in a given time.

Now if we shrink that container to half its size, the crowding of the particles increases the rate of collisions with the walls by a factor of two, thus doubling the pressure.

Here's what it looks like graphically. If we plot pressure (P) vs. volume (V), the graph is that of the rational function f(V) = c/V, where c is just a proportionality constant. It's there because we can't say for sure that P = 1/V.

Below are a few examples of how Boyle's law can be used.

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### Pascals (Pa)

The Pascal (Pa) is the SI unit of pressure.

1 Pa = 1 N·m-2 = 1 Kg·m-1s-2

Here's a picture of the situation before and after (a picture is often a great way to organize your work). This piston arrangement is a nice way to visualize gas-volume changes:

If we call the pressure and volume before the compression P1 and V1, and those after P2 & V2, then we can rearrange Boyle's law to solve for what's missing, P2:

Then it's just a matter of plugging in what we know:

... and calculating the result:

### Example 2

##### The pressure relief valve on a cylinder fitted with a piston activates at a pressure of 7.5 atm. If the initial pressure and volume of the cylinder are P1 = 1.2 atm and V1 = 10 liters, to what volume can the gas be compressed before the relief valve (a safety device to prevent over-pressurization of the cylinder) activates?

We often engineer gas-handling systems with pressure-relief valves to protect from over-pressurization, which could cause failure of the system and could hurt someone. Here's the sketch of what's going on:

The

We're looking for V2, so we can rearrange Boyle's law to get it (I like to rearrange formulas before plugging in numbers – then I know I'm on the right track):

Now plugging in what we know gives

So the volume can be reduced to

before the relief valve opens.

### Charles' law

Charles' law says that, if we keep the number of particles and the pressure constant, then the ratio of volume (V) to temperature (T) remains constant:

If we hold the number of particles (n moles or N particles) and the pressure (P) constant, then comparing state 1 of a gas system with state 2 gives us Charles' law:

If we rearrange the first expression to V = ct, where c is some constant of proportionality, and plot a graph of V vs. T, we see that the relationship is linear.

It makes sense, if volume changes, the temperature must change proportionally to keep the ratio constant.

Some examples of how to use Charles' law are given below.

### Example 3

##### How much (in percent) does the volume of a fixed amount of gas, held at a constant pressure, expand when the temperature is raised from 30˚C to 37˚C

Here's the situation. It might seem like we don't have enough information, but we do. We don't need the initial volume because we're only trying to find how much any volume of gas would expand (at constant pressure) if the temperature is raised from 30˚ to 37˚C.

We rearrange Charles' law to find the final volume:

Now plug in the temperatures, letting x be the initial volume. The Celsius temperatures are fine because we're calculating a ratio of temperatures, and because the size of the Celsius degree is the same as the Kelvin.

The ratio of temperatures gives us our result.

So for an initial volume of 1 liter, the final volume would be 1.233 liters, for a 23.3% change in volume.

### Example 4

##### If a gas is expanded to twice its original volume while keeping the pressure constant, by how much, and in which direction (warmer / cooler) must the temperature change?

Here is the picture, gas expanding in a cylinder with a movable piston:

In this case it seems like we have even less information to go on, but it's still OK. Take a look.

Charles' law can be rearranged to give the final temperature like this:

Then we can plug in the unknown volumes, but their ratio has to be ½ – that's all that's needed.

So the final temperature will be half of the initial temperature, whatever that was.

The cooling of gases when they expand is the basic idea behind refrigeration and air conditioning.

### Gay-Lussac's law

The Gay-Lussac law of gases says that if we hold volume and number of particles (or moles) of gas constant, then the ratio of pressure to temperature will be constant:

For a gas in state 1, with P1 and T1, we can make this statement, analogous to Charles' law:

We can rationalize the Gay-Lussac law by considering what happens to gas atoms or molecules when we raise the temperature. As T increases, the average velocity of particles increases, therefore collisions with the container walls exert more force on the walls, and the collisions are more frequent. That all adds up to higher pressure.

As for Charles' law, this relationship is linear. We would expect a graph of P vs. T to be linear, with a slope equal to the proportionality constant.

### Example 5

##### Calculate the pressure drop we would expect to see if a quantity of a gas is is cooled from 42˚C to 20˚C at constant volume.

Here's the picture for this problem. Remember that we're forcing volume to remain constant, so we're not letting the piston move in this case.

The Gay-Lussac law can be rearranged to solve for the final pressure, P2, like this:

We don't need to know the initial pressure, just that the equation shows that it will be reduced (in this case) by the ratio of the final and initial temperatures:

The result is:

... so this reduction of temperature would result in the pressure dropping to about 48% of its original value.

Finally, Avogadro showed that if we hold pressure and temperature constant, the ratio of volume to number of moles of gas (or number of particles, N) is constant:

Avogadro actually used this observation to develop the concept of the mole in chemistry.

Just like we did for the previous gas laws, we can rewrite Avogadro's law as an equation involving initial and final states of a gas – in this case as gas is added or removed from the container

Examples of using Avogadro's law are very similar to the previous examples involving ratios.

### The ideal gas law

Finally, we can put all of these observations together to develop the ideal gas law. It says that the pressure-volume product, PV, is proportional to the product of the number of particles (or moles) and the temperature. The constant of proportionality really just helps us get the units right.

The ideal gas law can be expressed in two main ways, in terms of particles

,

where N is the number of particles, and in terms of moles

where n is the number of moles, k = 1.381 x 10-23 J/K is the Boltzmann constant and R = 8.314 J/(mol·K) is the molar gas constant in SI units.

Often it's more convenient not to use the SI units of pressure and volume (Pascals and m3), and to use atmospheres (atm) and liters (L) instead. The gas constant R = 0.08314 L·atm/(mol·K) makes that easy.

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