# Ideal Gas Law

### Written by tutor Dave D.

In order to discuss the Ideal Gas Law, we might start by discussing matter. Matter primarily comes in three different phases. They are solids, liquids and gases. * Each of these three phases have their own distinct characteristics. A good introductory exercise to talking about this subject would be to, on a sheet of paper, make three lists of characteristics which are unique to each of the three phases of matter.

One obvious physical difference that you might notice just by looking at these phases in a common solution like water is the differences of each in volume and shape. Solids have a definite shape and volume that retains itself. Liquids are unique in that they have the quality of being amorphous which means they take on the shape of whatever containment you put them in. Gases, on the other hand, are unique in that they do not have any or take on any definite size or shape. An additional unique property of gases is that they attempt to expand uniformly to FILL whatever container you put them in. The word “uniformly” is important to emphasize here.

A nice way to illustrate uniform expansion is to blow some soap bubbles with a play bubble wand and ask yourself why bubbles are shaped like a sphere. The gas you blow into the soap bubbles has the quality of constantly expanding equally in all directions. This is because a sphere is the only shape where every point on the surface is the exact same distance from the center. (This distance is known as the radius of the sphere.) Hence, all bubbles naturally “want” to be spherical and grow spherically when there are no other forces of tension or compression subjected to their surface.

So, what is matter doing in each of these different phases that make its properties different? This was first discovered in 1827 by a guy named Robert Brown who was looking at very, very small grains of pollen under a microscope. Brown observed that, at a very small level, these grains were jiggling around with a slight jerky motion. Brown named this motion after himself calling it Brownian motion and it revolutionized the idea that all matter at a very small level is IN MOTION.

Now, does this mean that pollen grains are cells or molecules, or atoms… you ask? Actually, this is not quite the case. BUT the way you might picture what’s going on is by doing the following simple experiment. Float a deep plastic bowl in a tub of water and try to throw sunflower seeds in the bowl. Do you see the bowl move around at all in the water when you do this? I’m sure your answer in this case, is “no”. However, if you now try to throw marbles into the same bowl you might find the bowl is now “jerking” around the surface of the water every time a marble hits it. Once again, ask the question, “why?” I hope you considered the obvious fact that the size or more specifically, mass of the marbles that hit the side of the bowl is much larger relative to the mass of the bowl than the mass of each sunflower seed is. This is precisely what Brown inferred about the moving particles that were inside the grains of pollen. There were indeed jerky, moving mysterious particles which the pollen grains were comprised of. These “atoms” or “molecules” were large enough to cause their collisions with each other to wiggle the tiny, tiny pollen grains. When we consider ALL matter being in Brownian motion ALL of the time, then, at a very small level, we can easily picture all of the changes which occur in matter as it goes through these different phases, occur as the result of this motion.

Now, for the sake of illustration for the next few paragraphs, let’s think about our tiny particles as being pool balls bouncing around a pool table. A moving pool ball has kinetic energy as it rolls. A group of pool balls all rolling around on a pool table have a total “average” kinetic energy of all of the balls. This average kinetic energy of the pool balls in motion is called the temperature of the matter. As we increase the average kinetic energy (temperature) of the pool balls (molecules), they collide MORE and start to spread out more as a result of more pool ball (molecular) collisions. MORE collisions also cause our pool balls to end up in more random, less uniform places, and the matter (the group of pool balls) has a harder time retaining its original group-like shape.

If I had the ability to control the size and shape of my pool table such that I could cause it to shrink AROUND my moving pool balls, what would happen? As the pool table shrank around the moving pool balls (without shrinking the pool balls too), would there be MORE or LESS collisions with the sides of the table? The obvious answer here is that there would be MORE. If we had some way to measure the force of each pool ball colliding with the sides of the pool table and add all of these forces together, we would have a measurement of something we call pressure. Pressure would increase as the amount of collisions with the sides of the pool table (container) increased.

Finally, there exists only one small adaptation to our pool ball analogy that is worth mentioning here. This is the fact that pool balls moving on pool tables are technically only moving in two dimensions and molecules in matter are actually moving and colliding in three dimensions rather than on the flat plane of the pool table. So, if we could now, in order to have a more clearer picture of our molecules, picture a THREE dimensional pool table…or, for imaginations sake, a big cubical pool “room”. If we were to measure the amount of water that would “fill” up this 3 dimensional pool room, we could call this measurement, the volume of the container.

Now that we have adequately discussed some important concepts, we switch our focus to the character of a 17th century man named Robert Boyle. Although Robert Boyle’s time in history precedes much of the theory of the Ideal Gas Law, because of the great honor given to him in the science world, he is given credit for performing the experimentation and making the observations which were the foundation for the laws concerning this 3rd phase of matter. Boyle’s original lab experiments were with nothing more than some small strips of paper and a small amount of the element mercury in a J-shaped, cylindrical glass tube, yet his simple observations and notes taken from the suspension of mercury in the tube led to conclusions that were ahead of their time. Though these observations actually only directly gave way to one law known as Boyle’s Law. Two other closely related laws actually combined with Boyle’s Law are often termed the “Ideal Gas Law”. In some circles, these three laws are separately called Boyle’s Law (the pressure-volume relationship), Gay Lussacs Law (the pressure-temperature relationship), and Charles Law (the temperature-volume relationship) repectively. Often, at the college level, and in college textbooks, these three relationships are merely combined into a single catch-all equation which incorporates ALL three at once and called the “Ideal Gas Law”. For the sake of examination, we shall conceptualize them separately, in three separate equations as listed above and then look at how they combine to form the Ideal Gas Law.

In our three dimensional pool ball analogy, ** Boyle’s Law** is the relationship between the number of pool ball collisions with the sides of the pool table and the increasing or decreasing pool table size. The 1’s denote the initial pressure (P

_{1}) and volume (V

_{1}) before the conditions are changed and the 2’s represent the final pressure (P

_{2}) and final volume (V

_{2}) after the conditions are changed.

The equation is:

**P**

_{1}V_{1}= P_{2}V_{2}
When using the equation in this form, the implication is that mass and temperature are both being held constant. (This means these quantities do not change through the event we are observing.)

Units for pressure include: atmospheres (atm) , pascals (Pa), or kilopascals (kPa)
Units for volume include: cubic centimeters (cm^{3}), cubic meters (m^{3}), or liters (l).

In our three dimensional pool ball analogy, Gay Lussac’s Law is the relationship between the number of pool ball collisions with the sides of the pool table and the average kinetic energy of the pool balls in motion. The 1’s denote the initial pressure (P_{1}) and initial temperature (T_{1}) before the conditions are changed and the 2’s represent the final pressure (P_{2}) and the final temperature (T_{2}) after the conditions are changed. The equation is:

When using the equation in this form, the implication is that volume and temperature are both being held constant. (This means these quantities do not change through the event we are observing.)

Units for pressure include: atmospheres (atm), pascals (Pa), or kilopascals (kPa)

Units for temperature must be in Kelvin (K). If temperatures given in the problem are in degrees Celsius (°C) or degrees Fahrenheit (°F), they must first be converted to Kelvin (K) before proceeding with the problem.

In our three dimensional pool ball analogy, ** Charles’ Law** is the relationship between increasing or decreasing the pool table size and the average kinetic energy of the pool balls in motion. Once again, the 1’s denote the initial volume (V

_{1}) and the initial temperature (T

_{1}) before the conditions are changed and the 2’s represent the final volume (V

_{2}) and the final temperature (T

_{2}) after the conditions are changed. The equation is:

When using the equation in this form, the implication is that the pressure and mass are both being held constant. (Once again, this means that these quantities do not change through the event we are observing.) Units for volume include: cubic centimeters (cm^{3}), cubic meters (m^{3}), or liters (l). As in Gay Lussac’s Law, units for temperature must be in Kelvin (K). If temperatures given in the problem are in degrees Celsius (°C) or degrees Fahrenheit (°F), they must first be converted to Kelvin (K) before proceeding with the problem.

Finally, we come to the ** Ideal Gas Law**, which combines all three of these laws into a single equation for gases. The term “ideal gas law” has nothing to do with this law being just the perfect law for gases. Rather, we take an ideal gas to be a gas that has sufficiently low density such that the molecules of the gas in question are so far apart that they do not interact except with effectively elastic collisions. The equation is:

**PV = nRT**

P is the pressure of the gas, V is the volume of the gas, n is the number of moles of the gas and R is called the * universal gas constant* and has been experimentally determined to be 8.31 J/(mol *K) for any real gas with low enough density.

*Actually, if we are talking about the universe as a whole, we may include a fourth phase of matter known as plasma. Approximately 98% of the known universe consists of this stuff called “plasma” of which we know very little about because it only exists at super high temperatures that are higher than anything we might experience on our planet’s surface. This is the result of the fact that plasma makes up the core of all of the stars in our universe including our own sun. This accounts for much, much more mass than all of the mass on our planet combined. However, because plasma exists only under extremely high temperatures, this becomes a totally different discussion that we shall not address here and now. So, for OUR purposes, and the systems we deal with in our environment, there only exist 3 states of matter.

## Practice Problems for the Ideal Gas Law

Here are a few practice problems at the novice level using the Boyle’s Law equation, Gay Lussac’s equation and Charles’ Law equation. They are shown using the general problem solving method. This is followed by a few more difficult problems using the Ideal Gas Law.

**Practice 1.)** You pump 20.0 liters of air at atmospheric pressure (101.3 kPa) into a soccer ball that has a volume of 4.5 liters. What is the pressure inside the soccer ball assuming the temperature does not change?

**Practice 2.)** A constant volume of gas is heated from 20.0°C to 80.0°C. If the gas pressure starts at 1 atmosphere, what is the final pressure of the gas?

**Practice 3.)** A balloon holds 15 liters of helium at 20.0°C. If the temperature increases to 60.0°C, and the pressure is held constant, what will be the new volume of the balloon?

Try doing these problems using the same methods of setting up a problem as the practice problems are done above.

1.) A gas occupies a volume of 25 cubic meters at 8000 pascals. If the pressure is lowered to 4000 pascals what volume will the gas occupy?

2.) At 350 K, a volume of a gas has a pressure of 0.45 atmospheres. What is the pressure of this gas at 273 K?

3.) A 15 liter scuba tank holds oxygen at a pressure of 212 .8 kPa. What is the original volume at 101.3 kPa that is required to fill the scuba tank?

The next set of problems gives examples of Ideal Gas Law problems at the college textbook level. These are a little trickier because one has to think of the system as a whole, before and after an event. One must also use all well-known constants like Avagadro’s number and the Periodic Table atomic masses etc. I’ve worked out solutions for a few of these below and given you a few more to try. Good luck!

**Practice 4.)** A bicycle tire whose volume is 4.1 x 10^{-4} m^{3} has a temperature of 296 K and an absolute pressure of 4.8 x 10^{5} Pa. A cyclist brings the pressure up to 6.2 x 10^{5} Pa without changing the temperature or volume. How many moles of air must have been pumped into the tire?

**Practice 5.)** A Goodyear blimp typically contains 5400 m^{3} of helium (He) at an absolute pressure of 1.1 x 10^{5} Pa. The temperature of the helium is 280 K. What is the mass (in kg) of the helium in the blimp?

## Ideal Gas Law Practice Quiz

Here are a few more problems to try on your own.

A 0.030 m^{3} container is initially evacuated. Then 4.0 grams of water is placed in the container and after some time, all of the water evaporates. If the temperature of the water vapor is 388 K, what is its pressure? Give the answer in Pascals.

If it takes 0.16 g of Helium (He) to fill a balloon, how many grams of nitrogen (N_{2}) would be required to fill a balloon to the same pressure, volume and temperature?

In a diesel engine, the piston compresses air at 305 K to a volume that is 1 /16th of the original volume and a pressure that is 48.5 times the original pressure. What is the temperature of the air after the compression?

**Bibliography**

1. Shamos, M.H., Great Experiments In Physics: Firsthand Accounts from Galileo to Einstein, Dover Publishers, New York, N.Y, 1959. (p36-41). (History Concerning Robert Boyle’s Experimentation)

2. Cutnell J.D. & Johnson K.W., Physics: Volume One, 7th Edition, John Wiley & Sons Inc., Hoboken, NJ, 2007. (p437 – p438). (College Textbook Problems)