The Gas Laws
Lots of students have trouble when doing gas
law calculations. Typically, these problems aren't so much about doing
the math once you have all of the right variables. Instead, the problem
involves figuring out which equation to use for any particular problem.
In this section of the Helpdesk, we'll discuss how to figure out which gas laws
need to be used, as well as how to use them and common pitfalls that people
encounter doing gas laws calculations.
Getting started: some handy terms you'll need to know
Before doing gas law calculations, you've got to figure out what all of the appropriate terms, symbols, and variables are. For those of you having trouble figuring it out, here are some of the more common ones:
ideal gas: An imaginary model of a gas that has a few very important properties. These properties are that the particles of the gas are assumed to be infinitely small, the particles move randomly in straight lines until they bash into something (another gas molecule or the side of whatever container they're in), the gas particles don't interact with each other (they don't attract or repel one another like real molecules do) and the energy of the particles is directly proportional to the temperature in Kelvins (in other words, the higher the temperature, the more energy the particles have). We make these assumptions because a) They make the equations a whole lot simpler than they would be otherwise, and b) Because these assumptions dont' cause too much deviation from the ways that actual gases behave.
kelvins: A temperature scale in which the degrees are the same size as degrees Celsius but where "0" is defined as "absolute zero", the temperature at which molecules are at their lowest energy. To convert from degrees Celsius to Kelvins, add 273. By the way, we don't say "degrees Kelvin", we just say "Kelvins". Go figure.
pressure: A measure of the amount of force that a gas exerts on whatever container you've put it into. Imagine a shaving cream can. When the pressure is very high in there, the gas in the can pushes very hard on the walls of the can, which is why the cans are made much stronger than soda cans. Units of pressure include atmospheres (1 atm is the average atmospheric pressure at sea level), Torr (which are equal to 1/760 of an atmosphere), millimeters of mercury (1 mm Hg is the same as 1 Torr, or 1/760 atm), and kilopascals (there are 101.325 kPa in 1 atm).
standard temperature and pressure : A set of conditions defined as 273 K and 1 atm.
temperature: A measure of how much energy the particles in a gas have. Units of temperature that you'll run into include degrees Celsius (which you shouldn't use when doing gas law calculations for reasons we'll talk about later) and Kelvins (which is equal to 273 plus the degrees Celsius).
volume: The amount of space that some object occupies. The unit of volume can be cubic centimeters (abbreviated "cc" or "cm3"), milliliters (abbreviated "mL" - 1 mL is the same as 1 cubic centimeter), liters (abbreviated as "L" and equal to 1000 mL), or cubic meters (abbreviated "m3" - there are one million cubic centimeters in a cubic meter).
Amadeo Avogadro was a man with a dream. He knew that if he studied and practiced and worked really hard, he could be the world's most famous tuba player, renowned throughout the modern world for all time.
Tragically, his dream died during a freak tuba accident in the early 1800's - though he was able to regain the use of his tuba hand, it never recovered enough for him to gain superstar status*. Heartbroken, he turned to science, coming up with a law that we use today, called, straightforwardly enough, Avogadro's Law.
Here's what Avogadro said: If you have the same volumes of two gases, they'll have the same number of molecules. For example, one liter of carbon dioxide will have just as many molecules in it as a liter of nitrogen. He figured this out through a series of experiments using the very best equipment of the day.
Of course, this equipment wasn't all that great and the error masked the fact that this law isn't, in fact, true. However, it's mostly true, which allows us to assume that all gases (roughly) behave the same under the same conditions of volume, temperature, and pressure. Without this law, gas law calculations would be very, very inconvenient.
* The tuba story is a total lie. I made it up because I got bored of only talking about gas laws.
Robert Boyle was another man with a dream. He wanted to be the first man to eat 100 hard boiled eggs in a 24-hour period. Unfortunately, some of the other chemists got jealous - let's just say that considerable ugliness ensued and Boyle's dream was permanently derailed. However, Boyle was a man of many talents, and was able to come back from his humiliating egg fiasco* to come up with a gas law of his own.
Here's what Boyle did: He put a gas into a container in which he could change the volume and measure the pressure. When he multiplied the volume of the gas times it's pressure, he found it was equal to some arbitrary number (let's call it k, because he did). If he changed the pressure of the gas, he found that the volume also changed, which isn't really surprising (if you push on something, it gets smaller). What is surprising is that if you multiply the new pressure by the new volume, the answer is the same arbitrary number that you had in the first place (k!). From this, we can make the following statement:
P1V1 = P 2V2
In this equation, P1 is the initial pressure of the gas and V1 is the initial volume of the gas. P2 is the final pressure of the gas and V2 is the final volume of the gas. This way, if you know the initial pressure and volume of a gas and know what the final pressure will be, you can predict what the volume will be after you put the pressure on it. Let's see an example.
Question: If we have 4 L of methane gas at a pressure of 1.0 atm, what will be the pressure of the gas if we squish it down so it has a volume of 2.5 L?
Answer: Let's plug the numbers we've been given into the problem. P1 is 1.0 atm and V1 is 4 L. After we squish the gas, the volume (V 2) is 2.5 L. When we put all of these numbers into the equation, we get:
(1.0 atm)(4 L) = (x atm)(2.5 L)
x = 1.6 atm
* I also made up the egg incident.
Jaques Charles was a disturbing and scary guy. Though he came up with a really handy law for determining what the relationships between the volume and temperature of a gas are, his private life was far more bizarre. Some say that if you go by the old Charles mansion at the edge of town, you can still hear the moaning and wailing of his ghost, forever roaming the night.*
Anyhow, what Charles determined through his studies was that when you change the temperature of a gas, the volume changes. Not surprising - you probably know already that if you heat something, it tends to get bigger. What he found, though, was that if you divide the volume by the temperature of a gas at one temperature, you get a constant. Just like Boyle found, if you change the volume or temperature of this gas, you get the same constant. From this, Charles came up with this statement:
V1/T1 = V 2/T2
Where the subscript "1" indicates the initial volume and temperature and the subscript "2" indicated the volume and temperature after the change. Temperature, incidentially, needs to be given in Kelvins and not in Celsius - this is because if you have a temperature below zero degrees Celsius, the calculation works out so the volume of the gas is negative, and you can't have a negative volume.
Let's see an example of this equation in action:
Question: If we have 2 L of methane gas at a temperature of 40 degrees Celsius, what will the volume be if we heat the gas to 80 degrees Celsius?
Answer: The first thing we have to do is convert the temperatures to Kelvins (by adding 273), because Celsius can't be used in this equation. To do this, we get that the initial temperature is 40 + 273 = 313 K and the final temperature is 80 + 273 = 353 K. We're now ready to start sticking these numbers into the equation:
2 L / 313 K = x L / 353 K
x = 2.26 L
*This story isn't true.
P1/T1 = P2/T2
This gas law explains how if you increase the temperature of a container with fixed volume, the pressure inside the container will increase. This explains why you shouldn't leave cans of spray paint in your trunk - the pressure might get so high that the propellant will blow the can up.
The Combined Gas Law
Imagine a world in which you didn't need to memorize the three laws above. Instead, there was one big law that covered both of them. Hey, that's the world you live in now, and the law you need to know is the combined gas law:
(P1V1) / T 1 = (P2V2) / T2
In this equation, all of the terms are exactly the same as in the preceding equations. The way you can use this equation is that whenever you're changing the conditions of pressure, volume, and/or temperature for a gas, you just plug the numbers into this equation. However, let's imagine that the temperature of the gas didn't change while you were making your change. Since the first temperature term and the second are the same, they cancel out. As a result, if one of these variables isn't mentioned in the problem, just ignore it entirely. Let's see an example:
Question: If we have two liters of a gas at a temperature of 420 K and decrease the temperature to 350 K, what will the new volume of the gas be?
Answer: To solve this problem, use the combined gas law to find the answer. Since pressure was never mentioned in this problem, just ignore it. As a result, the equation will be:
V1/T1 = V 2/T2
Which is the same thing as Charles's law. To solve, the initial volume is 2 L, the initial temperature is 420 K, and the final temperature is 350 K. The final volume, after solving the equation, should be 1.67 L.
The Ideal Gas Law
What happens if you don't change the conditions of a gas, but just want to find out what a gas is like when it's sitting in a container, not doing much? Well, the equations above won't help you much, because they're equations which depend on making a change and comparing the conditions before the change and after the change to make determinations about what the gas is like.
The ideal gas law is an equation of state, which means that you can use the basic properties of the gas to find out more about it without having to change it in any way. Because it's an equation of state, it allows us to not only find out what the pressure, volume, and temperature are, but also to find out how much gas is present in the first place.
Here it is:
PV = nRT
Where P is the pressure of the gas (either in atmospheres or kilopascals), V is the volume (in liters), n is the number of moles, R is the ideal gas constant, and T is the temperature (in Kelvins).
There are two common values for the ideal gas constant. One of them is 0.08206 L x atm / mol x K , and the other is 8.314 L x kPa / mol x K. The question is, which one do you use?
The value of R used depends on the pressure given to you in the problem. If the pressure is given to you in atmospheres, use the 0.08206 value because the unit at the end of it contains "atmospheres". If the pressure is given to you in kilopascals, use the second value because the unit at the end contains "kPa".
Another good thing about this law: It allows us to figure out how many grams and moles of the gas are present in a sample. After all, "moles" is the "n" term in the equation, and we already know how to convert grams to moles (if you've forgotten how, click here ).
Let's see an example:
Question: If I have 4 liters of a gas at a pressure of 3.4 atmospheres and a temperature of 300 K, how many moles of gas are present?
Answer: The first thing you need to do is figure out what value of the ideal gas constant should be used. Since pressure is given to you in "atmospheres", use the first one, 0.8206 L x atm / mol x K. After plugging in the given terms for pressure, volume, and temperature, you end up with:
atm)(4 L) = n (0.08206 L x atm / mol x K)(300 K)
And it's as easy as that! For a practice worksheet with the ideal gas law, click here .
There's a whole bunch of other gas laws, but since I've found that students usually have troubles figuring out these, I've decided to stick to these. Well, that, and because my fingers got tired of typing. You know how it goes.
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