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## Group Activity: Gas Laws

### Introduction

Without even knowing what gases are made of (OK, they’re made of atoms and molecules) we can understand how they behave on a macroscopic level. Gases are a form of ordinary matter that is much less dense than liquids or solids. Because of this they tend to fill completely any containers they are in and are very compressible.

Everyone has experience working with gases and taking advantage of their properties. If you have ever blown up a balloon, used a straw, inflated a tire, complained about the weather, or taken a deep breath then you already know how this topic relates to “real life”.

There are four mathematical variables used to describe the behavior of gases. Pressure (P), Volume (V), Temperature (T), and amount (n). There are some common units of measurement for each of these variables so your first task will be to become familiar with them.

One more thing: Using these variables will help you to understand the ideal gas laws. Why are they called ideal? It’s not because they are the best possible laws gases could follow. Nor is it because they are the best scientific laws anyone ever found. No, it is because the gases we will discuss are ‘idealized’. That is, they are not real gases and no real gases act the way these equations say they will. But, and this is important, almost all gases come very, very close to acting exactly according to the ideal gas laws. So even though they are not perfect, they are very useful.

Here is the next Gas Laws Activity

### Some Units We Will Use

 Volume Unit Symbol Convert Example Cubic Meter m3 1000 L a large fish tank Liter L 0.001 m3 1000 mL a 2 L bottle of soda Milliliter mL 1/1000 L 0.001 L a drop from an eyedropper Pressure Unit Symbol Convert Example Atmosphere atm 760 Torr 14.7 lb/in2 The ‘normal’ amount of pressure exerted by Earth’s atmosphere Torr torr 0.00132 atm 0.0193 psi Atmos. pressure can drop to 740 torr during a storm lbs. per square in. psi lb/in2 0.0680 atm 51.7 torr A car tire might be rated for 35 psi Temperature Unit Symbol Convert Example Fahrenheit °F °C · 9/5 + 32 32°F water freezes 212°F water boils Celsius °C (°F -32) · 5/9 0°C water freezes 100°C water boils Kelvin K °C + 273 77 K liquid nitrogen boils 273 K water freezes 373 K water boils Amount Unit Symbol Convert Example Mole mol 1 mol of C = 6.02 x 1023 atoms of C 1 mol of C atoms weighs exactly 12 g

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First, a helpful note about using the units given above. You will need to be able to change a number given in one unit into another unit. For instance, you might have a problem to solve in which a volume is given in cubic meters (m3). But the formula you will end up using requires that all volumes be given in liters. How do you go about changing over? Use a technique called Unit Analysis. Here’s how it works:

```           1000 L                Put the unit you need on top of the unit factor
1.2 m3 x  -------- =  1,200 L (1.2 x 103 L)
1 m3                 Put the unit you’re getting rid of on the bottom
```

OK, what’s a unit factor?

It is just another name for a conversion factor: a factor you multiply by number in one unit to make it a number in another unit. The word unit is a bit of a pun in the phrase ‘unit factor’. First, it means you’re dealing with units: liters (L), atmospheres (atm), or whatever. Second, it means that the factor is equal to the number one. Let me show you what I mean:
```                                                         1 m
1 m = 100 cm   -->   divide both sides by 100 cm   -->  -----  = 1
100 cm
Equivalently:
100 cm
1 m = 100 cm   -->   divide both sides by 1 m   -->     -----  = 1
1 m
```

What happens when you multiply by one? You get the same number again, right? So using a unit factor does not change the quantity! Just the units.

The key is picking the right unit factor. The hint you need is in the first example above. Find out what one unit (say liters) is equivalent to in another unit (say milliliters). In this case 1 L = 1,000 mL. That means the unit factor is either (1 L)/(1,000 mL) or (1,000 mL)/(1 L). If the number you need to convert is in mL and you want L then use the first factor. If the number you need to convert is in L and you want mL use the second factor. Here, try a few problems to see if you get what I’m talking about:

### Unit Analysis

1. Pressure Units
1. Give 2.1 atm in torr
2. Give 778 torr in atm
3. Give 14.7 psi in atm
4. Give 35 psi in torr
1. Volume Units
1. Give 5.2 L in mL
2. Give 10 m3 in L
3. Give 1,345 L in m3
4. Give 4,321 mL in L
1. Temperature Units
1. Give 350 K in °C
2. Give 35 °F in °C
3. Give 55 °C in °F
4. Give 32 °F in K

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### Boyle’s Law (1662)

Now that you know a bit about the units you will be using you are ready to start working with the mathematical laws that we will use to build up the Ideal Gas Law. (This should sound momentous and important.) The first law we will examine is known as Boyle’s Law and was first quantified and mathematically modelled by Robert Boyle is about the year 1662. The law, in words, says the following:

At a given constant temperature (T) and number of moles of gas (n),
the pressure and volume
of a gas are inversely proportional.

That is, the higher the pressure, the smaller the volume; the lower the pressure, the larger the volume. This relationship can be expressed in the formula:

```P · V = constant    or more algebraically: P · V = k  (only when T and n are constant)
```

By the way, what is pressure? The key to understanding pressure is to think about force per unit area. Which unit expresses that idea most clearly? What do you think causes the force?

As you work through the following problems try to stop briefly after finding each solution and imagine what the answer means physically. Many people say that chemistry is very abstract, and it is, but with these problems you can very often think about the answers in a clear, physical way. If you do this, you will also have a leg up on making sure your solution is correct. If your volume is a negative number and you try to imagine a negative volume, then you will know you did something wrong!

1.      First, in order for the formula to be of any use we need to put it into a form that tells us about changes in pressure and volume. For example, say you have a balloon filled with air at 1 atm. Say also that the balloon has a volume of exactly 1 L. What would the pressure inside the balloon be if you squished it down to 0.75 L?
1. First, find the constant k for the initial P and V.
2. Now, plug the new volume and the constant (k) into the expression above (PV = k) and use those two variables to find the new pressure.
3. Call the first P and V values (1 atm and 1 L) P1 and V1. Their product is by definition a constant value. So the new P you found multiplied by a V of 0.75 L should equal the same number. Let’s call these two new values of P and V, P2 and V2. Show arithmetically that the product P1· V1 equals P2· V2.
4. What does this bit of arithmetic allow you to write algebraically? Write down the new formula which allows you to quickly and easily find a new V or P given the other three variables.
2. Apply this new formula to the following problem: An inflated balloon has a volume of 0.55 L at sea level (1 atm) and is allowed to rise to a height of 6.5 km, where the pressure is about 0.40 atm. Assuming that the temperature remains constant, what is the final volume of the balloon? For practice with conversion, convert your answer to cubic meters (1 m 3 = 1000 L). (Source: Chemistry, Raymond Chang)
3. A sample of chlorine gas occupies a volume of 946 mL at a pressure of 726 torr. Calculate the pressure of the gas (in torr) if the volume is reduced at constant temperature to 154 mL. Convert your answer to atm (1 atm = 760 torr). (Source: Chemistry, Raymond Chang)

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### Charles’s Law (1787)

The next important law you will learn is called Charle’s Law. Mr. Charles was an Englishman and played around with hot air balloons. He never established a mathematical law describing the relationship between temperature and volume but some English historian liked his name better than the name of the man who did. The man who actually figured out this law of nature is Mr. Gay-Lussac, a Frenchman. Whoever figured it out, the law we’re interested in is:
At a given constant pressure (P) and number of moles of gas (n),
the volume and temperature
of a gas are directly proportional.
In other words, the more you raise the temperature, the larger the volume of gas (as long at it remains at the same pressure and you don’t add or subtract any molecules). Temperature must be expressed in Kelvins, not degrees Celsius (x°C = x + 273 K). This relationship can be expressed in the formula:
```V                    V
— = constant   or    — = k  (only when P and n are constant)
T                    T
```
1. Can you write down the formula relating T1 and V1 to T2 and V2? Hint: it is very similar to the formula you discovered relating pressure and volume. Show the same sort of steps you worked out in that problem.
2. Now that you have that formula, find the new volume in L when a 452 mL sample of fluorine gas is heated from 22°C to 187°C at constant pressure. Reminder! The only acceptable temperature unit is Kelvins!(Source: Chemistry, Raymond Chang)
3. A sample of carbon monoxide gas occupies 3.20 L at 125°C. Calculate the temperature at which the gas will occupy 1.54 L if the pressure remains constant. Be sure to express your answer in kelvins. (Source: Chemistry, Raymond Chang)
4. Under constant pressure conditions 9.6 L of hydrogen gas initially at 88°C is cooled to -15°C. What is its volume after the cooling is complete? (Source: Chemistry, Raymond Chang)

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This next law uses the unit Mole (mol) which you may never have heard of. For now it is enough to know that it refers to the amount of a substance in terms of the number of atoms or molecules. We will discuss it in more detail later. To give you some idea of its application to gases you should know that 1 mol of any gas (just about) will have a volume of 22.4 L at 273 K (0 °C) and 1 atm. That 1 mol of gas will contain an incredibly large number of particles: 6.02 x 1023 of them (that 602 followed by 21 zeros). At any rate, the law that Mr. Avogadro proposed is the following:
At a given constant temperature (T) and constant pressure (P),
the volume and the number of moles, n, of a gas are directly proportional.
In other words, the more gas you have, the bigger the volume (as long as you compare volumes at the same pressure and temperature). The mathematical expression of this (obvious) law is:
```V                    V
— = constant   or    — = k  (only when T and P are constant)
n                    n
```
Note: you cannot change the number of moles just by changing the volume! For that you have to add or subtract gas particles.

1. This should be very easy by now: relate V1 and n1 to V2 and n2. Show your work, please!
2. One mole of an ideal gas has a volume of 22.4 L at 1 atm and 273K. (This set of conditions is known as STP: Standard Temperature and Pressure). What is the volume of 0.50 mol of gas? Of 0.01 mol of gas?
3. A balloon containing 2.1 mol of O2 has a volume of 36 L at some temperature and pressure. You pump out 1.5 mol. What is the new volume?
4. As an aside, the word Mole comes from the German Molekül. Its abbreviation to ‘mol’ is to save you the hardship of writing that ‘e’ every time.

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### Combined Gas Laws

So what happens if two variables change at once? Say the temperature and the pressure both change and you need to find the volume: are you stuck, unable to find an answer? Fortunately, no. You can use the combined gas laws:

```PV = nRT is the combined form of the ideal gas law.
This formula is good for static situations

how about when variables are changing?
This formula combines all the gas laws we have used up to now.

P1V1      P2V2        P1V1      P2V2
R = ------ = ------  or  ------ = ------ because moles usually don't change
n1T1      n2T2         T1        T2       (ex., n1 = n2 = 1)

```

R is called the so-called ‘gas constant’ and has a value of 0.0821 L·atm/K·mol

1. Sulfur hexafluoride (SF6) is a colorless, odorless, very unreactive gas. It is often used to fill the space between the panes in double-paned windows. Find the pressure (in atm) of 1.82 mol of SF6 in a steel vessel of volume 5.43 L at 69.5°C. (Which expression of the combined gas law is appropriate here?)
2. What is the volume (in L) of 3.2 mol of N2 (nitrogen gas) at 3040 torr and 86°C?
3. A bubble rising from the bottom of a deep lake starts with a temperature of 8°C and a pressure of 6.4 atm. When it gets to the surface, just before it bursts, the temperature is 25°C and the pressure is 1 atm. If the initial volume was 2.1 mL what is the final volume?
4. A gas starts with V1 = 4,000 mL, P1 = 1.2 atm, and T1 = 66°C. If no gas is added or removed and the V2 = 1.7 L and T2 = 42°C, what is P2?
Remember to use only Kelvins in your calculations!
Last updated: Jun 23, 2006      Keller Home  |