Introduction to the Mole

Before you can do molar math successfully, you must first know what a mole IS. If you are NOT completely clear about what a mole IS, quickly browse the page at the following link. Then come back when you think you understand enough to go on. What is a Mole?

We use the unit of a "mole" in science to talk about extremely large numbers of extremely tiny countable objects, such as molecules, atoms or electrons, objects which are obviously way too small to see.

We use the unit "mole" just like we use the unit "dozen." One dozen means "12 of something." In chemistry, one "mole" means "602,200,000,000,000,000,000,000 of something." ("Six-hundred-two-sextillion-two-hundred-quintillion of something.") The "something" can be anything which is "countable" by individual units. "Countable" as opposed to something without a definite particle nature, such as light or energy. The unit of a "mole" is usually used to count molecules, atoms or electrons, but technically it can also be used as a numerical unit even for something like apples, if you really wanted to take the time to count that long. (Actually you could not count that many apples in an entire lifetime, even with the help of many friends!)

The numerical value of the "mole" is frequently written using scientific notation as 6.022 x 10^23. ("^23" means "to the 23rd power, or something multipled by 10 and then multipled by 10 again and again and again, until 23 times.)

So basically, a "mole" is just a very, VERY large number.

Four Simple Steps to Mastering Molar Math

Molar Math is actually VERY EASY! With a little patience, you can learn the 4 simple steps listed below. We will present these concepts to you one by one so you do not get overwhelmed. Be sure you understand each step before going onto the next one, so you can build a solid foundation. Otherwise you may get confused.

(1) Be Able to Find Starting and Ending Units in a Problem

What is a unit and why is it important? A unit is one of whatever it is you are counting or measuring. For example, if you have 10 gallons of gasoline, "gallons" is the unit you are counting. If you drive 40 miles, "miles" is the unit of distance you are measuring. Units are extremely important in science. Without units, our numbers don't mean anything. If I said, "I have 30," what would your question be? Wouldn't it be "30 of what?" Reporting WHAT we are measuring or counting is essential to understanding. So we must always use units.

All problems in molar math are really just "unit conversion" problems.

Let's take a non-chemistry example first. Say I want to convert units of "feet" to units of "inches." My starting units are "feet" and my ending units are "inches." Using this information, I would set up a conversion structure as shown below, leaving space for a "conversion factor" in between.

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We will talk about "conversion factors" later. But first you need to be sure you can find starting and ending units from the wording of problems. Practice with the following examples. We will use nonsensical units so you can concentrate on the LANGUAGE which gives you clues to the starting and ending units.

(1) How many bars are in a plick? Ask yourself, "what are you trying to find?" The answer is "bars." Which words give you the clue that these are your ending units? "How many." What are your starting units? "plick" Which words give you the clue? "are in." So you would set up the structure this way:

(2) Convert jupos to babs. What are your ending units? "babs." Which word gives you the clue? "to." What are your starting units? "jupos." Which word gives you the clue? "Convert." Set up the structure like this:

(3) Given 15 beebos, how many gibbers do you have? What are your ending units? "gibbers." What word clue tells you that? "how many." What are your starting units? "beebos." What word clue tells you that? "Given." Set up the structure like this:

You may find other language variations in problems, but you should always be able to find the ending units and starting units and set up a structure as we have done above.

(2) Be Able to Set Up Conversion Ratios to cancel the unwanted starting units and leave you with the desired ending units.

What is a Conversion Ratio?

A "conversion ratio" is a ratio which converts one unit into another.

Here is an example of a conversion ratio using feet and inches.

We can use this conversion ratio to convert feet to inches as follows:

Notice that using the conversion ratio, "12 inches/1 foot," the starting units of "feet" cancel, leaving the ending units, "inches." Placing ratios correctly to cancel units is the second basic skill needed to solve ALL mole problems in chemistry.

Turning Conversion Ratios Upside Down in Order to Cancel Units

For example, the ratio "12 inches/1 foot," may be turned "upside down" as shown below.

The second form of the ratio is used when we want to convert from inches to feet, instead of from feet to inches. We set up the structure to START with inches and END with feet, as shown below.

It is important to ALWAYS choose the correct form of the ratio to cancel the units we DON'T want and end up with the units we DO want. To convert inches to feet, which of the above forms of the ratio should we use?

Use ONLY Horizontal Fraction Bars

You also have to know is that it is FAR better to write all ratios in science with horizontal fraction bars, rather than diagonal bars. This keeps everything easy to see when you have a string of ratios in a problem.

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(3) Be able to find any of the 4 ratios related to the mole listed below. (There are ONLY 4!)

Ratio #1: The Number of Particles in a Mole (This value is always the same, regardless of the type of particle.)
Ratio #2: The Volume of a Mole of Gas at STP (This value is always the same, no matter which gas you have.)
Ratio #3: The Molar Mass of a Specific Substance (This value changes depending on which substance you have.)
Ratio #4: The Mole Ratio of Two Substances in a Chemical Equation (This value changes depending on the coefficients in your chemical equation.)

Ratio # 1: The Number of Particles in a Mole

602,200,000,000,000,000,000,000, or 6.022 x 10^23 in scientific notation, is the number of particles in a mole. While anything "countable" may be considered a particle, in chemistry, particles usually means molecules, ions or electrons. Without going into explicit details here, the numerical value of this ratio is given BY DEFINITION based on experiments in the real world. This relationship may be written in one of two ways:

And remember, it doesn't matter WHAT the particles are. The number of particles in a mole is ALWAYS the same.

Now let's use this ratio in solving two example problems. In the first few examples, I go into excruciating detail, but after you get a feel for this kind of problem solving, it goes very quickly and is really very EASY. (My assumption is that you already understand scientific notation and significant figures.)

Problem 1: Find the Number of Particles from Moles

You have 3.0 moles of substance. How many particles of substance do you have?

Analysis: You are given moles and are asked to find particles. So you start with units of moles and end up with units of particles in your answer. Before you even begin to worry about the numbers, be sure to place the starting and ending units where they need to go.

Notice that we placed the units where they go first, without even worrying about the numbers. We also left a blank space to write in the conversion ratio, once we determine what that is.

Next, we need to choose the correct version of the ratio between moles and particles to make the conversion work. Again, not worrying about numbers, concentrate on just finding the UNITS which will cancel out moles and leave us with particles. Below we see the two possible ways that units in the Conversion Ratio may be placed. Which one do you think is correct?

If you chose possibility "A," you are correct. That placement allows us to cancel the unit "moles" and keep the unit "particles."

Now that our units are placed correctly, we can insert the numbers. According to the problem, we start with 3.0 moles of substance. The number of particles in a mole is constant, so our conversion factor is constant. (I have also placed grey numeral "1's" into the problem to help us remember that even when there are no numbers or units written, the number "1" is always there.)

Our final steps are to check the unit cancellation, do the calculation, determine significant figures and put the answer in final form.

Problem 2: Find the Number of Moles from Particles

Instead of being given moles as in Problem 1, you are now given a number of particles and must convert back to the units of moles.

You have 2.56 x 10^25 particles of substance. Find out how many moles you have.

Analysis: You are given particles and are asked to find how many moles you have. So this time you start with particles and end up with units of moles in your answer. Follow the same pattern as before. Begin by placing your units first.

Next, choose which version of the conversion ratio you need to cancel out the units you DON'T want and end up with the units you DO want.

In this case, if you chose possibility "B," you are correct. Particles will cancel and you will end up with moles, which is what we want.

Now put the numbers in, cancel units, do the calculations, determine significant figures and report the answer in final form.

Review of the Four Steps to Solve Mole Problems

The number of particles in a mole is a constant, 6.022 x 10^23. This is ALWAYS true. In doing simple conversions back and forth between the number of particles and the mole, we can use the conversion ratio either "right side up" or "upside down," depending on what we're converting FROM and what we're converting TO. The steps to solving these kinds of problems are:

1. Place the starting and ending units in their correct locations.

2. Choose the correct conversion ratio to make unwanted units cancel.

3. Plug in the numbers and do the calculation.

4. Put the answer in final form. EASY!

Ratio # 2: The Volume of a Mole of Gas at STP

This quantity is used specifically ONLY for particles in the gaseous state. Its value is FIXED at 22.4 Liters per mole at STP (Standard Temperature and Pressure). STP was arbitrarily set and agreed upon by the scientific commmunity to mean 25 degrees C and 1 atmosphere pressure, the pressure at sea level. Note: At DIFFERENT temperatures and pressures, the volume occupied by one mole of gas is different and can be calculated using the laws of relationship known as the "gas laws." Nevertheless, for many calculations in chemistry, volume is measured and compared at STP, so this is a useful ratio. The ratio representing the volume of a mole of gas at STP may be written in either of two ways:

Problem 3: Find the Number of Liters from Moles

You have 4.0 moles of pure nitrogen gas at STP. How many Liters of gas do you have?

Analysis: You are given moles and are asked to find Liters. You start with units of moles and end up with units of Liters. Place your units as shown below.

Choose the correct conversion ratio to make the unwanted units cancel.

If you chose "A," you are correct. Next, insert the numbers, cancel units, do the calculation, round to the correct number of significant figures and report the answer in final form.

Problem 4: Find the Number of Moles from Liters

You have 49.7 Liters of oxygen gas at STP. How many moles do you have?

Analysis: You are given Liters and are asked to find moles. You start with units of Liters and end up with units of moles. You should place your beginning and ending units as shown below.

From here, which conversion factor do you choose? The one which has Liters on top or moles on top? Choose the correct conversion factor, plug in the numbers and proceed through to the final answer.

The Secret to Solving ALL Mole Chemistry Problems

Hopefully by now you find it EASY to choose the correct conversion ratio to cancel the unwanted units and leave us with the units we want.

Here is the Secret! ALL mole chemistry problems you have from here on out are simply extensions of the skill of being able to find the correct conversion ratio which cancels unwanted units and leaves wanted units. EASY!

Using Two Ratios in One Problem

Problem 5: Find the Number of Molecules from Liters.

In this problem, you will be using TWO conversion factors to get the answer.

How many molecules of hydrogen gas are in 54.6 Liters of pure hydrogen at STP?

Analysis: You are asked to start with Liters and end with molecules. (The clue that you end with molecules is given from the words "how many.") While we have no direct one-shot conversion between Liters and molecules, we DO have "Liters to moles" and "moles to molecules." So this time we can use TWO conversion factors to get our answer.

Set it up like this:

Next, find the conversion factors needed to do the conversion.

Finally, plug in the numbers and do the calculation.

Notice that the entire strategy for solving problems like this, even with more than one conversion factor, is simply setting up the starting and ending units and then filling in the gaps with ratios either "right side up" or "upside down" to make unwanted units cancel and keep the units you want in the answer. EASY!

Ratio #3: The Molar Mass of a Specific Substance

Molar mass is different from the first two ratios, because it is NOT a single constant value which never changes. Although any given substance has the same unchanging, constant molar mass, each different substance has its own molar mass. Look at the three examples below.

A mole of hydrogen molecules (which are made up of two hydrogen atoms bonded together) has a mass of 2 grams.

A mole of helium gas has a mass of 4.0 grams.

A mole of water has a mass of 18.0 grams.

If you do not know how to find the molar mass of a substance, click here. [Not active yet.]

Now let's use molar mass to solve problems.

Problem 6: Find Moles from Grams.

How many moles of water are in 45.0 grams of water?

Analysis: The words "how many" point to the needed "ending units." The words "are in" indicate the "starting units."

For water, the two possible ratios are:

Now choose which of the above conversion ratios can be used to solve this problem.

Hopefully you chose the conversion factor shown below.

Problem 7: Find Grams from Moles.

How many grams of diatomic hydrogen gas are in 5.7 moles of hydrogen?

Analysis: The words "how many" point to the ending units. The words "are in" point to the starting units.

Start by setting up the units

Choose the correct conversion ratio.

Write in the numbers, cancel units, do the calculation, determine significant figures and report the answer in final form.

Using Two Ratios in One Problem
(Part 2)

Now let's use molar mass ratios along with the first two molar ratios to solve problems.

Problem 8: Find Liters at STP from Grams.

How many Liters of space will 25.3 grams of methane gas occupy at STP?

Analysis: The words "how many" indicate the ending units. What are the starting units? And what are the conditions? (STP) The condition of STP is important, because the conversion ratio of gas volume at STP is valid ONLY at standard temperature and pressure. That's why is must be stated explicitly in the problem.

Set up the problem.

Find and choose the correct conversion ratios.

Put in the numbers, cancel units, do the calculation, determine correct number of significant figures and report the final answer.

Ratio #4: The Mole Ratio of Two Substances in a Chemical Equation

How to Find Mole Ratios from a Chemical Equation

Here is the balanced equation for the formation of water. From this equation, we know that 2 molecules of hydrogen plus 1 molecule of oxygen combine to make 2 molecules of water. The coefficients (large numbers) in front of the chemical formulas represent a ratio of components. Thus, we have a 2:1:2 ratio between the different species in the reaction.

From the above equation, we get 3 different "mole ratios." We get the mole ratios from the coefficients. Study the diagram below and see if you can figure out how to find mole ratios.

Using Mole Ratios to Solve Problems

The following problems will give you practice in solving mole ratio problems.

Introducing the Multiple Conversion Flow Chart

Problem 9: Find Grams of Water from Liters of Hydrogen.

Given 17.3 Liters of diatomic hydrogen gas, how many grams of water will be produced, if there is an excess of oxygen?

Analysis: Start units are "Liters of hydrogen." End units are "grams of water."

Set up the problem. This time there will be 3 conversion ratios. The middle ratio will be one of the mole/mole ratios from the balanced chemical equation.

This kind of problem can be made EASY by charting the "conversion flow" on the "Multiple Conversion Flow Chart" as shown below.

Now we will show how to use the "Multiple Conversion Flow Chart" with the problem above.

(1) Write the starting units and ending units under the particular chemical species to which they pertain. We start with 17.3 Liters of hydrogen gas, so we must place 17.3 Liters in the bottom box directly under hydrogen in the equation. We are looking for grams of water, so we must write "grams" with a question mark under water, or H2O.

(2) Draw an arrow directly up from the start unit to the mole/mole box and write the word "moles." (For this problem, it will mean "moles of H2.")

(3) Next, draw an arrow directly down from the mole/mole box pointing towards the ending units, which in this case are "grams of water." Write the word "moles" in the mole/mole box where the down arrow begins. This will mean "moles of water."

(4) Finally, draw an arrow from the moles of the starting chemical (in this case, hydrogen gas) to the moles of the ending chemical (in this case, water).

You have now graphically set up the flow of the conversion factors you will use.

(5) Once you have these units correctly placed, writing the conversion ratios is simply a matter of following the arrows.

(6) Finally, input the numbers, cancel units, do the calculation, determine significant figures, and write the final answer in correct form.

Problem 10: Find Liters H2 at STP from Grams O2.

If you want to use up 56.7 grams of oxygen to make water, how many Liters of hydrogen gas at STP will you need?

Analysis: Start units "56.7 grams of oxygen." (The words "use up" are the clue.) End units "Liters hydrogen gas."

Set up the equation showing start and end units, leaving space for the conversion factors.

Next, set up the "Multiple Conversion Flow Chart" with the equation as shown below. We are starting with 56.7 grams of oxygen and ending with Liters of hydrogen gas at STP. (Notice that the direction of the arrow in the mole/mole box goes in the opposite direction from the previous problem.)

Next, set up the conversion factor ratios with the units in the correct places for cancelling.

Now input the numbers.

If you are not sure how to find the grams per mole of any substance, such as that there are 32 grams per mole of diatomic oxygen gas, click here. [Not active yet.]

Finally, cancel units to make sure you have them correctly placed, do the calculation, round to the correct number of significant figures and report the answer in final form.

Recognize that in the real world of doing chemistry, we do not go to all the trouble of actually drawing the "Multiple Conversion Flow Chart." We just remember the structure and apply it. So my actual figuring of the above problem on paper would look more like this:

From here, I would cancel the units and do the calculation.

Hopefully this has been helpful. If you see something confusing in this presentation, or if you find errors, or if you have questions, please contact me, Lynda Jones, at sing-smart.com. My goal is to make chemistry EASY! How am I doing?

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