This group activity is intended to introduce you to two very important parts of the study of chemistry. First, you will learn the basics of using scientific notation. This is a way of writing very large and very small numbers with a minimum of fuss and without having to count a lot of zeros. Second, you will learn about the scientific unit for ‘amount of substance’. You may have encountered this unit before and in this activity it will be made clear exactly what is meant by it.
Scientific notation is exceptionally useful for dealing with numbers like 137,760,000,000 or 0.000 000 000 679. The first number is a whole number and can be rewritten as 1.3776 x 1011. The second number is a fraction and can be rewritten as 6.79 x 10–10. The first way that I wrote both of these numbers is ungainly and requires you to count out zeros every time you write the number. And if you miscount your zeros you get a much different number.
Numbers in scientific notation have two parts. The first part is a number between 1 and 10. The second part is the number 10 raised to some whole-number exponent or power. You can see how this works by looking at this general model for a number in scientific notation: N x 10x. To express a number in scientific notation is really as simple as counting the number of digits. For example, look at the number given above: 137,760,000,000. Write this number down but put a decimal point between the 1 and the 3. Now, starting with the three, count digits to the right until you get to the last zero before the original location of the decimal point. You will find these are 11 digits. So 11 is the x in the expression N x 10x and N is 1.3776. So there you have 1.3776 x 1011. The same thing works for very small numbers: take the small number given above and put the decimal point between the first two non-zero digits. Starting with the first non-zero digit count digits to the left until you get to the original position of the decimal point. You will find there are 10 digits so the exponent in N x 10x is –10 and the number is 6.79 x 10–10. For fractions you count to the left and the exponent is negative. For whole numbers you count to the right and the exponent is positive. Here is a summary of a few numbers and how they are written using scientific notation:
smaller bigger
Fraction 1/100 1/10
Decimal notation 0.01 0.1 1 10 100 1,000
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Scientific notation 10-2 10-1 100 101 102 103
Here are some problems to make scientific notation more clear:
Examples 101 = 10 105 = 100,000 10–3 = 0.001 10–7 = 0.0000001Write the number indicated as shown in the example.
Examples 104 —— 109 10–4 —— 10–9Put a box around the larger number in each pair.
Examples 2 x 103 = 2,000 4.1 x 10–5 = 0.000041 2.9 x 1014 = 290,000,000,000,000Write the following numbers in decimal form.
Examples 602,200,000,000,000,000,000,000 = 6.022 x 1023 0.00132 = 1.32 x 10–3 0.0193 = 1.93 x 10–2Write the following numbers in scientific notation.
Now you should know how to write numbers using scientific notation. If you do not yet understand any of the exercises on the previous page, please ask for help before going on.
Numbers written in scientific notation are easy to multiply and divide but a bit harder to add and subtract. So we’ll start with addition and subtraction.
The first thing to note when adding and subtracting numbers in scientific notation is what order of magnitude the numbers are in. Order of magnitude refers to the difference between two exponents of 10. The number 1.2 x 102 is one order of magnitude smaller than the number 2.4 x 103. Likewise, 6.02 x 1023 is 20 orders of magnitude larger than 6.0 x 103. Numbers should only be added together (or subtracted from one another) when they are within 2 or at most 3 orders of magnitude. If you need to add two numbers that differ by more than 3 orders of magnitude (like 104 and 109) just write down the larger number. It is so much larger that the smaller number adds too small an amount to matter. If you must subtract a smaller number from a larger one (say 102 – 10–3) just write the larger number as the answer: the number you are subtracting is too small to matter.
Note that you must make the exponents the same in order of magnitude to add or subtract. To make a number have a larger exponent, you must write the number smaller. To make a number have a smaller exponent, you must write the number larger. That just means moving the decimal point:
These numbers all mean the same thing! (1.25 if not using scientific notation) 0.0125 x 102 0.125 x 101 1.25 x 100 12.5 x 10–1 125 x 10–2 1,250 x 10–3 Write the following numbers so that they have the exponent of 10 shown.
Addition and Subtraction Examples
(3 x 104) + (2 x 103) = (3 x 104) + (0.2 x 104)
= 3.2 x 104
= 32,000
(3 x 104) - (2 x 103) = (30 x 103) - (2 x 103)
= 28 x 103
= 2.8 x 104
Change numbers so that both in each problem have the same exponent, then
add or subtract. If numbers differ by more than 2 orders of magnitude,
then write down the larger number without performing any operation.
Now multiplication and division are actually quite easy. When multiplying, multiply the numbers and add the exponents. When dividing, divide the numbers and subtract the exponents. Finally, once you have an answer in scientific notation change the answer so that the number multiplied by a power of 10 is between 1 and 10:
Multiplication and Division Examples (2 x 102) × (6 x 103) = (6 x 103) ÷ (2 x 102) = 12 x 105 = 1.2 x 106 3 x 101 (2.7 x 109) × (4.2 x 103) = (2.16 x 102) ÷ (1.2 x 106) = 11.34 x 1012 = 1.134 x 1013 1.8 x 10–4 (5.6 x 10–9) × (2.3 x 103) = (4.5 x 102) ÷ (9.0 x 10–5) = 12.88 x 10–6 = 1.288 x 10–5 0.50 x 107 = 5.0 x 106 Multiply or divide as indicated. Simplify all answers so that the number multiplied by a power of 10 is between 1 and 10.
Atoms are too small to weigh one at a time. Not only are there no scales that could handle the extremely tiny mass of one atom but it would take forever to get anything done if you had to weigh 1.2 x 1023 atoms one at a time. So chemists have a short-cut and it is called the mole or Avogadro’s Number.
First, a word or two about the number. Avogadro’s Number is 6.02 x 1023 and is also called ‘the mole’. This is such a large number that it is hard to visualize, but here are a few tries: if you had a watermelon the size of the moon, it might have a mole of seeds inside it. One mole of chicken eggs would cover the entire surface of the Earth…four miles deep. This is a seriously large number.
Second, an introduction to some things you need to know to make use of this number. Think of the mole like you think of the words ‘pair’, ‘dozen’, and ‘gross’. A pair has 2 objects in it. A dozen has 12. A gross is a dozen dozen or 144 items. Well, a mole is just 6.02 x 1023 items. Mostly, a mole is only useful for counting very small and very numerous items. I feel a bit sorry for all the chickens (and those who have to clean up after them) that would be needed to lay a mole of eggs.
The value of Avogadro’s number was not randomly chosen but instead was determined experimentally. You know from previous lessons that a carbon-12 atom (12 6C) weighs exactly 12 amu. Scientists have experimentally determined the number of carbon atoms in exactly 12 grams of carbon to be 6.022 137 x 1023. This means that when you weigh out 12 g of carbon on a scale you have a mole of carbon atoms. The usefulness of this comes when you realize that you can relate the mass of an atom or molecule in amus to the mass of a mole of that atom or molecule in grams. The mass of a mole of atoms or molecules in grams is the same number as the mass of those atoms or molecules in atomic mass units. The mass of a mole of atoms or molecules is called the molar mass and has units of g/mol.
Examples (using the Periodic Table) 1 Cu atom has a mass of 63.55 amu 1 mole of Cu atoms has a mass of 63.55 g: copper has a molar mass of 63.55 g/mol 1 S atom has a mass of 32.065 amu 1 mole of S atoms has a mass of 32.065 g: sulfur has a molar mass of 32.065 g/mol Find the formula mass of the following atoms using your periodic table. Express your answer using both amu and g/mol.
In the same way that you find the molar mass of an atom, you can find the molar mass of a chemical compound. Compounds are most commonly written in a formula notation that shows what kinds of atoms are in the molecule (or formula unit) and how many of each. For example, in the formula for sugar (glucose) you can see that there are 6 carbon atoms, 12 hydrogen atoms, and 6 oxygen atoms: C6H12O6. The small number down and to the right of each atomic symbol is called a subscript and shows how many of that kind of atom can be found in each molecule of the substance. The molar mass of a compound is the sum of the molar masses of each atom in the compound multiplied by its subscript.
Examples C6H12O6 6 × 12.011 g/mol + 12 × 1.0079 g/mol + 6 × 15.999 g/mol = 180.16 g/mol Write the formula for each of the following compounds and find the formula mass. Express the mass in amu and in g/mol.
Use the following types of conversion factors for these problems:
6.02 x 1023 atoms/mol 6.02 x 1023 molecules/mol 6.02 x 1023 things/mol
1 mol 6.02 x 1023 things
or ——————————————————— or ———————————————————
6.02 x 1023 things 1 mol
Express your answers using scientific notation.
A chemist makes use of all this by being able to find the mass of a chemical and be able to tell how much, that is, how many moles, of the chemical she has. Given the molar mass of a chemical, she can use a mass in grams to figure out how many moles she has. Or if she needs a certain numbers of moles, she can calculate how many grams that would be.
Examples
Moles from grams
C6H12O6 has a molar mass of 180.16 g/mol
1 mol
360.0 g of C6H12O6 × —————— = 1.998 mol
180.16 g
Grams from moles
NaCl has a molar mass of 58.44 g/mol
58.44 g
2.700 mol of NaCl × —————— = 157.8 g
1 mol
Find the molar mass of each compound (hint, see your own answers above), then follow the instructions given. Convert to moles
Convert to grams