The Math of Chemistry
Yup! That’s right. Math! There are a few basic math skills that we want to brush up before we ask you to use
them for chemistry. Today we will cover three: scientific notation, metric conversions, and significant figures.
REVIEW: Algebraic Manipulation of equations
Rearrange the following equations to solve for the variable that is in bold/italics.
PV = nRT
D=
Mass V E = ½ mv2
v1 = √ M2 √ M1 v2 Scientific Notation
• What is it? Scientific notation is the way that scientists easily handle very large numbers or very small
numbers. For example, instead of writing 299,800,000 we write 2.998 x 108 OR instead of
0.0000000056, we write 5.6 x 10-9.
• So, how does this work?
o We can think of 5.6 x 10-9 as the product of two numbers: 5.6 (the digit term) and 10-9 (the
exponential term).
o As you can see, the exponent of 10 is the number of places the decimal point must be shifted to
give the number in long form. All empty spots are filled in with zeros.
§ A positive exponent shows that the decimal point is shifted that number of places to the
right.
• 2.998 x 108 = 299,800,000
§ A negative exponent shows that the decimal point is shifted that number of places to the
left.
• 5.6 x 10-9 = 0.0000000056
o Examples: Convert the following numbers into scientific notation.
a) .00000357
o
a) 6.357 x 10-3
•
b) 12345.678
c) 100
Examples: Convert the following from scientific notation into numbers.
b) 1.234 x 105
c) 2.3 x 10-8
How to do calculations:
o Multiplication
§ The digit terms (numbers out front) are multiplied just like normal and the exponents are
added together. If necessary, the new digit term is changed so that there is only one
nonzero digit to the left of the decimal.
§ Example: (3.4 x 106)(4.2 x 103) = (3.4)(4.2) x 10(6+3) = 14.28 x 109 = 1.4 x 1010
§ Examples:
• (6.73 x 10-5)(2.91 x 102) =
o
Division
§ The digit terms (numbers out front) are divided just like normal and the exponents are
subtracted. The quotient is changed (if necessary) so that there is only one nonzero digit
to the left of the decimal.
§ Example: (6.4 x 106)/(8.9 x 102) = (6.4)/(8.9) x 10(6-2) = 0.719 x 104 = 7.2 x 103
§ Examples:
• (3.2 x 103)/(5.7 x 10-2) =
o
On your calculator
§ Make sure that the number in scientific notation is put into your calculator
correctly. This will make your life SO much easier.
§ Graphing calculators (yellow)
• Type in the digit term (the numbers out front)
• Hit the second button followed by the comma button (right above the #7)
o This should now show and E on the screen.
o The E stands for the “x 10”
• Now type in the exponent (if it’s negative, make sure to enter the negative sign
THEN the number)
• Your final number should look like this: 5.6 x 10-6 à 5.6E-6
§ Scientific calculators (blue)
• Type in the digit term (the numbers out front)
• Hit the second button followed by the x-1 button (right above the #7)
o This should now show and E on the screen.
o The E stands for the “x 10”
• Now type in the exponent (if it’s negative, make sure to enter the number THEN
the negative sign)
§ Your final number should look like this: 5.6 x 10-6 à 5.6E-6
§ When entering scientific notation into QUEST, you will need to type it as it appears in the
calculator (5.6E-6).
Metric Conversions
• Conversion is the changing of one unit to another.
• Metric units are the universally excepted units around the world (except for the US)
o They are the easiest to covert because everything is based on multiples of 10.
§ The basic units of length, volume and mass can be multiplied or divided by multiples of
10 to measure objects that are bigger or smaller.
• Length – meters
• Volume – liters
• Mass – grams
Metric Prefixes:
Prefix
Giga
Symbol
Scientific Notation
G
Mega
M
Kilo
K
1 x 10
9
1 x 10
6
2
Decimal
1,000,000,000
1,000
Hecto
H
1 x 10
Deca
Da
1 X 10
1
Base Units – g, L, m
If you know this chart, the scientific notation will always be with the base unit and the 1 will be with the prefix!
Deci
d
.1
•
Centi
c
Milli
m
1 x 10
-2
1 x 10
-6
.001
Micro
μ
Nano
n
.000000001
Pico
p
.000000000001
Convert 350 mL into liters.
350 mL
1L
=
1000 mL
0.35 L
•
Examples:
a) Convert 378.4 cm to meters
-4
b) Convert 4.32 x 10 g to micrograms
c) Convert 88.1 km to meters
Significant Digits
• Please remember that, in science, most numbers are based upon measurements. Since all
measurements are somewhat uncertain, we must only use those numbers that are meaningful.
o Not all of the digits have significance and, therefore, should not be written down.
o In science, only the numbers that are derived from accurate measurements are written.
§ For example, a common ruler cannot measure something to be 22.4072643 cm long, so
we shouldn’t write down all of those digits because that would be claiming that you FOR
SURE know all of those decimal places.
• Significant figures are critical when reporting scientific data because they give the
reader an idea of how well you could actually measure/report your data.
• QUEST will require you to answer in proper sig figs, so if you’re missing a question, double check your
sig figs!
• Rules for determining the number of significant digits:
o Non-zero digits are always significant
o Sandwiched zeros are significant (a zero between non-zero digits)
o Trailing zeros do not count unless they are after a decimal
o Leading zeros do not count (Read from the left and start counting when you encounter the first
non-zero digit)
o Examples: Determine the number of sig figs for the following numbers.
a) .0000404
•
c) 100.0
d) 4500023
Rule for addition/subtraction problems
o When adding or subtracting numbers, count the NUMBER OF DECIMAL PLACES to determine
the number of significant figures. The answer cannot CONTAIN MORE PLACES AFTER THE
DECIMAL POINT THAN THE SMALLEST NUMBER OF DECIMAL PLACES in the numbers
being added or subtracted.
o Examples: Calculate the following problems and round the answers to proper sig figs.
a) 287.99 + 303.001 + 1000 =
•
b) 1000
b) 24.6 + 605.004 + 77.7 + 99.99 =
Rules for multiplication/division problems
o When multiplying or dividing numbers, count the NUMBER OF SIGNIFICANT FIGURES. The
answer cannot CONTAIN MORE SIGNIFICANT FIGURES THAN THE NUMBER BEING
MULTIPLIED OR DIVIDED with the LEAST NUMBER OF SIGNIFICANT FIGURES.
o Examples: Calculate the following problems and round the answers to proper sig figs.
a) 102.5 x 100 x 1002 =
b) 1000.0 x 23.56 x 45.1002 =