CPChemistry
Math Tool Kit
Study and Practice Packet
I. SI Units
II. Derived Units
III. Metric Prefixes
IV. Scientific Notation
V. Significant Figures
VI. Counting Significant Figures
VII. Calculations with Significant Figures
VIII. Dimensional Analysis
IX. Conversion Factors
X. Calculating Percent Error
CPChemistry - Math Tool Kit - Study and Referemce Packet
I. Units of Measurement: 7 Fundamental SI Units
Property / SI Unit and Standard of Measurement / SymbolLength / meter / m
Mass / kilogram / kg
Time / second / s
Temperature / Kelvin / K
Amount of Substance / mole / mol
Current / ampere / A
Luminous Intensity / candela / cd
II. Derived Units
Property / Meaning / Derived Unit / SymbolArea / l x w / square meter / m2 / m x m
Volume / l x w x h / cubic meter / m3 / m x m x m
Volume (liquid) / cubic decimeter / dm3 / dm x dm x dm
Force / mass x acceleration / newton / N / 1N = 1kg-m/s2
Pressure / force/area / pascal / Pa / 1Pa = 1N/m2
Energy / force x distance / joule / J / 1J = 1N-m
Frequency / cycles/second / hertz / Hz / 1Hz = 1wave cycle/second
Density / mass/volume / m/s, km/hr, m/min, etc.
Speed / distance/time / kg/m3, g/cm3, g/mL, etc
III. Table of Metric Prefixes
Prefix: / Symbol: / Magnitude: / Meaning (multiply by):Tera- / T / 1012 / 1 000 000 000 000
Giga- / G / 109 / 1 000 000 000
Mega- / M / 106 / 1 000 000
kilo- / k / 103 / 1000
hecto- / h / 102 / 100
deka- / da / 101 / 10
- / - / - / -
deci- / d / 10-1 / 0.1
centi- / c / 10-2 / 0.01
milli- / m / 10-3 / 0.001
micro- / u (mu) / 10-6 / 0.000 001
nano- / n / 10-9 / 0.000 000 001
pico- / p / 10-12 / 0.000 000 000 001
femto- / f / 10-15 / 0.000 000 000 000 001
IV. Scientific Notation
The speed of light is approximately 300,000,000 meters per second. Working with a large number such as this can become cumbersome so we use scientific notation to represent very large and very small numbers. 300,000,000 m/s can also be written as: 3 x 100,000,000 or 3 x 108, where 8, the exponent, is the number of zeros.
Positive exponents
Large numbers can be written in scientific notation by moving the decimal point to the left. For example, Avogadro's number, 602,200,000,000,000,000,000,000, is central to chemistry. The decimal point that you don’t see is to the right of the last zero in the measurement.
The decimal point is moved left until you have a number between 1 and 10. In the example above, the decimal point was moved 23 places to the left. That number is now the positive exponent of the base 10.
Negative exponents
Numbers less than 1 can be expressed in scientific notation by moving the decimal to the right. In this instance, the decimal point needs to move to the right by 4 places to the first non-zero number. For every place we move the decimal to the right we decrease the power of ten by one, starting from zero. That number can be written as 7.2 x 10-4
Review: Power of 10 notation
For any positive whole number, n, 10n is 1 followed by n zeros. Remember a positive power of 10 means a large number, greater than 1.
100 = 1
101 = 10
102 = 100
103 = 1000
When 10 is raised to a negative power, the exponent tells you how many places after the decimal point to place the 1. Remember, a negative power of 10 means a small number, less than 1.
10-1 = 0.1
10-2 = 0.01
10-3 = 0.001
10-4 = 0.0001
N YOU TRY!!! N Practice Problems: Scientific Notation
Express the following in scientific notation. Remember to retain the same significant figures.
Ordinary notation / Scientific Notation1. / 137,000,000
2. / 0.000290
3. / 0.00000158
4. / 738
5. / 0.020
6. / 4200
7. / 7.050 x 10-3
8. / 4.00005 x 107
9. / 2.3500 x 104
10. / 1.15 x 10-3
V. Significant Figures
Some numbers are exact and some are not. For example, your family has exactly 5 people, your class has exactly 21 students, and there are exactly 100 centimeters in one meter. The last example is a conversion factor. There is no uncertainty in a conversion factor.
In chemistry lab this year you will be making measurements of mass, volume and temperature. Numbers that are obtained by making measurements are not exact. There is always uncertainty as a result of the limitations of the instrument scale and the skill of the technician reading the scale. Calculations made with measured values must be rounded off properly to the appropriate number of significant figures. Careful measurements together with rounding correctly make your reported measurements reliable.
The significant figures in a measurement are all of the digits known with certainty (those for which there is a marking on the scale) plus one digit which is estimated between the smallest markings.
VI. Counting Significant Figures
In numbers written with decimal points, count significant figures from the left beginning with the first nonzero digit. In numbers written without decimal points, count from the right beginning with the first nonzero digit.
Examples:
Measurements / Number of SFs0.0370 / 3
200 / 1
20. / 2
20.0 / 3
400,900 / 4
0.00990 / 3
NYOU TRY!!! NPractice Problems: Counting SGs
measurement / SFs1. / 0.23100
2. / 23100
3. / 23100.
4. / 7.203
5. / 0.00231
6. / 2000
VII. Calculating With Significant Figures:
☞ When you use your measurements in calculations, your answer may only be as exact as your least exact measurement!
RULE FOR ADDITION AND SUBTRACTION: Round to fewest decimal places.
Example / Unrounded answer / Rounded answer / Explanation4.1cm + 0.07cm / 4.17cm / 4.2cm / 4.1 has one decimal so answer rounded to tenths place
18.3m – 11m / 7.3m / 7m / 11 has no decimals so answer rounded to ones place
8.120g-7.090g / 1.03g / 1.030g / Both measurements have three decimals so answer should have three decimals
RULE FOR MULTIPLICATION AND DIVISION: Round to fewest significant figures. (abbreviated SFs)
Example / Unrounded answer / Rounded answer / Explanation4.1cm x 0.07cm / 0.287cm2 / 0.3 cm2 / 0.07 has one SF, answer rounded to one SF
7.079cm0.53s / 13.356603774cms / 13cms / 0.53 has two SFs, answer rounded to two SFs
8.120m x 7.090m / 57.5708m2 / 57.57 m2 / Both measurements have four SFs so answer should have four SFs
☞ Notice that units of measurement are carried through the calculations and shown in all results.
NYOU TRY!!! NPractice Problems: Calculations with SFs
Practice Problems: Using Significant Figures in Calculations
Example / Unrounded answer / Rounded answer / Explanation1 / 45.71cm x 0.20cm
2 / 10.2s0.4
3 / 100. mm – 1.6 mm
4 / 4302g + 0.837g
5 / 87.3cm – 1.655cm
6 / 2.099g + 0.05681g
7 / 2.4gmL x 15.82mL
8 / 105.725g39.1mL
VIII. Dimensional Analysis:
Units in science are sometimes called dimensions. Keeping track of units in calculations is called dimensional analysis.
When you multiply or divide numbers with units (measurements) you also multiply or divide the units.
Examples:
1. Area = length x width = 3cm x 2cm = 6cm2
2. 32 s4 s=8 (The time units, seconds, have divided out.)
3. Density = massvolume = 853.76g310.1mL = 2.7531763947 gmL = 2.753 gmL
(The units haven’t changed in the calculation and so are brought out in the result. The answer is rounded to the same number of SFs as the measurement with the fewest, the volume, which has four SFs.)
4. Convert 3.72 hours to seconds:
?s = 3.72h x 60min1h x 60s1min = 13,392s = 13,400s (Hours & minutes divided out)
IX. Conversion Factors
Conversion Factors can be used to convert from one unit of measurement to another. The ability to convert between units of measurement with confidence is essential in the lab, research, industry, and hospital settings. Another name for a conversion factor is Unit Equality. That’s because a conversion factor is equal to ONE; the original quantity will not be changed when you multiply it by a Unit Equality. The reciprocal of a unit equality is also equal to one! Conversion factors are written like fractions.
Example: Convert the height of a student from inches to centimeters:
? cm =63in x 2.54cm1in=160cm ?cm=63in x 1cm0.3937in =160cm
X. Calculating Percent Error (Percent Difference)
Use the equation: Percent Error=measured value-accepted valueaccepted value x 100
Example:
Your teacher asks you to demonstrate your skill on the balance by massing an object in the lab. You measure its mass 6.878g but the teacher’s measurement was 6.085g. If the teacher’s value is the accepted value, what is the percent error in your mass measurement?
Percent Error=6.878-6.0856.085 = 0.7936.085=13.0% with 3SFs
It’s important to show the intermediate step above because this step determines the number of SFs in your answer. If your answer is a negative value it simply means that your result is lower than the true value.
N YOU TRY!!! N Practice Problems: Percent Error
1. What is the percent error of a length measurement of 0.229cm if the correct value is 0.225cm?
2. A handbook gives the density of calcium as 1.54 gmL. Lab measurements resulted in a density of 1.25 gmL. What is the percent error?
YOUR NAME:
Practice Problems: Unit conversions using dimensional analysis
Convert 0.000830m to cm.Convert 7.56kg to g.
Convert 4.02 hours to seconds.
Add 9.78m to 245cm. Express your answer in cm.
Convert 10km to miles.
Convert 36.7miles to kilometers.
The distance from wing to wing is 0.75 mi. How many cm is this?
A Motrin tablet is 200. mg. How many ounces is this?
The price of gas in Germany is $2.118 per liter. What is the price per gallon?
12