Math Concepts

How can a chemist achieve exactness in measurements? Significant Digits/figures. Significant Digits/figures. Sig figs = the reliable numbers in a measurement and at least one estimated digit. Sig figs = the reliable numbers in a measurement and at least one estimated digit.

Make readings for the following measurements using significant figures. Make readings for the following measurements using significant figures.

Rules for significant figures 1. All non-zero numbers or digits are significant. Ex: 23 g 1. All non-zero numbers or digits are significant. Ex: 23 g 2. All zero in-between 2 non-zero numbers are significant. Ex: 2.002g 2. All zero in-between 2 non-zero numbers are significant. Ex: 2.002g 3. When working with a small decimal number, work your way over to the right until you get to your 1 st non-zero number - anything from there over is significant. Ex: g 3. When working with a small decimal number, work your way over to the right until you get to your 1 st non-zero number - anything from there over is significant. Ex: g 4. Final zeros 25.00g are significant. 4. Final zeros 25.00g are significant. 5. When working with large numbers (no decimals), look for your 1 st non-zero number – anything from there to the beginning of the number are significant. Ex: g 5. When working with large numbers (no decimals), look for your 1 st non-zero number – anything from there to the beginning of the number are significant. Ex: g 6. A line/bar over or under a zero designates it as significant. 6. A line/bar over or under a zero designates it as significant. 7. Exact numbers = numbers that you are use to working with are unlimited in terms of significant figs. Ex: there are 12 men on the football field. = unlimited. 7. Exact numbers = numbers that you are use to working with are unlimited in terms of significant figs. Ex: there are 12 men on the football field. = unlimited.

Significant Figures An easy way to count the number of significant figures in any number is: An easy way to count the number of significant figures in any number is: DOT LEFT – NOT RIGHT *If there is a visible decimal, look all the way to the left of the value and move to the right. Begin counting digits after your first non-zero digit. Any numbers that follow a non-zero digit are significant. EX: = 4 sig figs = 5 sig figs

*If there is no visible decimal, look all the way to the right of the value and move to the left. Begin counting digits after your first non-zero digit. Any numbers that precede a non-zero digit are significant. EX: 2500 = 2 sig figs = 1 sig figs 5001 = 4 sig figs

Examples How many sig figs are in the following: How many sig figs are in the following: 20 kg 20 kg 1 sig fig 1 sig fig g g 2 sig figs 2 sig figs 11 m 11 m 2 sig figs 2 sig figs s s 2 sig figs 2 sig figs 90.4˚C 90.4˚C 3 sig figs 3 sig figs cm cm 1 sig fig 1 sig fig kg 2 sig figs l 3 sig figs 100.0˚C 4 sig figs 20 cars unlimited cm 6 sig figs 5310 g 3 sig figs m 4 sig figs

If an exponential number, look at coefficient only. If an exponential number, look at coefficient only. If decimal at end all numbers are significant. If decimal at end all numbers are significant. A line over a zero indicates that zero as the last significant digit. A line over a zero indicates that zero as the last significant digit. Use decimal or line, not both. Use decimal or line, not both. No lines over nonzero digits. No lines over nonzero digits.

Calculations using sig figs Adding or subtracting: Adding or subtracting: Look at the decimal places. Choose the given information that has the least number of decimal places. Make sure to put your answers in the least number of decimals. Your calculator does not do this! Your final measurement can not be more specific than your least specific measurement! Look at the decimal places. Choose the given information that has the least number of decimal places. Make sure to put your answers in the least number of decimals. Your calculator does not do this! Your final measurement can not be more specific than your least specific measurement! Multiplying or dividing: Multiplying or dividing: Identify sig figs for each number in your information. Your answer needs to be altered to the least number of sig figs used when solving the problem. (for the same reason) Identify sig figs for each number in your information. Your answer needs to be altered to the least number of sig figs used when solving the problem. (for the same reason)

Addition Division Multiplication Subtraction

Practice: 1. Give the correct number of significant figures for: = ? = ? 4. (1.3 x 103)(5.724 x 104) = ? 5. (6305)/(0.010) = ?

Scientific Notation Why is it that we use scientific notation in science? because many of the numbers, amount, etc. that we use are either really big or very small. Examples: Distance from the Earth to the Sun, size of an atom, the mass of an electron, proton, or even neutron…..

Scientific Notation If the number is large – you will have a positive exponent If the number is large – you will have a positive exponent If the number is very small – you will have a negative exponent. If the number is very small – you will have a negative exponent. Exponent decides which direction and how many spots you will move the decimal Exponent decides which direction and how many spots you will move the decimal EX: = 1 x = 4.4 x Must honor sig figs in original value Must honor sig figs in original value Root number or coefficient is the only number that is significant (exponent does not count) Root number or coefficient is the only number that is significant (exponent does not count) EX: x sig figs 7.8 x sig figs 2 sig figs

Examples What is the correct scientific notation for: What is the correct scientific notation for: What is the correct standard notation for: What is the correct standard notation for: 1.98 x x x x x x x x x x 10 -3

Calculations with scientific notation Multiplication: multiply the coefficients(roots) and add your exponents Multiplication: multiply the coefficients(roots) and add your exponents Division: divide the coefficients(roots) and subtract your exponents Division: divide the coefficients(roots) and subtract your exponents Add or subtract: Change your exponents to equal (largest one), then add and put back into correct scientific notation. OR put your numbers in standard notation +/- and then place back into scientific notation Add or subtract: Change your exponents to equal (largest one), then add and put back into correct scientific notation. OR put your numbers in standard notation +/- and then place back into scientific notation

Practice: (2.68 x ) x (4.40 x ) (2.68 x ) x (4.40 x ) (2.95 x 10 7 ) ÷ (6.28 x ) (2.95 x 10 7 ) ÷ (6.28 x ) (8.41 x 10 6 ) x (5.02 x ) (8.41 x 10 6 ) x (5.02 x ) (9.21 x ) ÷ (7.60 x 10 5 ) (9.21 x ) ÷ (7.60 x 10 5 ) (4.52 x ) + (1.24 x ) + (3.70 x ) + (1.74 x ) (4.52 x ) + (1.24 x ) + (3.70 x ) + (1.74 x ) (2.71 x 10 6 ) - (5.00 x 10 4 ) (2.71 x 10 6 ) - (5.00 x 10 4 ) (4.56 x 10 6 ) + (2.98 x 10 5 ) + (3.65 x 10 4 ) + (7.21 x 10 3 ) (4.56 x 10 6 ) + (2.98 x 10 5 ) + (3.65 x 10 4 ) + (7.21 x 10 3 ) (3.05 x 10 6 ) x (4.55 x ) (3.05 x 10 6 ) x (4.55 x )

How can you decide if your experiments are accurate/precise? Percent error = calculations that will give you a percent deviation from the true value. Percent error = calculations that will give you a percent deviation from the true value. Formula: l True – experimental l x 100 Formula: l True – experimental l x 100 True True

Example A student measured the density of an object to be g/ml, the true density of the object is 2.699g/ml. What is the percent error of the experiment? Is the student accurate? ANSWER: 7.040% error, anything below 10% is acceptable as accurate. The closer to 0% the better!

Metric Conversions

Conversion Practice: *honor sig figs Conversion Practice: *honor sig figs 550 cm  m 1500 mL  liters 3500 mg  g liters  mL 1.50 m  cm liters  mL 40 mL  liters 1500 mL  cm cm 3  mL 2560 cm 3  liters

Dimensional Analysis Used to convert between units of measurement using equivalent values. Used to convert between units of measurement using equivalent values. EX: Convert grams into pounds Step 1: Place given value over 1. Step 2: Select appropriate conversion factor (454 grams = 1 lb) and place in parenthesis so that the unit of the given will cancel with the same unit in the conversion factor. Step 3: Continue conversion factors until the only unit remaining is the one that you want. Step 4: Divide the product of the numerator by the product of the denominator. Step 5: Express answer in correct sig figs and unit.

800.0 g ( 1 lb ) =1.762 lb 1(454 g ) 1(454 g ) Common Conversion Factors: Common Conversion Factors: 2.54 cm = 1 inch16 oz = 1 lb 454 grams = 1 pound12 inch = 1 ft 5280 feet = 1 mile1 L = 1.06 quarts 4 quarts = 1 gallon

Convert: Convert: 25.0 inches  cm 2.45 pounds  grams 2500 grams  pounds cm  inches 750 cm  feet cm  miles years  minutes *45 miles/hour  km/minute