Chapter-2: Analyzing Data (measurements) Dr. Chirie Sumanasekera CHEMISTRY Lecture_4 Dr. Chirie Sumanasekera 8/30/2018 TOPICS covered today: Scientific Notation Exponent-Math review Dimensional Analysis

Scientific Notation Scientists have to often work with extremely large or extremely small numbers EX: Astrophysicists calculating the distance from earth to the sun in meters =149600000000000 m Or the height in km of a person who is 5 feet 9 inches tall = in cm = 175.3 cm in m = in km = 1.753 m 0.001753 km Can we write the person’s height in km and the distance from the earth to the Sun in meters in a easier way? Yes We CAN, using Scientific Notation!

1.851 x 10-8 m Coefficient Exponent Scientific Notation Base Scientific notation = all numbers are expressed as a coefficient between 1 and 10 multiplied by 10 raised to a power. Note 1.851 x 10-8 m Coefficient Exponent The coefficient must have a single digit to the left of the decimal point. Base *In scientific notation, the exponent of a number tells you how many spaces you have to move the decimal. Ex: Write the following regular notation numbers in scientific notation: 149600000000000 m = 175.3 cm = 1.753 m = 0.001753km = 1.496 x 1014 m 1.753 x 102 cm 1.753 x 101 m 1.753 x 10-3 m

Math Review: 1. EXPONENTS Exponent: a number written as a superscript that indicates a power Exponent Base Power Meaning Actual value 103 10 3 10 x 10 x 10 1000 102 2 10 x 10 100 101 1 10-1 -1 1/10 0.1 10-2 -2 1/10x10 0.01 10-3 -3 1/10 x 10 x 10 0.001

Math Review: 1. EXPONENTS

Math Review 2. Division and Multiplication in Scientific Notation For division and multiplication can be performed as long as the base has the same number 1. Dividing exponents: when the base is the same we subtract denominator exponent by numerator exponent: Ex-1: 103 = 103 – (101) = 10 (3-1) = 102 101 Ex-2: (8 x 104) ÷ (2 x 10-2) = 8 ÷ 2 (104-(-2)) = 4 x (104+2) = 4.0 x 106 Ex-3: (2 x 104) ÷ (4 x 104) = 2 ÷ 4 (104) = 0.5 x 104 = 0.5 x 101 x103 = 5.0 x 103 2. Multiplying exponents: when the base is the same we add the powers : Ex-1: 103 x 101 = 10 (3+1) = 104 Ex-2: 103 x 10-4 = 10 3+(-4) = 10 [3-4] = 10 -1 Ex-3: (5 x 105) x (3.1 x 10-4) = (5 x 3.1)x 10 5+(-4) = 15.5 x 10 (5-4) = 15.5 x 101 = 1.55 x 101 x 101 = 1.55 x 101+1 = 1.55 x 102

Math Review 3. Addition and subtraction in Scientific Notation For addition and subtraction, both the base and the exponents have to be identical 3. Adding exponents: add only when power of 10 is the same: Ex-1: (1.2 x 103) + (4 x 103) = 1.2 + 4 (103) = 5.5 x 103 Ex-2: (1.2 x 103) + (4 x 10-1) = (1.2 x 104 x 10-1) + (4 x 10-1) = = 12000 + 4 (10-1) =12004 x 10-1 = 1.2004 x 104 x10-1 = 1.2004 x 10 [4+(-1)] = 1.2004 x 10 (4-1) =1.2004 x 103 EX-3: (1.2 x 103) + (4 x 105) = (1.2 x 103) + (4x 102 x 103) =1.2 + 400 (103) = 401.2 (103) = 4.012 x 102 x 103= 4.012 x 10 (2+3) = 4.012 x 105 4. subtracting exponents: subtract only when power of 10 is the same: Ex-1: (1x 103) – (2x 101) = (1x 100 x 101) - (2 x 101) = 100 -2 x (101) = 98 x (101)= 9.8 x 102 Ex-2: (6x103 ) - (2x103 ) = 6 -2 x (103)= 4.0 x 103 Ex-3: (6x103 ) - (2x10-2 ) = (6x 105x10-2)- (2x10-2) = [6x105 – 2] x(10-2) = 600000 -2 (10-2)= 599998 x (10-2)= 5.99998 x 105x 10-2= 5.99998 x105 +(-2) = 5.99998 x 103

Math Review: Dimensional analysis Solve these: Number of seconds are in 2.0 years 2. Convert 4.5 mg/ml into Kg/L

Dimensional Analysis Dimensional Analysis (Unit Factor Method) is a problem-solving method that uses concept that any number (or factor) can be multiplied by 1 without changing its value. Unit factors may be made from any two terms that describe the same or equivalent "amounts" of what we are interested in. Ex: 1 inch = 2.54 cm There are exactly 2.540000000... cm in 1 inch. We can make two unit factors from this information:

Solving Problems with Dimensional Analysis Write down what you need to find with a question mark. Then set it equal to the information that you are given. The problem is solved by multiplying the given data and its units by the appropriate unit factors so that only the desired units are present at the end after the other units cancel out. (1) How many centimeters are in 6.00 inches? (2) Express 24.0 cm in inches.

Solving Problems with Dimensional Analysis You can also string many unit factors together. (3) How many seconds are in 2.0 years?  (4) Convert 50.0 mL to liters. (This is a very common conversion.)

Solving Problems with Dimensional Analysis (5) How many atoms of hydrogen can be found in 45 g of ammonia, NH3? We will need three unit factors to do this calculation, derived from the following information: 1 mole of NH3 has a mass of 17 grams. 1 mole of NH3 contains 6.02 x 1023 molecules of NH3. 1 molecule of NH3 has 3 atoms of hydrogen in it.

Today we examined how to perform calculation using: Scientific Notation Dimensional Analysis