# Significant figures. Significant figures. Rounding off numbers. How many significant figures in these measurements? inches. 4.

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1 Significant figures All non-zero numbers are always significant 2.38 has three significant figures 25 has two significant figures Are zeros significant? It depends on their position in the number. A zero is significant when it is: between non-zero digits 205 has three significant figures has four significant figures at the end of a number that includes a decimal point 1.50 has three significant figures 30. has two significant figures Significant figures All non-zero numbers are always significant 2.38 has three significant figures 25 has two significant figures Are zeros significant? It depends on their position in the number. A zero is not significant when it is: before the first non-zero digit (leading zeros) 0.03 has one significant figure has two significant figures at the end of a number without a decimal point 2500 has two significant figures 30 has one significant figures How many significant figures in these measurements? 4.5 miles inches Rounding off numbers When we do calculations, we often obtain answers that have excess digits -- i.e., more digits than the correct number of significant figures gallons F 15 pencils centimeters 1000 inches 1.0 x 10 3 inches x 10 3 inches 1 x 10 3 inches Example: = inches We must drop the excess digits so the answer will have the correct number of significant figures rounding off numbers -- the process of determining the value of the last digit that is retained after dropping the excess (non-significant) digits

2 Rules for rounding off numbers Rule 1: If the first of the digits to be dropped is less than 5 (i.e., 0, 1, 2, 3, or 4), drop the excess digits and don t do anything to the last retained digit. Example: Round off inches to 3 significant figures inches Rules for rounding off numbers Rule 2: If the first of the digits to be dropped is 5 or greater (i.e., 5, 6, 7, 8, or 9), drop the excess digits and increase the last retained digit by one. Example: Round off inches to 3 significant figures inches less than 5 drop these digits greater than or equal to 5 drop these digits Do not round up: 5.38 inches Round up: 8.32 inches CAUTION: Common mistakes involving zeroes when rounding off numbers When you are just putting a number into scientific notation, do NOT change the number of significant figures Example: Express the number 62,301 in scientific notation 62,301 (5 significant figures) x 10 4 (5 significant figures) NOT x 10 4 (4 significant figures) 6.23 x 10 4 (3 significant figures) CAUTION: Common mistakes involving zeroes when rounding off numbers After rounding off, do not drop zeroes that are significant Example: Round off inches to 4 significant figures greater than or equal to inches Round up: inches drop these digits (4 significant figures -- Correct) 5.63 inches (3 significant figures -- Incorrect) x 10 4 (5 significant figures)

3 CAUTION: Common mistakes involving zeroes when rounding off numbers After rounding off, do not drop zeroes that are needed as place holders Example: Round off 354 inches to 2 significant figures 354 inches To help you remember this, think of an example involving money... Your friend owes you exactly \$5268, but you both agree to settle the debt to the nearest \$100 (two significant figures in this case) Round off \$5268 to 2 significant figures (i.e., to the hundreds place) \$5268 less than 5 drop this digit greater than or equal to 5 drop these digits Do not round up: 350 inches (2 significant figures -- Correct order of magnitude) 35 inches (2 significant figures -- Incorrect order of magnitude) Round up: \$5300 (2 significant figures -- Correct order of magnitude) \$53 (2 significant figures -- Incorrect order of magnitude) Even though the number of significant figures is correct, the value of the measurement is now off by a factor of 10 Even though the number of significant figures is correct, the value of the measurement is now off by two factors of 10 Significant figures in calculations The results of a calculation based on measurements can not be more precise than the least precise measurement. For multiplication and division: The answer must contain the same number of significant figures as the measurement with the least number of significant figures cm 3 x 4.3 g/cm 3 = g Significant figures in calculations The results of a calculation based on measurements can not be more precise than the least precise measurement. For addition and subtraction: The answer must be rounded to the same number of decimal places as the measurement with the lowest number of decimal places inches inches = inches 3 sig figs 2 sig figs Answer must have 2 significant figures 3 decimal places 1 decimal place Answer must be rounded to one decimal place Round to 95 g (or 9.5 x 10 1 g) Round to inches

4 Significant figures in calculations 3.1 miles/gallon x 84.5 gallons = Unit conversions To convert a measurement from one type of unit to another type of unit you must use a conversion factor Example: How may centimeters are there in 5.30 inches? gallons gallons = 1 in = 2.54 cm 1. To obtain conversion factor, start with a known equality 3.12 grams / 7.0 milliliters = meters meters = 1 in = 2.54 cm 2. Divide both sides by the quantity on the right (or left) 1 in 1 in hand side of the equation 3. Cancel out numbers and units 2.54 cm 1 = 1 in 4. Multiply this conversion factor (which is equal to one) by your measurement 2.54 cm 5.30 in( 1 in ) = 13.5 cm and cancel units to get an equivalent value with the desired units Unit conversions To convert a measurement from one type of unit to another type of unit you must use a conversion factor Example: How may liters are there in 10.0 gallons? Density density -- the ratio of the of a substance to the occupied by that -- i.e., the per unit of a substance 1 gal = L 1. To obtain conversion factor, start with a known equality 1 gal = L 1 gal 1 gal 1 = L 1 gal L 10.0 gal ( 1 gal ) = L = 37.9 L 2. Divide both sides by the quantity on the right (or left) hand side of the equation 3. Cancel out numbers and units 4. Multiply this conversion factor (which is equal to one) by your measurement and cancel units to get an equivalent value with the desired units 2 cm = 16 g 2 cm = 8 cm 3 2 cm density (!) = / = 16 g / 8 cm 3 = 2 g / cm 3

5 Density and weight are not the same Q: What weighs more, a ton of feathers or a ton of lead? A: Neither -- they both weigh the same ( 1 ton ) Density and weight are not the same density (!) = / For a given, the lead will weigh more since it has a greater density 1 ton 1 m 3 1 ton 1 m 3 Density and weight are not the same So you can say that lead has a greater density than feathers (density is a characteristic property of a substance) but you can t say that lead weighs more than feathers (weight depends on how much of the substance is present) Note: Density varies with temperature The of a substance (especially liquids and gasses) varies with temperature -- i.e., increases as temperature increases density (!) = / 1 lb 1 ton = 100 g = 100 ml 1 g / ml = 100 g = 150 ml 0.67 g / ml

6 Density calculations Density calculations For problems involving density: you will typically be given the values for two of these variables (or the information to calculate their values) you then have to solve for the value of the third variable Example: A cube of metal has a of 8.0 g and a of 2.0 cm 3. What is the density of the metal? ( 8.0 g ) ( 2.0 cm 3 ) = 4.0 g / cm 3 A cube of silver has sides long and weighs 10.5 g = 10.5 g = 3? What is the density of silver? / = 10.5 g / 3 = 10.5 g / cm 3 =? = g / cm 3 Density calculations A cube of silver has sides long. The density of silver is 10.5 g / cm 3. What is the of the cube? 10.5 g = ( 3 ) 10.5 g / cm 3 = ( 3 ) 3 Determining density by observations / To find the density of any object, determine its and -- then divide! -- can be determined using a balance 50.0 g -- can be determined by measuring the dimensions of the object 2.0 cm 5.0 cm = 1.0 cm x 5.0 cm x 2.0 cm = 10. cm g / 10. cm 3 = 5.0 g / cm cm

7 Determining density by observations Sample density problems / To find the density of any object, determine its and -- then divide! -- can be determined using a balance -- can also be determined by water displacement 20 ml 20 ml Example: Methanol has a density of g / cm 3. What is the of 10.0 ml of methanol? g / cm 3 = 10.0 cm g 10 ml 10 ml 10.0 cm 3 x g / cm 3 = x 10.0 cm cm 3 = 20. cm cm 3 = 10. cm g / 10. cm 3 = 5.0 g / cm g = Sample density problems Sample density problems Example: An empty beaker has a of 20.0 g. After pouring some water in the beaker and putting in back on the balance, you find that its is 50.0 g. Water has a density of 1.00 g / cm 3. What is the of the water in the beaker? Example: An empty beaker has a of 20.0 g. After pouring some water in the beaker and putting in back on the balance, you find that its is 50.0 g. Water has a density of 1.00 g / cm 3. What is the of the water in the beaker? x 1.00 g / cm 3 = 30.0 g x x 1.00 g / cm 3 = 30.0 g of beaker and water of beaker = of water 1.00 g / cm g / cm g 20.0 g = 30.0 g = 30.0 cm 3 or 30.0 ml

8 Non-miscible fluids with different densities Note: If two fluids are not miscible (i.e., they form two separate phases), the fluid with lower density will float on top of the fluid with higher density. Mixtures with water ( density of water = 1.00 g / ml ) Homework Assignment Chapter 2 Problems: 2.26, 2.34, 2.44, 2.45, 2.46, 2.47, 2.49, 2.50, 2.55, 2.56, , 2.74 Homogeneous mixture of water and ethanol cyclohexane water water methylene chloride Ethanol Density = 0.79 g/ml Soluble in water Cyclohexane Density = 0.78 g/ml Insoluble in water Methylene chloride Density = 1.32 g/ml Insoluble in water

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