Measurements and Calculations Chapter 2a Measurements and Calculations

2.1 Scientific Notation 2.2 Units 2.3 Measurements of Length, Volume, and Mass 2.4 Uncertainty in Measurement 2.5 Significant Figures

Measurement Quantitative observation. Has 2 parts – number and unit. Number tells comparison. Unit tells scale.

Technique used to express very large or very small numbers. Expresses a number as a product of a number between 1 and 10 and the appropriate power of 10.

Using Scientific Notation Any number can be represented as the product of a number between 1 and 10 and a power of 10 (either positive or negative). The power of 10 depends on the number of places the decimal point is moved and in which direction.

Using Scientific Notation The number of places the decimal point is moved determines the power of 10. The direction of the move determines whether the power of 10 is positive or negative.

Using Scientific Notation If the decimal point is moved to the left, the power of 10 is positive. 345 = 3.45 × 102 very large number If the decimal point is moved to the right, the power of 10 is negative. 0.0671 = 6.71 × 10–2 very small number In Webassign homework use format: 345 = 3.45e02 0.0671 = 6.71e-02

Concept Check Which of the following correctly expresses 7,882 in scientific notation? 7.882 × 104 788.2 × 103 7.882 × 103 7.882 × 10–3 The correct answer is c. The decimal point should be moved three places to the left to be correctly expressed in scientific notation.

Concept Check Which of the following correctly expresses 0.0000496 in scientific notation? 4.96 × 10–5 4.96 × 10–6 4.96 × 10–7 496 × 107 The correct answer is a. The decimal point should be moved five places to the right to be correctly expressed in scientific notation.

Precision vs. Accuracy good precision poor precision good precision poor accuracy good accuracy good accuracy

There is no such thing as a totally accurate measurement! Measurement Accuracy How long is this line? There is no such thing as a totally accurate measurement!

Nature of Measurement Measurement Quantitative observation consisting of two parts. number scale (unit) Examples 20 grams 6.63 × 10–34 joule·seconds If a CHP asks you what do you have and you answer I have 3 kilos, you may go to jail. You should have said I have 3 kg of doughnuts for my chemistry instructor.

Measurement in Chemistry lll Measurement in Chemistry Length Mass Volume Time meter gram Liter second Km=1000m Kg=1000g KL=1000L 1min=60sec 100cm=1m 1000mg=1 g 1000mL=1L 60min=1hr 1000mm=1m SI System http://www.kickstarter.com/projects/52746223/the-state-of-the-unit-the-kilogram-documentary-fil Foot pound gallon second British 12in=1ft 16oz=1 lb 4qt=1gal (same) 3ft=1yd 2000 lb=1 ton 2pts=1qt 5280ft=1mile

Conversion between British and SI Units Measurement in Chemistry Conversion between British and SI Units 2.54 cm = 1 in 454 g = 1 lb 1 (cm)3 = 1 cc = 1 ml = 1 gwater 1.06 qt = 1 L

Prefixes Used in the SI System Prefixes are used to change the size of the unit.

Length Fundamental SI unit of length is the meter.

Volume Measure of the amount of 3-D space occupied by a substance. SI unit = cubic meter (m3) Commonly measure solid volume in cm3. 1 mL = 1 cm3 1 L = 1 dm3

Mass Measure of the amount of matter present in an object. SI unit = kilogram (kg) 1 kg = 2.2046 lbs 1 lb = 453.59 g

A gallon of milk is equal to about 4 L of milk. Concept Check Choose the statement(s) that contain improper use(s) of commonly used units (doesn’t make sense)? A gallon of milk is equal to about 4 L of milk. A 200-lb man has a mass of about 90 kg. A basketball player has a height of 7 m tall. A nickel is 6.5 cm thick. A basketball player cannot be 7 m tall (but rather 7 feet tall). There are about 3 feet in a meter so it’s more likely that the player has a height of 2.3 m. A nickel cannot be 6.5 cm thick. There are 10 mm in every cm, so it’s more likely that the nickel could be about 3 mm thick. Copyright © Cengage Learning. All rights reserved

A digit that must be estimated is called uncertain. A measurement always has some degree of uncertainty. Record the certain digits and the first uncertain digit (the estimated number).

Measurement of Length Using a Ruler The length of the pin occurs at about 2.85 cm. Certain digits: 2.85 Uncertain digit: 2.85 Estimate between smallest division! Copyright © Cengage Learning. All rights reserved

Numbers that measure or contribute to our accuracy. Significant Figures Numbers that measure or contribute to our accuracy. The more significant figures we have the more accurate our measurement. Significant figures are determined by our measurement device or technique. Copyright © Cengage Learning. All rights reserved

Rules of Determining the Number of Significant Figures 1. All non-zero digits are significant. 234 = 3 sig figs 1.333 = 4 sig figs 1,234.2 = 5 sig figs 2. All zeros between non-zero digits are significant. 203 = 3 sig figs 1.003 = 4 sig figs 1,030.2 = 5 sig figs

Rules of Determining the Number of Significant Figures 3. All zeros to the right of the decimal and to the right of the last non-zero digit are significant. 2.30 = 3 sig figs 1.000 = 4 sig figs 3.4500 = 5 sig figs 4. All zeros to the left of the first non-zero digit are NOT significant. 0.0200 = 3 sig figs 0.1220 = 4 sig figs 0.000000012210 = 5 sig figs

Rules of Determining the Number of Significant Figures Zeros to the right of the first non-zero digit and to the left of the decimal may or may not be significant. They must be written in scientific notation. 2300 = 2.3 x 103 or 2.30 x 103 or 2.300 x 103 2 sig figs 3 sig figs 4 sig figs

Rules of Determining the Number of Significant Figures 6. Some numbers have infinite significant figures or are exact numbers. 233 people 14 cats (unless in biology lab) 7 cars on the highway 36 schools in town

How many significant figures are in each of the following? 1) 23.34 4 significant figures 2) 21.003 5 significant figures 3) .0003030 4 significant figures 4) 210 2 or 3 significant figures 5) 200 students infinite significant figures 1, 2, 3, or 4 significant figures 6) 3000

Measurements and Calculations Chapter 2b Measurements and Calculations

2.5 Significant Figures 2.6 Problem Solving and Dimensional Analysis 2.7 Temperature Conversions: An Approach to Problem Solving 2.8 Density

Using Significant Figures in Calculations Addition and Subtraction Line up the decimals. Add or subtract. Round off to first full column. 23.345 +14.5 + 0.523 = ? 23.345 14.5 + 0.523 38.368 = 38.4 or three significant figures

Using Significant Figures in Calculations Multiplication and Division Do the multiplication or division. Round answer off to the same number of significant figures as the least number in the data. (23.345)(14.5)(0.523) = ? 177.0368075 = 177 or three significant figures

Rules for Rounding Off 1. If the digit to be removed is less than 5, the preceding digit stays the same. 5.64 rounds to 5.6 (if final result to 2 sig figs)

Rules for Rounding Off 1. If the digit to be removed is equal to or greater than 5, the preceding digit is increased by 1. 5.64 rounds to 5.6 (if final result to 2 sig figs) 3.861 rounds to 3.9 (if final result to 2 sig figs)

Rules for Rounding Off 2. In a series of calculations, do within the parenthesis first and determine the significant figures and use that answer to calculate and find the significant figures after the multiplication and/or division.

Concept Check You have water in each graduated cylinder shown. You then add both samples to a beaker (assume that all of the liquid is transferred). How would you write the number describing the total volume? 3.08 mL What limits the precision of the total volume? 2.80 1st graduated cylinder + .280 2ndgraduated cylinder 3.080 or 3.08 ml The total volume is 3.1 mL. The first graduated cylinder limits the precision of the total volume with a volume of 2.8 mL. The second graduated cylinder has a volume of 0.28 mL. Therefore, the final volume must be 3.1 mL since the first volume is limited to the tenths place.

Example #1 A golfer putted a golf ball 6.8 ft across a green. How many inches does this represent? To convert from one unit to another, use the equivalence statement that relates the two units. 1 ft = 12 in The two unit factors are:

Example #1 A golfer putted a golf ball 6.8 ft across a green. How many inches does this represent? Choose the appropriate conversion factor by looking at the direction of the required change (make sure the unwanted units cancel). Copyright © Cengage Learning. All rights reserved

Correct sig figs? Does my answer make sense? Example #1 A golfer putted a golf ball 6.8 ft across a green. How many inches does this represent? Multiply the quantity to be converted by the conversion factor to give the quantity with the desired units. Correct sig figs? Does my answer make sense? Copyright © Cengage Learning. All rights reserved

Example #2 An iron sample has a mass of 4.50 lb. What is the mass of this sample in grams? (1 kg = 2.2046 lbs; 1 kg = 1000 g) 454 g OR 4.50 lbs x -------------- = 2043g = 2.04x103 g 1 lb

Sample Answer: Concept Check What data would you need to estimate the money you would spend on gasoline to drive your car from New York to Los Angeles? Provide estimates of values and a sample calculation. Sample Answer: Distance between New York and Los Angeles: 2500 miles Average gas mileage: 25 miles per gallon Average cost of gasoline: $3.25 per gallon This problem requires that the students think about how they will solve the problem before they can plug numbers into an equation. A sample answer is: Distance between New York and Los Angeles: 2500 miles Average gas mileage: 25 miles per gallon Average cost of gasoline: $3.25 per gallon (2500 mi) × (1 gal/25 mi) × ($3.25/1 gal) = $325 Total cost = $325 = $(3.3x102)

Three Systems for Measuring Temperature Fahrenheit Celsius Kelvin Gabriel Fahrenheit Lord Kelvin Copyright © Cengage Learning. All rights reserved

The Three Major Temperature Scales F = 1.8C + 32 C = (F-32)/1.8 K = C + 273 What is 35oC in oF? 95 oF What is 90oF in oC? 32oC What is 100K in oC? -173oC

a) 373 K b) 312 K c) 289 K d) 202 K Exercise The normal body temperature for a dog is approximately 102oF. What is this equivalent to on the Kelvin temperature scale? a) 373 K b) 312 K c) 289 K d) 202 K C = (F-32)/1.8 = (102-32)/1.80 = 38.9oC K = C + 273 = 38.9 + 273 = 312 K The correct answer is b. (102 – 32) / 1.80 = 39°C 39 + 273 = 312 K

At what temperature does C = F? Exercise At what temperature does C = F? The answer is -40. Since °C equals °F, they both should be the same value (designated as variable x). Use one of the conversion equations such as °C = (°F-32)(5/9), and substitute in the value of x for both °C and °F. Solve for x. Copyright © Cengage Learning. All rights reserved

Use one of the conversion equations such as: Solution Since °C equals °F, they both should be the same value (designated as variable x). Use one of the conversion equations such as: Substitute in the value of x for both T°C and T°F. Solve for x. Copyright © Cengage Learning. All rights reserved

Solution 1.80x = x -32 0.80x = -32 x = -32/0.80 So –40°C = –40°F Copyright © Cengage Learning. All rights reserved

Mass of substance per unit volume of the substance. Common units are g/cm3 or g/mL. Copyright © Cengage Learning. All rights reserved

Measuring the Volume of a Solid Object by Water Displacement

Example #1 A certain mineral has a mass of 17.8 g and a volume of 2.35 cm3. What is the density of this mineral? Copyright © Cengage Learning. All rights reserved

Example #2 What is the mass of a 49.6 mL sample of a liquid, which has a density of 0.85 g/mL? OR g

Exercise If an object has a mass of 243.8 g and occupies a volume of 0.125 L, what is the density of this object in g/cm3? a) 0.513 b) 1.95 c) 30.5 d) 1950 The correct answer is b. Density = mass/volume. First convert 0.125 L to cm3. 0.125 L × (1000 mL/1 L) × (1 cm3/1mL) = 125 cm3 Density = 243.8 g / 125 cm3 = 1.95 g/cm3

Using Density as a Conversion Factor How many lbs of sugar is in 945 gallons of 60.0 Brix (% sugar) orange concentrate if the density of the concentrate is 1.2854 g/mL? 4 qt 1 gal 1 L 1.06qt 1000 mL 1 L 1.2854 gT 1 mL 60.0 gS 100 gT 1 lbs 454gS 945 gal = 6057.865514lbs = 6.06 x 103 lbs sugar lbs of what? Coffee? Cocaine?

Using Density as a Conversion Factor Using the Formula How many lbs of sugar is in 256 L of 60.0 Brix (% sugar) orange concentrate if the density of the concentrate is 1.2854 g/mL? M D = V Solve for Mass DV = M (1.2854 g/mL)(256,000 mL) = 329062.4 gT = 3.29 x 105 gT 1 lbT 454 gT 60.0 lbsS 100 lbsT = 434.8017621 lbsS 3.29 x 105 gT = 4.35 x 102 lbsS = 435 lbsS

Concept Check a) 8.4 mL b) 41.6 mL c) 58.4 mL d) 83.7 mL Copper has a density of 8.96 g/cm3. If 75.0 g of copper is added to 50.0 mL of water in a graduated cylinder, to what volume reading will the water level in the cylinder rise? a) 8.4 mL b) 41.6 mL c) 58.4 mL d) 83.7 mL The correct answer is c. Using the density and mass of copper, determine the volume of metal present. 75.0 g × (cm3/8.96 g) = 8.37 cm3 Since the density of water is 1 g/1 mL, the volume of the metal can be determined by displacement. Therefore, the water level will rise to 58.4 mL (50.0 + 8.37 mL). 8.37 mL Cu + 50.0 mL water = 58.4 mL