math and measurement

Math and Measurement

Unit 2

10 mL Graduate What is the volume of liquid in the graduate?

_ . _ _ mL6 26

Self TestExamine the meniscus below and determine the volume of liquid contained in the graduated cylinder.

The cylinder contains:

_ _ . _ mL7 6 0

Reading the Thermometer

Determine the readings as shown below on Celsius thermometers:

_ _ . _ C _ _ . _ C 8 7 4 3 5 0

Numbers…which ones are important?

What is 13/7?

Is it 1.8571428?

Or…is it 1.86? Or 1.9? Or 2?

Where do we round?

Significant Digits

Rules for Counting Significant Rules for Counting Significant Figures - DetailsFigures - Details

• Exact numbersExact numbers have an infinite have an infinite number of significant figures. number of significant figures.

How many beakers are on the How many beakers are on the shelf?shelf?

Exactly 5. Exactly 5.

Rules for Counting Significant Rules for Counting Significant Figures - DetailsFigures - Details

• Nonzero integersNonzero integers always count as always count as significant figures. significant figures.

34563456 hashas

44 sig figs.sig figs.

Rules for Counting Significant Rules for Counting Significant Figures - DetailsFigures - Details

• Zeros Zeros -- Leading zerosLeading zeros do not count as do not count as

significant figuressignificant figures..

• 0.04860.0486 has has

33 sig figs. sig figs.

Rules for Counting Significant Rules for Counting Significant Figures - DetailsFigures - Details

• Zeros Zeros -- Captive zerosCaptive zeros always count always count

as as significant figures.significant figures.

• 16.0716.07 has has

44 sig figs. sig figs.

Rules for Counting Significant Rules for Counting Significant Figures - DetailsFigures - Details

• Zeros Zeros Trailing zerosTrailing zeros are significant only if are significant only if the number contains a decimal the number contains a decimal point. point.

9.3009.300 has has

44 sig figs. sig figs.

Sig Fig Practice #1Sig Fig Practice #1How many significant figures in each of the following?

1.0070 m 5 sig figs

17.10 kg 4 sig figs

100,890 L 5 sig figs

3.29 x 103 s 3 sig figs

0.0054 cm 2 sig figs

3,200,000 2 sig figs

Rules for Significant Figures in Rules for Significant Figures in Mathematical OperationsMathematical Operations

• Multiplication and DivisionMultiplication and Division:: # sig figs in # sig figs in the result equals the number in the least the result equals the number in the least precise measurement used in the precise measurement used in the calculation. calculation.

6.38 x 2.0 = 6.38 x 2.0 =

12.76 12.76 13 (2 sig figs)13 (2 sig figs)

Sig Fig Practice #2Sig Fig Practice #2

3.24 m x 7.0 m

Calculation Calculator says: Answer

22.68 m2 23 m2

100.0 g ÷ 23.7 cm3 4.219409283 g/cm3 4.22 g/cm3

0.02 cm x 2.371 cm 0.04742 cm2 0.05 cm2

710 m ÷ 3.0 s 236.6666667 m/s 240 m/s

1818.2 lb x 3.23 ft 5872.786 lb·ft 5870 lb·ft

1.030 g ÷ 2.87 mL 0.3588850174 g/mL 0.359 g/mL

Rules for Significant Figures in Rules for Significant Figures in Mathematical OperationsMathematical Operations

• Addition and SubtractionAddition and Subtraction: The number : The number of decimal places in the result equals the of decimal places in the result equals the number of decimal places in the least number of decimal places in the least precise measurement. precise measurement.

6.8 + 11.934 = 6.8 + 11.934 =

18.734 18.734 18.7 ( 18.7 (3 sig figs3 sig figs))

Sig Fig Practice #3Sig Fig Practice #3

3.24 m + 7.0 m

Calculation Calculator says: Answer

10.24 m 10.2 m

100.0 g - 23.73 g 76.27 g 76.3 g

0.02 cm + 2.371 cm 2.391 cm 2.39 cm

713.1 L - 3.872 L 709.228 L 709.2 L

1818.2 lb + 3.37 lb 1821.57 lb 1821.6 lb

2.030 mL - 1.870 mL 0.16 mL 0.160 mL

Practice Question

Questions 1-2 refer to the following sets of numbers.

A.1.023 gB.0.0030 mLC.40,500 m

1.Is a number containing three significant figures2.Is a measure of mass

In science, we deal with some very In science, we deal with some very LARGELARGE numbers:numbers:

1 mole = 6020000000000000000000001 mole = 602000000000000000000000

In science, we deal with some very In science, we deal with some very SMALLSMALL numbers:numbers:

Mass of an electron = Mass of an electron = 0.000000000000000000000000000000091 kg0.000000000000000000000000000000091 kg

Scientific NotationScientific Notation

Imagine the difficulty of calculating the Imagine the difficulty of calculating the mass of 1 mole of electrons!mass of 1 mole of electrons!

0.000000000000000000000000000000091 kg 0.000000000000000000000000000000091 kg x 602000000000000000000000x 602000000000000000000000 ???????????????????????????????????

Scientific Notation:Scientific Notation:A method of representing very large or very small A method of representing very large or very small numbers in the form: numbers in the form:

M x 10M x 10nn

MM is a number between is a number between 11 and and 10 10 nn is an integer is an integer

Step #1: Insert an understood decimal pointStep #1: Insert an understood decimal point

.

Step #2: Decide where the decimal must end Step #2: Decide where the decimal must end up so that one number is to its leftup so that one number is to its left

Step #3: Count how many places you bounce Step #3: Count how many places you bounce the decimal pointthe decimal point

123456789

Step #4: Re-write in the form M x 10Step #4: Re-write in the form M x 10nn

2 500 000 000

2.5 x 102.5 x 1099

The exponent is the number of places we moved the decimal.

0.00005790.0000579

Step #2: Decide where the decimal must end Step #2: Decide where the decimal must end up so that one number is to its leftup so that one number is to its left

Step #3: Count how many places you bounce Step #3: Count how many places you bounce the decimal pointthe decimal point

Step #4: Re-write in the form M x 10Step #4: Re-write in the form M x 10nn

1 2 3 4 5

5.79 x 105.79 x 10-5-5

The exponent is negative because the number we started with was less than 1.

If the number is larger than 1, the exponent is positive.If the number is smaller than 1, the exponent is negative

Pause for a Cause Scientific Notation #1. Write the following numbers in

scientific notation:a. 0.000 673 0b. 50 000.0c. 0.000 003 010

#2. The following numbers are in scientific notation. Write them in ordinary notation.a. 7.050 X 103 gb. 4.000 05 X 107 mgc. 2.350 0 X 104 mL

Multiplying and Dividing in Scientific Notation

1. Multiply or divide the “M” values

2. If multiplying, add the exponents

3. If dividing, subtract the exponents.

4. If necessary, adjust to put back in scientific notation.

Example #1

(1.35 x 104) x (2.35 x 105)

Example #2

(2.6 x 108) / (4.6 x 103)

You try1. (6.00 X 106) x (4.0 X 10-3)

2. (3.2 x 104) x (4.5 x 105)

3. ( 4.5 x 10-5) / (9 x 10-3)

Nature of MeasurementNature of Measurement

Part 1 – Part 1 – numbernumber

Part 2 – Part 2 – unitunit

Examples: Examples: 2020 gramsgrams

6.63 x 106.63 x 10-34-34 Joule secondsJoule seconds

Measurement – quantitative observation Measurement – quantitative observation consisting of 2 partsconsisting of 2 parts

Accurate or Precise?Accurate or Precise?Accurate measurements are close to the actual or accepted value.

Precise measurements are close to one another.

More than one measurement must be taken to determine if the measurements are precise.

Pause for a Cause Accuracy and Precision

A handbook gives the density of calcium as 1.54 g/cm3. A student runs three experiments and determines the density to be 2.25 g/cm3, 2.28 g/cm3 and 2.20 g/cm3. Discuss this student’s accuracy and precision.

A student measures the mass of a sample as 9.67 grams, 9.99 grams and 8.85 grams. The actual mass is 7.50 grams. Discuss this student’s accuracy and precision.

The Fundamental SI UnitsThe Fundamental SI Units (le Système International, SI)(le Système International, SI)

SI Prefixes Common to ChemistryPrefix Abbr. Exponent MeaningGiga G 109 1,000,000,000

Mega M 106 1,000,000

Kilo k 103 1,000

Deci d 10-1 1/10

Centi c 10-2 1/100

Milli m 10-3 1/1000

Micro 10-6 1/1,000,000

Nano n 10-9 1/ 1,000,000,000

Now Let’s Do Metrics the Right Way!

Practice Conversion Factors

Express a mass of 5.712 grams in milligrams.

Given: 5.712 g

Unknown: mass in mg

quantity given × conversion factor = quantity sought

Conversion Factor 1 g = 1,000 mg

5.712 g 1000 mg

1 g= 5712 mg

Practice Conversion Factors

Express a mass of 5.712 grams in kilograms.

quantity given × conversion factor = quantity sought

5.712 g 1 kg

1000 g= 0.005712 kg

Let’s do the following metric conversions using unit cancellation.

1. 2500 grams = ___ kilograms

2. 5600 centimeters = ____ meters

3. 500 liters = ____ milliliters

4. 75000 milligrams = ____ kilograms

5. 25 meters = ____ decimeters

6. 4.5 kilograms = ____ grams

7. 45 decimeters = ____ centigrams

Pause for a Cause 2 Now Its your turn!

1. 1500 centigrams = ___ grams

2. 6.00 milliliters = ____ liters

3. 700 meters = ____ kilometers

4. 750 milligrams = ____ centigrams

5. 250 meters = ____ decimeters

6. 0.25 kilograms = ____ grams

7. 0.75 decimeters = ____ millimeters

8. 47 grams = ____ centigrams

9. 2.5 meters = ____ millimeters

10. 500 deciliters = ____ liters

11. 250 kilograms = ____ grams

12. 3500 centigrams = ____ kilograms

Which is more dense?Select the object that you think has the highest

density and write one sentence explaining your answer.

Styrofoam cup

Rock

Gatorade

Derived SI Units• Produced by multiplying or dividing standard

units.For Example:

Area = (Length)(Width)

Leng

th

Width

5 m

2.5 m

Area = (2.5 m)(5 m) = 12.5 m2

What does density describe?

Density describes how tightly

particles are packed within a

sample of matter.

Density

The ratio of mass to volume, or mass divided by volume.

Density = mass

volumeD =

mV

Density

• A measure of how closely matter is packed into a volume.

• Unique for each compound.– Density of water is 1.00 g/mL at 25˚C.– Increasing temperature decreases the density, so

densities are given with temperatures.• An intensive property.• Substances that are less dense float in

substances more dense.

Density Problems

D = mv

A sample of aluminum metal has a mass of 8.4 g. The volume of the sample is 3.1 cm3. Calculate the density

of aluminum.

D = 8.4 g

3.1 cm3

Don’t forget units!

D =

Don’t forget units!!!

Given:

Unknown:

Box answer!

m = 8.4 gV = 3.1 cm3

D = ?

Equation: 2.7 g/cm3

You try!

An unknown liquid is discovered at a crime scene. A volume of 2.3 mL has a mass of 4.1

grams, what the liquid’s density?

Density ProblemsDiamond has a density of 3.26 g/cm3. What is the mass

of a diamond that has a volume of 0.350 cm3?

Density Problems

A sample of metal is found to have a mass of 4.56 g and a density of 1.98 g/mL. What is the

volume of this metal?

Density Problem (No calculator)

The typical battery in a car is filled with a solution of sulfuric acid, which is approximately 39.9% sulfuric acid. If the density of this solution is 1.3 g/mL, determine the number of grams of acid present in 500.0 mL of battery solution.

1. What is the volume of 5 grams of this substance?

2. What is the approximate density of the substance?

Converting TemperaturesC = 5/9 (F-32)

F =

K = C + 273

C =

You Try!

Convert the following temperatures

1. 293 K to Celsius2. Room temperature to

Celsius3. Internal body

temperature to Kelvin

Answers

How many seconds are there in exactly 1 year?

Steps to complete these problems

Step 1: Read the problem CAREFULLY.Step 2: Determine the unit for the answerStep 3: Write down all values given in the problem

and retrieve any needed conversion factorsStep 4: Set up the problem (watch carefully as teacher

does this step)Step 5: Calculate—Multiply by numbers on the top

and divide by those on the bottom

The record long jump is 349.5 inches. Convert this to meters. There are 2.54 cm in an inch.

Practice #1

Practice #2

A car is traveling 55.0 miles per hour. Convert this to meters per second. One mile is equal to 1.61 km.

Practice #3How many mg are there is a 5.00

grain aspirin tablet?1 grain = 0.00229 oz.There are 454 grams/lb. There are

16 oz./lb

Practice #4

Convert 24 km/h to m/s (write out all steps before using calculator).

Practice #5

In 1980, the US produced 18.4 billion (18.4 X 109) pounds of phosphoric acid to be used in the manufacture of fertilizer. The average cost of the acid is $318/ton. (1 ton = 2000 lbs). What was the total value of the phosphoric acid produced?

1. Light travels at a speed of 300,000 km/sec. What distance in centimeters does light travel in a year?

2. A landfill can hold 4.8 X108 m3 of trash. If 250 000 000 objects averaging 0.060 m3 each are placed in the landfill each year, how many years will it take to fill the landfill?

3. A baker uses 1.5 tsp. of vanilla extract in each cake. How much vanilla in liters should the baker order to make 800 cakes? (1 tsp. = 5 mL)

4. A person drinks 8 glasses of water each day, and each glass contains 300 mL. How many liters of water will that person consume in one year?

5. What is the speed of a car in m/sec when it is moving 100. km/hr?

Pause for a CauseStudent Practice on Dimensional Analysis