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Measurement And Chemical Calculations

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MeasurementAnd

Chemical Calculations

Measurement in Every Day Life

Measurement: • How tall are you?• What is the temperature outside?• How many inches of rain did Lubbock get?

Exact number: No uncertainty• 7 days in a week • 24 hours in a day• 100 cents in a dollar

Depending on who is doing the measurement therecould be a difference in the reported value.

Types of Observations and

• We make QUALITATIVE observations of reactions — changes in color and physical state.

• We also make QUANTITATIVE MEASUREMENTS, which involve numbers.

• Use SI units — based on the metric system

UNITS OF MEASUREMENTUse SI units — based on the

metric systemLength Mass

Time

Meter, m

Kilogram, kg

Seconds, sCelsius degrees, ˚Ckelvins, K

Exponential Notation• Use exponents to represent very large

or very small numbers.

Table 3-1, p. 52

Table 3-2, p. 62

Table 3-3, p. 75

Fig. 3-3c, p. 63

Fig. 3-2, p. 63

Fig. 3-5, p. 63

Fig. 3-6, p. 64

Fig. 3-1, p. 61

English System:Units for measurement used in the United StatesMetric System:Units for measurement used in the rest of the worldSI Units:International system of units based on the metric system• Mass• Length• Temperature• Time• Amount

Connecting everything together are numbers and units!

Basic Units of Measure

Numbers and UnitsNumber:How much of something ie one dollar

Unit:How much of WHAT ie one dozen eggs, 100 bucks, etc

2 grams

number unit

• Mass (kilograms)• Length (meters)• Temperature (Kelvin)• Time (seconds)• Amount (moles)

Scientific NotationUsed to express both small and large numbers.

Example:The mass of a single He atom is

0.000000000000000000000000664 grams

24 leading zeros make this number very small and very difficult to enter into a calculator!

6.64 x 10-25

decimal part:value between 1 and 10

exponential

exponent: a whole number

(positive or negative)

Scientific NotationSingle digit between 1 and 10 before the decimal point

Multiply or divide the number by 10

Exponent equals number of “moved” decimal places

Positive exponent = move left to put into scientific notationNegative exponent = move right to put into scientific notation

Example:Express the following numbers in scientific notation.

Write each of these numbers in scientific notation.

Using Scientific Notation

0.000873

To get a number between 1-10, move the decimal how manyplaces, in what direction?

0.000873 = 8.73 x 10-4

4310000

To get a number between 1-10, move the decimal how manyplaces, in what direction?

4310000 = 4.31 x 106

Changing Back to DecimalsThe size of the exponent shows how to move the decimal

Positive exponent = large numberNegative exponent = small number

When switching back to decimal form, move in the opposite direction!

3.49 x 10-11 = 0.0000000000349

5.28 x 103 = 5280

6.72 x 10-1 = 0.672

1.29 x 108 = 129000000

ExponentsWhen two numbers with exponents are multiplied, the product is

the multiple of the base raised to a power equal to the sum of the exponents.

Example:10a x 10b = 10a+b

102 x 103 = 105

(3.54 x 107)(1.43 x 102)First, multiply 3.54 x 1.43 to get the value of the base

3.54 x 1.43 = 5.06Second, add the exponents together

7 + 2 = 9Finally, combine all the numbers together correctly

5.06 x 109 = 5060000000

A Harder Example:

ExponentsWhen two numbers with exponents are divided, the product is the

dividend of the base raised to a power equal to the difference of the exponents.

Example:10c ÷ 10d = 10c-d

106 ÷ 103 = 10(6-3) = 103

(7.35 x 106) ÷ (3.43 x 104)First, divide 7.35 ÷ 3.43 to get the value of the base

7.35 ÷ 3.43 = 2.14Second, subtract the exponents

6 - 4 = 2Finally, combine all the numbers together correctly

2.14 x 102 = 214

A Harder Example:

Combine the Ideas(9.41 x 103)(1.21 x 10-5)

(342)(2.66 x 10-7)

First, multiply 9.41 x 1.21 to get the value of the base on top9.41 x 1.21 = 11.4 = 1.14 x 101

Second, add the exponents together3 + -5 + 1 = -1

Third, convert 342 to scientific notation342 = 3.42 x 102

Fourth, multiply 3.42 x 2.66 to get the value of the base below3.42 x 2.66 = 9.10

Fifth, add the exponents together2 + -7 = -5

Sixth, rewrite the problem1.14 x 10-1

9.10 x 10-5

Seventh, divide 1.14 ÷ 9.10 to get the final base1.14 ÷ 9.10 = 0.125

Eighth, subtract the exponents(-1) – (-5) = 4

Finally, combine all the numbers together correctly0.125 x 104 = 1.25 x 103 = 1250

Very Tricky:

Adding Numbers with ExponentsIf adding/subtracting numbers without a calculator, align the digits

vertically. Adjust the coefficients and exponents so that all the numbers are raised to the same power.

39710000 + 19800

39729800

3.971 x 107 + 1.98 x 104 = ?

3.971 x 107 + 1.98 x 104 = 3.973 x 107

0.000105 - 0.000097

0.000008

1.05 x 10-4 – 9.7 x 10-5 = ?

1.05 x 10-4 – 9.7 x 10-5 = 8 x 10-6

Significant FiguresIn science, it is important to make accurate measurements

and record them correctly.

Every measurement has some degree of uncertainty (or error).However, depending on the measuring device,

the error can be reduced.

Every digit in a number is known accurately except the last digit, which is estimated (or uncertain).

Example:

A nut weighs 1.8 gramsWe record the weight as 1.8 ± 0.1 g

A nut weighs 1.81 gramsWe record weight as 1.81 ± 0.01 g

Board is ~2/3 the length of the meter stick, so length is 0.6-0.7 m

0.6 ± 0.1 m

Lines are added to the meter stick every tenth of a meter; the board is between 60-70 cm

Estimate closest tenth0.64 ± 0.01 m

Millimeter lines are added; the board is between

64.3-64.4 cm Best Estimate

0.643 ± 0.001 m

Showing Uncertainty

Centimeter lines are added to the meter stick; the board

is between 64-65 cmEstimate closest tenth

0.642 ± 0.001 m

Significant FiguresThe location of the decimal point has nothing to do with significant

figures.

0.643 m and 64.3 cm both have 3 sigfigs

Begin counting sigfigs at the first nonzero digit.All nonzero digits are significant!

345 has 3 sigfigs

All zeros between nonzero digits ARE significant.

305 and 3.05 both have 3 sigfigs

Zeros at the beginning of a decimal number are NOT significant.

0.000643 km has 3 sigfigs

leading zeros are NOT significant

Significant Figures: Decimal Points

643,000,000 nm has 3 sigfigs

trailing zeros are NOT significant—Use exponential notation to remove ambiguity

If the final zero were not significant, is should not be recorded.

Zeros at the end of a large number are NOT significant, though often ambiguous.

0.67 has 2 sigfigs while 0.670 has 3 sigfigs

Zeros written at the end of a number AFTER the decimal point ARE significant.

If you are not sure about a zero, write the number in scientific notation. All non-significant zeros will be eliminated.

546,000 = 5.46 x 105 = 3 sigfigs

Significant Figures: Practice Time!How many significant figures are in the following quantities?

1.002 L

36.4 cm

6.022 x 1023 atoms

2.88790 x 108 m/sec

0.003440 cm

4 sigfigs

3 sigfigs

4 sigfigs

6 sigfigs

4 sigfigs

How do we use sigfigs in calculations?

Exact NumbersSignificant figures do NOT apply to exact numbers.

Exact numbers have no uncertainty, they were not obtained by measurement.

Exact numbers have an infinite number of sigfigs.

Example:1 foot = 12 inches

1 dozen = 12 objects1 hour = 60 minutes

Any property based on a measurement is not exact!

Exact NumbersWhich of the following quantities represent exact numbers?

The density of water at 70 oC is 0.97778 g/mL.

14 people are going to the men’s BB game.

The distance from Lubbock to Amarillo is 124 miles.

The width of a human hair is 150 µm wide.

Not exact

Exact

Not exact

Not exact

There are 60 seconds in a minute.

Exact

Sigfigs: AdditionThe number of sigfigs is based on the position of the digits.

15.9994 + 1.00797

17.00737

The numbers being added only have 4 decimal places in common.

5281 + 18.05

13040.95

+7699 + 42.9

Align the decimal places

Final answer: 17.0074

Final answer: 13041

The sigfig stops at the decimal so only 5 numbers are significant.

Rounding NumbersShould the last significant digit remain the same, or be rounded to

the next highest number?

If the digit AFTER the last sigfig is less than 5, keep the last sigfig the way it is.

Four sigfigs: 13,672 becomes 13,670

If the digit AFTER the last sigfig is greater than 5, round the last sigfig to the next highest number.

Four sigfigs: 1.0058 becomes 1.006Three sigfigs: 3.799 becomes 3.80

If the digit AFTER the last sigfig is 5, round the last sigfig to the next highest number.

Four sigfigs: 6.7455 becomes 6.746

Sigfigs: SubtractionThe number of sigfigs is based on the position of the digits.

Same rules apply for addition and subtraction!

319.542 - 20.460

253.4181

- 0.0639- 45.6

Final answer: 253.4

If there are multiple steps in the calculation, only round the final answer.

Determine the proper number of sigfigs at the end of the calculation.

Sigfigs: MultiplicationThe number of sigfigs is based on the number of sigfigs of the

quantities being multiplied.

Final answer: 0.698 (3 sigfigs)

The answer should be limited to the lowest number of significant digits of the values used.

Exact numbers have an infinite number of sigfigs, so they do not affect the number of sigfigs in the final answer.

(38.6)(0.009037)(2.00) = 0.6979564

0.04201 x 68700 = 2886.087Final answer: 2890 (3 sigfigs) or 2.89 X 103

Sigfigs: DivisionThe number of sigfigs is based on the number of sigfigs of the

quantities being divided. Same rules as multiplication!

Final answer: 3.14 (3 sigfigs)

223.0 = 3.14084507042 71.0

Final answer: -0.0257 or -2.57 x 10-2 (3 sigfigs)

-(8.314)(298.15) = -0.02568724456 96,500

What if we combine addition, subtraction, multiplication or division?

Sigfigs: Everything Together

4.32 – 56.92 x (22.87 – 22.73)

Evaluate the following expression:

Solve the problem in parentheses first.

(22.87 – 22.73) = 0.14

Next, perform the multiplication.

56.92 x 0.14 = 7.9688

Finally, perform the subtraction.

4.32 – 7.9688 = -3.6488 = -3.65

only 2 sigfigs

Final answerlimited to the

hundredth place.

You can limit sigfigs at each step of a calculation, but that may lead you to a different answer at the end!

Sigfigs: Practice!

4.35 + 2.2975.1 – 1.66

1.97 x 3.904(8.42 + 11.2) x 1.6

5.11 / 3.0

6.647 = 6.65 (decimal places)

3.44 = 3.4 (decimal places)

7.691 = 7.69 (sigfigs)

a. 19.62 (can keep for now)

b. 31.392 = 31 (sigfigs)

1.70 = 1.7 (sigfigs)

Questions:

Answers:

Dimensional AnalysisIn a problem, identify the given and wanted quantities that are

related by a PER expression.

How many days are in 28 weeks?Example:

28 weeks x 7 days1 week

= 196 days

conversion factor: written as a fraction; used to change aquantity of one unit to an equivalent amount of the other unit.

7 days1 week

1 week7 days

Dimensional Analysis: Check Units!Always include units in your calculation setup.

If the units don’t make sense, the answer is wrong!

28 weeks x 1 week7 days

= 4 weeks2

How many days are in 14 weeks?Example:

days

The number of days must be larger than the number of weeks!

nonsense units!

When setting up a dimensional analysis problem, make sure the units cancel correctly!

Determine what information is given and what information is wanted.

Setting up Calculations

If a car travels at an average speed of 74 miles per hour, how far will it go in 8 hours?

Given: 8 hours Wanted: miles driven

8 hours x 74 miles1 hour

= 592 miles

How many dollars are in 1,624 quarters?

Given: 1,624 quarters Wanted: number of dollars

1,624 quarters x 1 dollar4 quarters

= 406 dollars

Proportional ReasoningProportional: Any change in either X or Y will result in a

corresponding change in the other.

Y X

Y = m x X

If X increases, Y increases.

proportionality constant

We can rearrange the equation to solve for the constant m.

m = YX

α

Inversely proportional: As one variable is increased, the other is decreased.

s 1t

Proportional Reasoning

Explore the relationship between the time (t) it takes to drive a given distance (d) at a certain speed (s).

Example:

Driving at a higher speed means it takes less time to get somewhere.

180 milesx 1 hour 40 miles

= 4.5 hours

180 milesx 1 hour 60 miles

= 3 hours

α s = d x 1t

d = s x t

If the pressure of a sample of gas is held constant, its volume (V) is directly proportional to the Kelvin temperature (T).

Proportional Reasoning

What volume will this gas sample occupy at a temperature of 392 K?

V = aT

For 28.6 g of CH4 gas at a pressure of 0.171 atm, V isobserved to be 248 L when T is 290 K. What is the value

of a, and what are its units?a = V = 248 L

T 290 K= 0.855 L/K

Write an equation for the proportionality between V & T,where a is the proportionality constant.

V = aT = 0.855 L/K x 392 K = 335 L

Metric Units: MassThe SI unit of mass is the kilogram, kg.

1 kg = 2.2 pounds

1 kg = 1000 grams

1 g = 0.001 kg

A kilogram is defined as the mass of a platinum-iridium cylinder stored in France.

The SI unit of length is the meter, m.

Metric Units: Length

A meter is defined as the distance light travels in a vacuum in 1/299,792,468 second.

1 m = 39.37 inches

1 km = 1000 meters

1 km = 0.621 miles

1 centimeter (cm) is the width of a fingernail

1 millimeter (mm) is the thickness of a dime

2.54 cm = 1 inch

The SI unit of volume is the cubic meter, m3.

Metric Units: Volume

A m3 is too large a volume in the laboratory, so chemists use the cubic centimeter, cm3.

Liquids and gases are not easy to weigh, so we measure the volume of space they occupy.

A teaspoon holds approximately 5 cm3.

1 L = 1000 cm3

1 L = 1000 mL

1 L = 1.057 quarts

Volumetric glassware

Conversions with the Metric System

How many mm are in 51.5 cm?

Given: 51.5 cm Wanted: mm

51.5 cm x 1 m100 cm

= 515 mm

Learn to convert quickly between metric units. Use dimensional analysis!

x 1000 mm1 m

51.5 cm = 515 mmx 1000 mm100 cm

OR combine conversion factors:

Let’s try another example!

Conversions with the Metric SystemSoda is sold in bottles that contain 2.00 L of fluid.

Express the volume in cubic centimeters and in quarts.

Given: 2 L Wanted: cm3

2 L x 1000 cm3

1 L= 2000 cm3

Given: 2 L Wanted: quarts

2 L x 1.057 quarts1 L

= 2.11 quarts

Check: Liters are larger than cm3 therefore there should be less liters.

Conversions Between SystemsLearn to convert between the United States Customary System

& the Metric System.

How many inches are in 23.65 cm?

Given: 23.65 cm Wanted: inches

23.65 cm x 1 inch2.54 cm

= 9.311 inches

How many ounces are in 124.3 grams?

Given: 124.3 g Wanted: ounces (oz)

124.3 g x 1 lb453.59 g

= 4.385 oz x 16 oz1 lb

TemperatureFahrenheit (°F): Water freezes at 32 °F and boils at 212 °F.

Celsius (°C): Water freezes at 0 °C and boils at 100 °C.

Kelvin (K): Water freezes at 273 K and boils at 373 K.

Temperature ConversionsWhat are the relationships between temperature scales?

Converting between °F and °C:

T°F -32 = 1.8 T°CFahrenheit temp Celsius temp

What is the temperature in Celsius when the thermometer at a picnic reads 65 °F?

Given: 65 °F Wanted: °C

T°C = T°F - 32

1.8

T°C = 65 °F - 32

1.8

= 18.3 °C = 18 °C

What are the relationships between temperature scales?

Converting between °C and K:

TK = T°C + 273Kelvin temp Celsius temp

What is the temperature in Celsius when the Kelvin temperature is 234 K?

Given: 234 K Wanted: °C

T°C = TK - 273

T°C = 234 K - 273 = -39 °C

The Kelvin scale is also called the absolute temperature scale

On the Kelvin scale, do not use a degree (°) symbol.

because it is based on zero as the lowest possible temperature.

Temperature Conversions

Temperature ConversionsConvert -118 °F to K:

T°C = -118 °F - 32

1.8

= -83.3 °C = -83 °C

TK = -83 °C + 273 = 190 K

Convert 32 K to °F:

T°C = 32 K - 273= -241 °C

T°F = 1.8 T°C + 32

T°F = 1.8 (-241 °C) + 32 = -401.8 °F = -402 °F

DensityThe ratio of mass to volume.

Density = mass

volume

Density can be thought of as the relative “heaviness” of a substance.

A block of iron is heavier than a block ofaluminum of the same size, due to the densities of the two substances.

If you weigh out 6.12 grams of cooking oil

Density = 6.12 g

8.14 mL= 0.752 g/mL

measuring cup, the density of the oil is:and it takes up a volume of 8.14 mL in the

Density ProblemsIn chemistry lab you are asked to identify a piece of

metal. You decide to calculate the density of the metal to determine its identity.

The piece of metal weighs 198.4 grams. When you drop it in a cup of water, the metal displaces 18.7 mL of water.

What is the density of the metal? What metal is it?

Density = 198.4 g

18.7 mL= 10.6 g/mL

Substance Density (g/mL) Substance Density (g/mL)

Water 1.00 Lead 11.34

Aluminum 2.72 Mercury 13.60

Chromium 7.25 Gold 19.28

Nickel 8.91 Tungsten 19.38

Copper 8.94 Platinum 21.46

Silver 10.50

Density ProblemsThe gasoline in an automobile gas tank has a mass of 80.0 kg

and a density of 0.752 g/cm3. What is the volume in L?

Given: 80.0 kg Wanted: volume (L)

80.0 kg x 1000 g1 kg

= 106 Lx 1 cm3

0.752 gx 1 L1000 cm3

What is the mass of a ball of mercury that has a volume of 1.32 mL and a density of 13.6 g/mL?

Given: 1.32 mL Wanted: mass 13.6 g/mL

Mass = Volume x Density = 1.32 mL = 17.9 g Hgx 13.6 g1 mL

Water: A Special Case

Frozen water floats onliquid water.

Frozen ethanol sinks inliquid ethanol.

Ice is LESS DENSE than water!A given volume of ice must have less mass than an equal

volume of liquid water.

Therefore, the molecules in water pack together tighter than the molecule in ice!

Typically the solid phase is MORE dense than the liquid phase!

1. Write each of the following numbers in scientific notation.

a. 56897b. 123c. 0.000678d. 789540e. 560000000

2. Perform the following calculations using proper sigfigs.

a. 3.65 + 4.2 =b. 8.6 – 2.34 =c. 15.6 x 22.34 =d. (9.7 - 3.48) x 2.3 =e. (6.0 x103) + (3.2 x104) =

3. Use dimensional analysis to convert between units.

a. Convert 46.2 cm to inches (2.54 cm = 1 in)b. Convert 27 inches to feet (1 ft = 12 in)

Practice At Home!

Practice At Home!

1. Rice Krispies comes in a travel boxes containing 0.88 ounces. How many grams of cereal is this?

2. An address label has the length of 2.12 inches. What is the length of the label in cm?

3. Mount McKinley in Alaska is 20,320 ft above sea level. Express this height in kilometers.

Practice At Home!

1. What is the temperature in Kelvin when it is 431 oC?

2. What is the temperature in oF when it is 39 oC outside?

3. What is the temperature in Kelvin when it is 92 oF outside?

1. Calculate the density of air if the mass of 15.7 L is 18.6 grams.

2. A rectangular block of iron 3.20 cm x 9.87 cm x 11.6 cm has a mass of 2.88 kg. Find its density in g/cm3.

3. Calculate the volume occupied by 32.4 grams of copper, which has a density of 8.94 g/mL.

Practice At Home!

1. A titanium bike has a mass of 3245 g and a density of 4.50 g/cm3. What is its volume?

2. An ice cube has a volume of 75.9 cm3

and a density of 0.92 g/cm3. What is its mass?

3. A glass ball has a mass of 4.5 g and volume of 1.73 cm3. What is its density in g/cm3?

Practice At Home!