- Ch.1 - Intro to General Chemistry
- Ch.2 - Atoms & Elements
- Ch.3 - Chemical Reactions
- BONUS: Lab Techniques and Procedures
- BONUS: Mathematical Operations and Functions
- Ch.4 - Chemical Quantities & Aqueous Reactions
- Ch.5 - Gases
- Ch.6 - Thermochemistry
- Ch.7 - Quantum Mechanics
- Ch.8 - Periodic Properties of the Elements
- Ch.9 - Bonding & Molecular Structure
- Ch.10 - Molecular Shapes & Valence Bond Theory
- Ch.11 - Liquids, Solids & Intermolecular Forces
- Ch.12 - Solutions
- Ch.13 - Chemical Kinetics
- Ch.14 - Chemical Equilibrium
- Ch.15 - Acid and Base Equilibrium
- Ch.16 - Aqueous Equilibrium
- Ch. 17 - Chemical Thermodynamics
- Ch.18 - Electrochemistry
- Ch.19 - Nuclear Chemistry
- Ch.20 - Organic Chemistry
- Ch.22 - Chemistry of the Nonmetals
- Ch.23 - Transition Metals and Coordination Compounds

**Significant Figures** are used to determine some level of accuracy within our recorded measurements.

**Concept:** Determining significant figures

We're going to say that we know that there is some level of accuracy and precision necessary with all of our calculations. But when we get an answer, how many digits does that answer have to have? That's when significant figures come into play and play a very important role in deciding the number of digits in our answer.

Now sig figs can be really easy as long as we remember two simple rules. It has to do with decimal places or no decimal places. Let's take a look at these two rules.

We're going to say rule one has to do with if you have a decimal point. We're going to say, if your number has a decimal point, we're going to move from left to right, so we're going to move from left to right. We're going to start counting once we get to our first non-zero number and keep counting until we get to the very end.

Now, rule number two is, if our number has no decimal point. If it has no decimal point, then you're going to start moving from right to left. We're going to start counting once we get to our first non-zero number and keep counting until you get to the very end.

Now, that we've seen these two simple rules, let's apply them to these examples.

**Example:** How many sig figs does each number contain?

0.0000185 m 749 mol

17.3 x 10** ^{3}** mL 100

0.0010050 kg 1560 mol

4m **Example:** Read the length of the metal bar to the correct number of significant figures.

**Concept:** Understanding calculations with Significant Figures

We’re going to say, when it comes to calculations we separate them into two categories. Multiplication and division go together; addition and subtraction will go together.

So we are going to say here when it comes to multiplication and division, measurements with the least number of significant figures, I’m just going to say sig figs, will determine our final answer. And when it comes to addition and subtraction we are going to say measurements with the least number of decimal places will determine our final answer.

So just remember when we are multiplication or division, we’re looking at the number of sig figs for our measurements. The one with the smallest number of sig figs will determine the number of sig figs in our final answer.

When we are doing addition and subtraction we want to get the fewest number of decimal positions that will give us our final answer.

The number of significant figures involved in a calculation depends on whether we are adding, subtracting, multiplying or dividing.

**Example:** Perform the following calculation to the right number of sig figs:

(3.16) x (0.003027) x (5.7 x 10** ^{-3}**)

Whenever we are adding or subtracting numbers with different exponents we must manipulate them to one common exponent value. We manipulate the values to get the least number of decimal places.

**Example:** Perform the following calculation to the right number of sig figs:

2.628 x 10^{6}

6.281 x 10^{4}

__+ 0.827 x 10__^{7}

Whenever we are dealing with a mixture of functions just remember your order of operations.

**Example:** Perform the following calculation to the right number of sig figs:

__(42.00 – 40.915) • (25.739 – 25.729)__

(11.50 • 1.001) + (0.00710 • 700.)

5mTranslate the following numbers into standard notation, maintaining the same number of significant figures.

a. 4.52 x 10^{3}

b. 8.11 x 10^{-2 }

^{}

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How many significant figures are in the number 0.80610?

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What is the correct answer to the following expression?

4.13 x 10^{-10} + 7.59 x 10 ^{-12 }

a. 4.21 x 10^{-10}

b. 4.2 x 10^{-10 }

c. 4 x 10^{-10 }

d. 4.206 x 10^{-10}

e. 4.2059 x 10^{-10}

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The number of significant figures in 0.066900 x 10 ^{-4} is

a. 5

b. 6

c. 7

d. 3

e. 4

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All 50 students agree that in the classroom there are 100 chairs. This number has:

a. 1 significant figure

b. 2 significant figures

c. 3 significant figures

d. Infinite number of significant figures

e. We cannot know with the available information

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There are 6 significant figures in the number 0.003720.

A. True

B. False

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Determine the number of significant figures for the result of the following arithmetic calculations.

(i) (332.41 + 711.59) (ii) (3.00 x 92.11) / 1.250 (iii) [(25.01 x 6.0) + 40.11]

A) 6, 3, 2

B) 4, 3, 3

C) 5, 3, 2

D) 5, 4, 3

E) 5, 3, 4

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How many significant figures should be used to express the value of x?

x = (23.714 + 86.97 - 0.8693) / 31.7400 ?

A) two

B) three

C) four

D) five

E) six

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How many significant figures are in the measurement, 20.300 m?

A) 3

B) 4

C) 5

D) 1

E) 2

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While standing on the summit, you discover there are 16 other people on the top of Africa with you. This number of people has how many significant digits?

A) 1

B) 2

C) 3

D) 4

E) unlimited

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Which measurement is correct when using a 30-cm ruler with 0.1 cm markings?

a) 10.1 cm

b) -0.05 cm

c) 5.225 cm

d) 3.45 cm

e) 29 cm

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What is the value for y, rounded correctly and to the proper number of significant figures when x = 0.320 in the following equation?

y = 7.3562 x - 0.14

a. 2.2

b. 2.214

c. 2.2140

d. 2.21

e. 2.2138

Watch Solution

Perform the following mathematical operations and express the result to the correct number of significant figures.

d. (3.8 x 10^{-12} + 4.0 x 10^{-13}) / (4 x 10^{12} + 6.3 x 10^{13})

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Perform the following mathematical operations and express the result to the correct number of significant figures.

c. 6.071 x 10^{-5} - 8.2 x 10^{-6} - 0.521 x 10^{-4}

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Consider multiplying 26.2 by 16.43. What would a mathematician say the answer is? What would a scientist say? Justify the scientist’s answer, not merely citing the rule, but explaining it.

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Consider the addition of 15.4 to 28. What would a mathematician say the answer is? What would a scientist say? Justify the scientist’s answer, not merely citing the rule, but explaining it.

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Do each calculation without using your calculator and give the answers to the correct number of significant figures.

[(1.36 X 10^{5})(0.000322)/ 0.082](129.2)

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Do each calculation without using your calculator and give the answers to the correct number of significant figures.

b. 1.87 X 10^{- 2} + 2 X 10^{- 4} - 3.0 X 10^{-3}

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Do each calculation without using your calculator and give the answers to the correct number of significant figures.

1.76 X 10^{- 3 }/ 8.0 X 10^{2}

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Calculate to the correct number of significant figures.

c. (9443 + 45 - 9.9) X 8.1 X 10^{6}

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Calculate to the correct number of significant figures.

d. (3.14 X 2.4367) - 2.34

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Calculate to the correct number of significant figures.

b. (568.99 - 232.1) ÷ 5.3

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Calculate to the correct number of significant figures.

a. [( 1.7 X 10^{6}) ÷ (2.63 X 10^{5})] + 7.33

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Calculate to the correct number of significant figures.

d. 5.99 - 5.572

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Calculate to the correct number of significant figures.

c. 19.6 + 58.33 - 4.974

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Calculate to the correct number of significant figures.

b. 17.6 + 2.838 + 2.3 + 110.77

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Calculate to the correct number of significant figures.

a. 43.7 - 2.341

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Calculate to the correct number of significant figures.

d. 453 ÷ 2.031

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Calculate to the correct number of significant figures.

c. 4.005 X 74 X 0.007

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Calculate to the correct number of significant figures.

b. (5.01 X 10^{5}) ÷ (7.8 X 10^{2})

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Calculate to the correct number of significant figures.

a. 89.3 X 77.0 X 0.08

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Round each number to three significan t figures.

a. 79,845.82

b. 1.548937 X 10^{7}

c. 2.3499999995

d. 0.000045389

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Round each number to four significant figures.

a. 156.182

b. 156.842

c. 156.849

d. 156.899

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Indicate the number of significant figures in each number. If the number is an exact number, indicate an unlimited number of significant figures.

a. 305,435,087 (2008 U.S. population)

b. 2.54 cm = 1 in

c. 11.4 g/ cm^{3} (density of lead)

d. 12 = 1 dozen

Watch Solution

How many significant figures are in each number?

a. 0.000312 m

b. 312,000 s

c. 3.12 X 10^{5} km

d. 13,127 s

e. 2000

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Read each measurement to the correct number of significant figures. Note: Laboratory glassware should always be read from the bottom of the meniscus. Digital balances normally display mass to the correct number of significant figures for that particular balance.

Watch Solution

Read each measurement to the correct number of significant figures. Note: Laboratory glassware should always be read from the bottom of the meniscus. Digital balances normally display mass to the correct number of significant figures for that particular balance.

Watch Solution

Read each measurement to the correct number of significant figures. Note: Laboratory glassware should always be read from the bottom of the meniscus. Digital balances normally display mass to the correct number of significant figures for that particular balance.

Watch Solution

What is the correct reading for the liquid in this buret?

(A) 32 mL

(B) 32.2 mL

(C) 32.26 mL

(D) 33.74 mL

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What is the correctly reported mass of water based on this data?

(A) 1.3 g

(B) 1.30 g

(C) 1.298 g

(D) 1.2980 g

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Solve and give the correct number of significant figures to the problem below.

5.80 x 10 ^{–1} – 3.4 x 10 ^{–2} =

A) 5.5 x 10 ^{–2}

B) 2.4 x 10 ^{2}

C) 5.46 x 10 ^{–1}

D) 2.4 x 10 ^{–3}

E) 5.5 x 10 ^{–1}

Watch Solution

Give the answer with correct significant figures.

- 4.56 / (7.49 + 5.493 - 5.345)

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How many significant figures are in the following?

- 6.022 x 10
^{23}= - 340384 =
- 10.00 =
- 8.1000 =
- 0.00000000003 =

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Calculate the following to the correct number of significant figures.

a) 2

b) 1.5

c) 1.51

d) 1.510

e) 1.5103

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What answer should be reported, with the correct number of significant figures, for the following calculation?

(965.43 × 3.911) + 9413.4136

A) 13189

B) 13189.2

C) 1.32 × 10^{4}

D) 1.3 × 10^{4}

E) 1.319 × 10^{4}

Watch Solution

How many significant figures are in 1009.630 mL?

A) 3

B) 4

C) 5

D) 6

E) 7

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Read the length of the metal bar with the correct number of significant figures.

A) 20 cm

B) 15 cm

C) 15.1 cm

D) 15.10 cm

E) 15.100 cm

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Read the length of the metal bar with the correct number of significant figures.

A) 20 cm

B) 15 cm

C) 15.1 cm

D) 15.10 cm

E) 15.100 cm

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Read the water level with the correct number of significant figures.

A) 5 mL

B) 5.4 mL

C) 5.42 mL

D) 5.420 mL

E) 5.4200 mL

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Which of the following numbers has the greatest number of significant figures?

A. 1.500 B. 2.4 x 10 ^{4} C. 0.00085 D. 752,000

Watch Solution

Perform the following calculation to the right number of sig figs:

a) 1

b) 2

c) 3

d) 4

Watch Solution

Give the correct number of significant figures for the following computation:

a) 1

b) 2

c) 3

d) 4

e) 5

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At 4.000°C water’s density is 1.000 g/cm ^{3.} For the range of temperatures associated with “room temperature” the density of pure water is:

Density (g/cm ^{3}) = (-0.00030 (g/°C • cm ^{3}) x Temperature (°C) + 1.0042 (g/cm ^{3})

Based on this thermometer reading, how many significant figures should you report for the calculated density?

a) One b) Two c) Three d) Four e) Five

Watch Solution

Every measurement or calculation we do with instruments in chemistry has some level of uncertainty called *experimental error*. Significant figures are necessary to communicate a level of accuracy and precision while dealing with this uncertainty.

**Rules for Number of Signficant Figures**

__Rule 1__: If your number has a ** decimal point** move from

*Trailing zeros* are a sequence of zeros after the decimal point once you’ve passed all non-zero integers. They would be significant.

__Rule 2: __If your number has ** NO decimal point** move from

If a value has no decimal point then the final zero would mean it has an infinite number of significant figures.

**Exact Numbers **

Exact numbers contain an infinite number of significant figures because they represent the amount of a particular item and not a measurement.

For example, when we say that a decade is equal to 10 years, there are 150 students in your chemistry class, or NH_{3} contains 1 atom of nitrogen and 3 atoms of hydrogen.

**Addition & Subtraction**

When dealing with addition or subtraction the least number of decimal places will determine the final answer.

^{6}, but the final answer should have only 1 decimal place and so the answer is rounded. Adding the two values gives the initial value of 5.375 x 10

**Multiplication & Division**

When dealing with multiplication or division the least number of significant figures will determine the final answer.

Multiplying the two values gives an initial value of 103.6, but you must round your final answer to significant figures.

Furthermore keep in mind when dealing with mixed operations we must follow the same arithmetic guidelines as we always do.

**Logarithm & Natural Logarithm**

When taking the log or ln of a value, the number of significant figures it has determines the number of digits ** after** the decimal point for the answer.

Using your calculator, the log of 1.53 x 10^{-3} gives an initial value of - 2.01531. The value of 1.53 x 10^{-3} has 3 significant figures and so the final answer should have 3 digits after the decimal point.

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