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Do you know how to take a measurement?

Do you know how to work with the measurements you take?

Chapter 2Significant Digits

Taking Measurements

•All measurements involve one estimation.

•If the measuring device is digital it will take the estimation for you.

Electronic Measuring Devices

• Digital readout measuring diameter of 1.0420 inches.

• The last zero is the estimated digit.

Taking Measurements

•All measurements involve one estimation.

•If the measuring device is scaled you must take the estimation yourself.

Scaled Measuring Devices


• Use the bottom ruler to measure the length of the metal bar.


•Bottom ruler gives a measurement of 8.? cm.

The Importance of Units

• The Mars Climate Orbiter was launched by NASA in December 1998 at a cost of $125 million dollars.

• The orbiter was to go into orbit around Mars on Sept. 23rd 1999.


• It entered Mars atmosphere 100 km (62 miles) lower than it was supposed to and was destroyed.

• The loss of the satellite cost not only millions of dollars and lost time but was because of a simple problem that all high school science students should understand.


• UNITS.

• Different groups of scientists failed to include units on calculations and therefore mixed English units with SI units.


• Engineers who built the spacecraft specified thrust in pounds, which is an English unit. Scientists who launched the satellite assumed the thrust was calculated in the metric unit newtons.

• But since the units were not specified the mistake was not caught until it was too late.


•Measure the length of the metal using the top ruler.

How to read a meniscus.

.

How to read a meniscus.

ml

Read the Volume in mL

Significant Digits

Rules for Determining the Number of Significant Digits in a Measurement

1. Digits other than zero are always significant.

96g 2 significant digits 61.42g 4 significant digits 0.538g 3 significant digits

2. One or more trailing zeros after the decimal point are always significant. 4.00000s 6 significant digits 0.05600s 4 significant digits 0.002 1 significant digit

3. Zeros between two other significant digits are always significant. 5.029km 4 significant digits 30600km 3 significant digits 0.050060km 5 significant digits

4. A zero which is not significant by any of the above rules can be made significant by placing a bar over it. However we usually write numbers such as these in scientific notation to show the number of significant figures.

1

0 ml 2 significant digits 1.0 x 102 ml

100

0 ml 4 significant digits 1.000 x 103 ml

10

0 000 ml 3 significant digits 1.00 x 105 ml When determining the number of significant digits in a measurement it is often helpful to use the following strategy.

1. Find the leftmost significant digit. It will always be your leftmost nonzero digit.

2. Find your rightmost significant digit. It will be the rightmost nonzero digit unless you have a zero which is further to the right and is (a) after the decimal point and at the end of the number or (b) has a bar over it.

3. All digits in between the leftmost significant digit and the rightmost significant digit are significant.

Examples:

70.12 L –270.8 °C 0.000800 mg 1000 mL 82.003 µm 42,729.00 cm 27.0 km 225 beans 50 people 99.294 dm 1.002cm 0.06900 m

2

0 0 kg 3,200,

0 00 µL

Rules for Significant Digits

• All non-zero digits are significant.

• “Trailing” zeros after the decimal point ARE significant.

• Zeros between significant digits are significant.

• All other zeros are NOT significant unless indicated to be so by having a bar placed over them.

How to Determine Significant Digits

•Underline the leftmost nonzero digit.

•Use the rules for significant digits to determine the rightmost significant digit.

•Every digit in between the leftmost and rightmost significant digits are significant as well.

Counting or Exact Numbers

Counting numbers: If there are 10 people in a room there are not 9.5 or 10.76 people in the room. Counting numbers are exact.

Ones in Conversion Factors: 1 kilometer = 1000 meters. Exactly 1 km is equal to exactly 1000m. The 1 is considered to be an exact number and so is the 1000.

• Since Counting numbers and metric conversions are exact they have an infinite number of significant digits.

Determine the Significant Digits(Examples in Notebook)

• 70.12 L • 0.000800 mg • 82.003 µm • 27.0 km • 50 people • 1.002 cm

• 200 kg

• -270.8 ºC • 1000 mL • 42,729.00 cm • 225 beans • 99.294 dm • 0.06900 m

• 3,200,000 µL

Determine the Significant Digits (Examples in Notebook)

• 70.12 L 4• 0.000800 mg 3• 82.003 µm 5• 27.0 km 3• 50 people infinite• 1.002 cm 4 • 200 kg 2

• -270.8 ºC 4• 1000 mL 1• 42,729.00 cm 7• 225 beans infinite• 99.294 dm 5• 0.06900 m 4 • 3,200,000 µL 5

Calculating in Chemistry Class

Multiplying and Dividing

Rule: When multiplying and dividing your answer should always contain the number of significant digits that are in the original number with the least number of significant digits.

Example: When multiplying 22.37 cm x 3.10 cm x 85.75 cm = 5946.50525 cm3

We look to the original problem and check the number of significant digits in each of the original measurements:

22.37 shows 4 significant digits.



Our answer can only show 3 significant digits because that is the least number of significant digits in the original problem. Our answer is 5946.50525 which has 9 significant digits, we must round this to show only three significant digits. To do this start at the leftmost significant digit and count 3 places to the right. This would place us at the “4” which occupies the tens place. We must therefore round our answer to the tens place in order to show only 3 significant digits. Our final answer becomes 5950 cm3.

Examples:

a) 24 mm x 31.8 mm =

b) 8.40 g ÷ 4.2 ml =

c) 200 dm x 3.58 dm =

d) 5500 km ÷ 55.0 s =

Math Operations with Significant Digits

• When multiplying and/or dividing your answer must reflect the smallest number of significant digits.



• (17.3 cm)(28 cm) = 484.4 cm2



• (17.3 cm)(28 cm) = 484.4 cm2 = 480 cm2



• (17.3 cm)(28 cm) = 484.4 cm2 = 480 cm2

• 708g ÷ 4.700ml = 150.63829 g/ml =



• (17.3 cm)(28 cm) = 484.4 cm2 = 480 cm2

• 708g ÷ 4.700ml = 150.63829 g/ml = 151g/ml

Adding and Subtracting

RULE: When adding or subtracting your answer should always contain the number of decimal places that are in the original number with the least number of decimal places.

Example: When we add 3.76 g + 14.83 g + 2.1 g = 20.69 g

We look to the original problem to see the number of decimal places shown in each of the original measurements. 2.1 shows the least number of decimal places. We must round our answer, 20.69, to one decimal place (the tenth place). Our final answer is 20.7 g

Examples:

a) 49.1 g + 8.001 g =

b) 81.350 m − 7.35 m =

c) 4.60 s + 3 s =

d) 67.5 cm − 0.009 cm =

Addition and/or Subtraction reflects the fewest decimal places

decimal places.

• 24.6 cm − 17.01 cm = 7.59 cm


decimal places.

• 24.6 cm − 17.01 cm = 7.59 cm = 7.6 cm


decimal places.

• 24.6 cm − 17.01 cm = 7.59 cm = 7.6 cm

• 8.5g + 1.32g + 0.18g =


decimal places.

• 24.6 cm − 17.01 cm = 7.59 cm = 7.6 cm

• 8.5g + 1.32g + 0.18g = 10g


decimal places.

• 24.6 cm − 17.01 cm = 7.59 cm = 7.6 cm

• 8.5g + 1.32g + 0.18g = 10g = 10.0g

Are Significant Figures Important?

• A student was given an assignment and a budget of $50 to complete it.

• Assignment: Get a cube of metal which has a mass of 83 grams.

• He knew the density of the metal was 8.6 g/mL, and used this to calculate the cube's volume.


• Believing significant figures were invented just to make life difficult for chemistry students and had no practical use in the real world, he calculated the volume of the cube to be about 9.651 mL.


• He then calculated that the edge of the cube had to be 2.129 cm. He took his plans to the machine shop where his friend had the same type of work done the previous year.

• The shop foreman said, "Yes, we can make the cube according to your specifications."


• He knew his friend had paid $35 for a similar metal cube, and he had been given $50 out of the school's research budget to get the job done so he assumed that he would have plenty of money.


• He returned the next day, expecting the job to be done. "Sorry," said the foreman. "We're still working on it. Try next week."

• Finally the day came, and our friend got his cube. It looked very, very smooth and shiny and beautiful in its velvet case.


• Seeing it, our hero had a premonition of disaster and became a bit nervous. But he summoned up enough courage to ask for the bill.


• "$500, and cheap at the price. We had a difficult job getting it right -- had to make three before we got one right."


• "But--but--my friend paid only $35 for the same thing!"

• "No. He wanted a cube 2.1 cm on an edge, and your specifications called for 2.129cm.


• We had yours roughed out to 2.1 cm the day that you brought it in.

• However it was the precision grinding and lapping to get it to the 2.129 cm specifications which took so long and cost the big money.

Homework

Worksheet – Significant Digits & Rounding.

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