Scientific Measurement• Investigation: Det’n of Thickness of Al foil• Counted versus Measured Quantities• Significant Digits and Measurement

- Rules for Significant Figures• Accuracy and Precision• Errors in Measurement• Scientific Notation• Rules for Calculations Involving Measured

Quantities• Investigation: Determination of the Density

of Water

Investigation: Determination of Thickness of Aluminum foil

Take note of what gauge of foil you’re given: regular, heavy duty, etc.

What equipment/info. do you need to find the thickness of the Al foil?

• square of foil

• mm ruler

• electronic balance

• access to PT at front of room

Brainstorm . . . and get busy!

Write your answer on the board.

Counted Quantities vs Measured Quantities

How many measuring spoons are illustrated?Four. Are you sure?Yes.

How many people arein this room?Do you have 100% confidence in this value?Yes.

Counted quantities have no uncertainty associated with them.

If there are 15 people in a room, this may also be expressed as

15.00000000000000000000000000—etc

There is an infinite number of significant figures associated with a

COUNTED QUANTITY.

Measured Quantities

Is it possible to use the illustrated devices to measure with absolute certainty?

No.

All measurements

have some

uncertainty.

Significant Figures and Measurement

When measuring, ALWAYS ESTIMATE BETWEEN THE TWO SMALLEST SCALE DIVISIONS to obtain the estimated (uncertain) digit.

eg. When using a mm ruler, estimate the

tenth of a mm.

When using an ungraduated metre stick, estimate to the

tenths of a metre.

When measuring temperature on a thermometer calibrated every degree, we estimate

the tenths of a degree.

How long is this line?____________

the length of the line is 2.08 cm.Remember—the “8” isestimated and is thereforeuncertain.

How long is this line?_____________________

The line is4.00 cm long.

Read the temperature indicated on the thermometer to the correct number of sig figs.

Ans: 32.6oC

What about a digital readout?

With a digital readout, assume that the

final—or rightmost—digit is estimated.

Significant Digits• All measured quantities have some

degree of uncertainty. • [cf. counted quantities have no uncert.]• The significant digits in a measurement

include all the digits that can be known precisely plus a final digit that must be

estimated. • Every measured quantity MUST have

ONE estimated, or uncertain, digit.

Rules for Significant Digits

1. All non-zero digits in a recorded measurement are significant.

The measurements 24.7 m, 0.743 m, and 714 m all express measures of length to

three significant digits.

2. Zeros appearing between non-zero digits are significant.

The measurements 7003 m, 40.79 m, and 1.503 m all have

four significant digits.

3. Zeros appearing in front of all nonzero digits are not significant. They are acting as place holders. The measurements 0.071 m, 0.42 m, and 0.000 099 m all have

two significant digits. Sci. not’n is your friend.

ie. 0.071 m =

7.1 x 10-2 m

0.000 099 m =

9.9 x 10-5 m

Check this . . .

orig. volume of coffee =

2.0 x 102 mL

“new” vol. of coffee =

2.0 x 102 mL

here’s why:

vol. before addition = 200 mL

+ 0.05 mL

“new” volume = 200.05 mL

Taking into account confidence limit,

“new” volume = 2.0 x 102 mL SAME

Accuracy

How close a measurement is to the accepted value.

e.g. the proximity of a dart to the “bullseye”

Precision

1. A precise instrument measures to more decimal places. e.g. a mm ruler is more precise than a cm ruler.

2. A precise set of data points are close to one another. eg. On a dartboard high precision has all

darts close together, but not necessarily at the bullseye.

Errors in Measurement

• Systematic Error: The same error occurs over and over.

eg. Taking a set of masses with an improperly calibrated balance.

• Random Error:

Errors that cannot be accounted for.

For which type of error can we most easily correct?

systematic . . . why?

Because you make the same error each time.

A Possible Ambiguity

The distance from Earth to moon is 382 000 km. How precise is this distance?

We don’t know.

How can we eliminate this ambiguity?

Use sci not’n: eg 3.82 x 105 km

Scientific NotationSome examples:

6807 =

6.807 x 103

0.000 000 000 813 =

8.13 x 10-10

To make it easy—

Always put values in scientific notation.ie. 0.00670 cm put in sci. not’n

6.70x10-3 cm, or 6.70E-3 cm

remember that the right-most zero is estimated, and is therefore a sig. fig.

Calculations Involving Measured Quantities

Overlying Principle:

A calculated value can have no more precision than the data used to calculate it.

You can’t create precision.

You can’t create a greater confidence level simply by doing a calculation.

“You can’t make a silk purse out of a sow’s ear.”

Calculations Involving Measured Quantities

1. Multiplication and Division of Measured Quantities

The measurement with the FEWEST number of significant figures that goes into a multiplication/division determines the number of sig figs in the final answer.

Fewest # s.f. “in” = # sf in answer.

Don’t forget to round the answer.

examples

a) Find the area of a room whose dimensions are 23 m x 7.582 m.

A = (23 m)*(7.582 m)

= 174.386 m2 says your calculator, or

A = 1.7 x 102 m2 to two sig figs.

b) Osmium, a metal, is the most dense element.

Calculate the density of Os if a 12.3 cm3 piece of Os has a mass of 277.98 g.

ρ = mass/volume

= 277.98 g/12.3 cm3

= 22.6 g/cm3 to three sig figs.

c) A vehicle travels 278 miles on 11.70 gallons of gasoline. What is the average fuel consumption in miles per gallon of the vehicle, to the correct number of sf.?

2. Addition and Subtraction of Measured Quantities

Remember that the final answer can have only ONE estimated digit. e.g. Add these:

2.45 m

4.5 m

+ 8.695 m

15.645 m . We must round this off to

one estimated digit, or

15.6 m

eg. d) Here are the masses of several plums.

56.4g, 65.5g, 62.34g, 102g, 77.8g.

Calculate the average mass of a plum.

56.4

65.5

62.34

102

+ 77.8

364.04 . . . con’t

364 g/5 plums =

72.8 g/plum says your calculator.

To the correct number of sf, the average mass of a plum is

73 g.

The least certainty was in the “units” digit of the 102 g plum.

But what about the 5 plums? This is a

counted quantity and therefore has

infinite sf.

f) Find the sum of 3.18 and 0.01315 to the correct number of sf.

Sample Problem

Using a mm ruler, determine the thickness of one sheet in your Chemistry textbook.

Solution:680 pages;thickness = 2.23 cm;

How many sig figs in each of these?∞ counted;

3 sf measured

680 pages/2 pages·sheet-1 = 340 sheets

2.23 cm/340 sheet

= 6.55882353 x 10-3

cm/sheet

= 6.56 x 10-3 cm/sheet, or

= 6.56 x 10-2 mm/sheet

Compare to thickness of Al foil

Al foil = 1.5 x 10-2 mm;

textbook sheet = 6.56 x 10-2 mm

Why is the textbook sheet’s thickness more

precisely known?

mass of Al foil 2 sf

number of sheets ∞ sf (counted quantity)

Rounding Off

When doing a series of calculations involving measured quantities, round off only the final answer.

Allow your calculator to carry a few more sig. digs throughout the (intermediate) calculations to

avoid compounding errors due to rounding.

Rounding “fives”

• Look at these numerals:• 1 2 3 4 5 6 7 8 9• What will happen if we always round a 5

“up” in a large set of data? • Average will be skewed slightly upward.• How can we overcome this problem?• We need to round a 5 up half the time, and

down the other half. • But how can we easily do this?

Rounding 5s (con’t)

• If the digit before the 5 to be rounded is odd, round “up”. e.g. 6.75 g

6.8 g

• If the digit before the 5 to be rounded is even, do not round “up”. e.g. 3.45 m

3.4 m

• If there are digits after the 5 that needs to be rounded, always round up. This makes sense, since we’re rounding something that’s greater than 5.

e.g. 4.6851 g (to 3 s.f.) 4.69 g

Investigation: Determination of the Density of Water

For each pair of students: Use the equipment provided to determine the density of tap water.

Write your raw data and your calculated density of water on the board.

Report your answer to correct number of

significant figures.