Chem 160- Ch # 2l.

Numbers from measurements.

Measurements

• Experiments are performed.

• Numerical values or data are obtained from these measurements.

12 inches = 1 foot100 centimeters = 1 meter

• Exact numbers have an infinite number of significant figures.

• Exact numbers occur in simple counting operations

Exact Numbers

• Defined numbers are exact.

12345

Form of a Measurement

70.0 kilograms = 154 pounds

numerical value

unit

Significant Figures

• The number of digits that are known plus one estimated digit are considered significant in a measured quantity

estimated5.16143

known

Significant Figures on Reading a Thermometer

Significant Figures on Reading a Thermometer

Temperature is estimated to be 21.2oC. The last 2 is uncertain.

The temperature 21.2oC is expressed to 3 significant figures.

Temperature is estimated to be 22.0oC. The last 0 is uncertain.

The temperature 22.0oC is expressed to 3 significant figures.

Temperature is estimated to be 22.11oC. The last 1 is uncertain.

The temperature 22.11oC is expressed to 4 significant figures.

461

All nonzero numbers are significant.

Significant Figures

461

All nonzero numbers are significant.

Significant Figures

461

3 Significant Figures

All nonzero numbers are significant.

Significant Figures

401

3 Significant Figures

A zero is significant when it is between nonzero digits.

Significant Figures

A zero is significant when it is between nonzero digits.

5 Significant Figures

600.39

Significant Figures

A zero is significant at the end of a number that includes a decimal point.

5 Significant Figures

000.55

Significant Figures

A zero is significant at the end of a number that includes a decimal point.

5 Significant Figures

0391.2

Significant Figures

A zero is not significant when it is before the first nonzero digit.

1 Significant Figure

600.0

Significant Figures

A zero is not significant when it is before the first nonzero digit.

3 Significant Figures

907.0

Significant Figures

A zero is not significant when it is at the end of a number without a decimal point.

1 Significant Figure

00005

Significant Figures

Roundingoff Numbers

Roundingoff Numbers

Rounding Off Numbers

• Often when calculations are performed extra digits are present in the results.

• It is necessary to drop these extra digits so as to express the answer to the correct number of significant figures.

• When digits are dropped the value of the last digit retained is determined by a process known as rounding off numbers.

80.873

Rule 1. When the first digit after those you want to retain is 4 or less, that digit and all others to its right are dropped. The last digit retained is not changed.

4 or less

Rounding Off Numbers

5 or greater

5.459672

Rule 2. When the first digit after those you want to retain is 5 or greater, that digit and all others to its right are dropped. The last digit retained is increased by 1.

drop these figuresincrease by 1

6

Rounding Off Numbers

Significant Figures in Calculations

Significant Figures in Calculations

Multiplication or DivisionMultiplication or Division

In multiplication or division, the answer must contain the same number of significant figures as in the measurement that has the least number of significant figures.

(190.6)(2.3) = 438.38

438.38

Answer given by calculator.

2.3 has two significant figures.

190.6 has four significant figures.

The answer should have two significant figures because 2.3 is the number with the fewest significant figures.

Drop these three digits.

Round off this digit to four.

The correct answer is 440 or 4.4 x 102

Addition or SubtractionAddition or Subtraction

The results of an addition or a subtraction must be expressed to the same precision as the least precise measurement.

The result must be rounded to the same number of decimal places as the value with the fewest decimal places.

Add 125.17, 129 and 52.2

125.17129.

52.2306.37

Answer given by calculator.

Least precise number.

Round off to the nearest unit.

306.37

Correct answer.

Scientific Notation of Numbers

Scientific Notation of Numbers

6022000000000000000000000.00000000000000000000625

• Very large and very small numbers like these are awkward and difficult to work with.

• Very large and very small numbers are often encountered in science.

602200000000000000000000

A method for representing these numbers in a simpler form is called scientific notation.

0.00000000000000000000625

6.022 x 1023

6.25 x 10-21

Scientific Notation• Write a number as a power of 10• Move the decimal point in the original number

so that it is located after the first nonzero digit.

• Follow the new number by a multiplication sign and 10 with an exponent (power).

• The exponent is equal to the number of places that the decimal point was shifted.

Write 6419 in scientific notation.

64196419.641.9x10164.19x1026.419 x 103

decimal after first nonzero

digitpower of 10

Write 0.000654 in scientific notation.

0.0006540.00654 x 10-10.0654 x 10-20.654 x 10-3 6.54 x 10-4

decimal after first nonzero

digitpower of 10