Measurement

a) Scientific Notationb) Precisionc) Rules for addition and subtractiond) Significant Digitse) Rules for multiplication and division

Scientific Notation

• Measurements in chemistry generate values that range from incredibly large to incredibly small values

• For example: 1 gram of iron contains about 10 000 000 000 000 000 000 000 atoms and 1 atom of iron weighs about 0.000 000 000 000 000 000 000 1 grams• Rather than write all those zeros, scientists write these

values in scientific notation • The general form for scientific notation is _._ _ x 10?

For numbers greater than one: move the decimal to the left until there is 1 digit in front of it. The number of places the decimal moves becomes the exponent on the 10.

15 000 000 000 000 000 000 000.= 1.5 x 1022

Try entering 15 000 000 000 000 000 000 000 into your calculator and then ask it to put it into scientific notationTry:56 000 000= 5.6 x 107

74.3 x 105

= 7.43 x 106

0.56 x 1010

= 5.6 x 109

500 000 000=5 x 108

For numbers less than one: move the decimal to the right until there is 1 digit in front of it. The number of places the decimal moves becomes the exponent on the 10, except that now the exponent is negative.

0.000 000 000 000 000 000 000 15= 1.5 x 10-22

Try:0.000 000 45= 4.5 x 10-7

36.0 x 10-10

= 3.60 x 10-9

0.545 x 10-6

= 5.45 x 10-7

0.000 000 5= 5 x 10-7

Precision• Depends on the graduations of the measuring device• Is the place value of the last measurable digit• For example, 12.57 cm is more precise than 125.7 cm

Low precision balance

High precision balance

Adding or subtracting measurements of different precision:

• The answer is rounded to the same precision of the least precise measurement

• For example:11.7cm + 3.29cm + 0.542cm. Your calculator will give the answer 15.532 cm BUT the correct answer is15.5cm

Try: 13.2 g + 18.18 g= 31.4g

Significant Digits

• The more digits that are in a measurement, the more certain you are about the measurement.

• For example, if the banana on the low precision balance were to be placed on the high precision balance, its mass may register 185.89 g rather than 186 g

• The certainty of the measurement is communicated by the number of significant digits recorded.

• There is more certainty in the second measurement because 5 significant digits (185.89) have been recorded rather than 3 (186).

• When zeros have been recorded in a measurement, they present a problem as to whether they are considered as significant digits

Rule to determine whether a digit is significant

• All non-zero digits are significant186 has 3 significant digits; 185.89 has 5 significant digits • Zeros between non-zero digits are significant1806 has 4 significant digits; 108006 has 6 significant digits• Any zero after a decimal point that also has significant digits

before the decimal point is significant623.0 has 4 significant digits; 6023.00 has 6 significant digits• Any zero before a decimal point and before other significant

digits is not significant0.632 has 3 significant digits; 0.00632 has 3 significant digits; 0.63200 has 5 significant digits

Measurements with no decimal and ending in zeros make it difficult to determine the precision of the measuring device.

• For example, 1860 could be considered to have 3 or 4 significant digits depending on the precision of the measuring device

• Writing the value in scientific notation makes it clearer: 1.860 x 103 or 1.86 x 103 • If the above example is not written in scientific notation, it will

be assumed that the zero is significant

Multiplying or dividing measurements of different degrees of certainty:

• The final answer has the same number of significant digits as the least number in the question

• For example: 2/6 = 0.3333….. would be written 0.32.1/6.3 = 0.332.00/6.0 = 0.3352 x 16 = 832 would be written 8.3 x 102 or 830