Chapter 3.1

Chapter 3

Scientific Measurement

Chapter Three: Table of Contents

3.1 Using and Expressing Measurements

3.2 Units of Measurement

3.3 Solving Conversion Problems

Section 3.1

Using and Expressing Measurements

How do you measure a photo finish?

Chemistry, just like a runner's time, requires accurate and precise measurements.

Measurement

• A quantity that has both a number and a unit.

• The number tells comparison, the unit tells scale.

• Examples of units: inches, years, liters, grams, °C

Scientific Notation

• Technique used to express very large or very small numbers.

• A given number is written as the product of two numbers: a coefficient and 10 raised to a power.

• Expresses a number as a product of a number between 1 and 10 and the appropriate power of 10.

• In scientific notation, the coefficient is always a number greater than or equal to one and less than ten. The exponent is an integer.

Writing Scientific Notation

• The power of 10 depends on the number of places the decimal point is moved and in which direction.

• The number of places the decimal point is moved determines the power of 10. The direction of the move determines whether the power of 10 is positive or negative. 

When writing numbers greater than ten in scientific notation, the exponent is positive and equals the number of places that the original decimal point has been moved to the left.

Numbers less than one have a negative exponent when written in scientific notation. The value of the exponent equals the number of places the decimal point has been moved to the right.

Multiple Choice

Which of the following correctly expresses 7,882 in scientific notation?

• 7.882 × 10⁴

• 788.2 × 10³

• 7.882 × 10³

• 7.882 × 10^–3

Multiplication

To multiply numbers written in scientific notation, multiply the coefficients and add the exponents.

Example:

 $(3×10^4)×(2×10^2)=(3×2)×10^4+2=6×10^6$  

Recall exponent rules:
 $(2.1×10^3)×(4.0×10^{-7})=(2.1×4.0)×10^{3+-7}=8.4×10^{-4}$  

Division

To divide numbers written in scientific notation, divide the coefficients and subtract the exponent in the denominator from the exponent in the numerator.

Example:

  $\frac{3.0\ \times\ 10^5}{6.0\ \times\ 10^2}\ =\ \left(\frac{3.0}{6.0}\right)\ \times\ 10^{5-2}\ =\ 0.5\ \times\ 10^3\ =\ 5.0\ \times\ 10^2$  

Addition and Subtraction

If you want to add or subtract numbers expressed in scientific notation and you are not using a calculator, then the exponents must be the same.

In other words, the decimal points must be aligned before you add or subtract the numbers.

Example:

 $(5.4×10^3)+(8.0×10^2)$  

 $=(5.4×10^3)+(0.80×10^3)$  
 $=(5.4+0.80)×10^3$  
 $=6.2×10^3$  

Multiple Choice

Solve the problem and express the answer in scientific notation.

 $(6.6×10^{−8})+(5.0×10^{-9})$  

•  $11.6\times10^{-8}$  

• 7.1

•  $7.1\times10^{-8}$  

• 11.6

Multiple Choice

Solve the problem and express the answer in scientific notation.

 $(9.4×10^{−2})−(2.1×10^{−2})$  

• 12.1

•  $7.3\times10^{-2}$  

•  $12.1\times10^{-2}$  

• 7.3

Multiple Choice

Calculate the following and give your answer in scientific notation:

 $\frac{6.6\times10^6}{\left(8.8\times10^{-2}\right)\times\left(2.5\times10^3\right)}$  

•  $3.0\times10^4$  

• .3

• 3000

•  $3.0\times10^{-1}$  

Why must measurements be reported to the correct number of significant figures?

The temperature shown on this Celsius thermometer can be reported to three significant figures.

Measurements must always be reported to the correct number of significant figures because calculated answers often depend on the number of significant figures in the values used in the calculation.

Significant Figures Rules

1. Every nonzero digit in a reported measurement is assumed to be significant.

2. Zeros appearing between nonzero digits are significant.

3. Leftmost zeros appearing in front of nonzero digits are not significant. They act as placeholders. By writing the measurements in scientific notation, you can eliminate such placeholding zeros.

Sig. Fig. Rules Cont'd.

4. Zeros at the end of a number and to the right of a decimal point are always significant.

5.  Zeros at the rightmost end of a measurement that lie to the left of an understood decimal point are not significant if they serve as placeholders to show the magnitude of the number.

6. There are two situations in which numbers have an unlimited number of significant figures. The first involves counting. A number that is counted is exact. (Ex. 23 people in the classroom, 1 hr = 30 min)