Significant Figures--Conversions

1.5 Uncertainty in Measurement

· Two types of numbers:

· Exact numbers (known by counting or definition)

· Inexact numbers (derived from measurement)

· All measurements have some degree of uncertainty or error associated with them.

Precision and Accuracy

· Precision: how well measured quantities agree with one another.

· Accuracy: how well measured quantities agree with the “true value.”

· Figure 1.25 is very helpful in making this distinction.

Significant Figures

· In a measurement it is useful to indicate the exactness of the measurement. This exactness is reflected in the number of significant figures.

· Guidelines for determining the number of significant figures in a measured quantity:

§ The number of significant figures is the number of digits known with certainty plus one uncertain digit. (Example 2.2405 g means we are sure that the mass is 2.240 g, but we are uncertain about the nearest 0.0001 g.)

· Final calculations are only as significant as the least significant measurement.

· Rules:

1. Nonzero numbers are always significant.

2. Zeros between nonzero numbers are always significant.

3. Zeros before the first nonzero digit are not significant. (Example: 0.0003 has one significant figure.)

4. Zeros at the end of the number after a decimal place are significant.

5. Zeros at the end of the number after a decimal place are ambiguous (e.g. 10,300 g).

· Method:

1) Write the number in scientific notation.

2) The number of digits remaining is the number of significant figures.

3) Examples:

· 2.50 x 104 cm has 3 significant figures as written

· 1.03 x 104 g has 3 significant figures

· 1.030 x 104 g has 4 significant figures

· 1.0300 x 104 has 5 significant figures

Significant Figures in Calculations

· Multiplication and Division

§ Report to the least number of significant figures

· (e.g. 6.221 cm x 5.2 cm = 32 cm2)

· Addition and Subtraction

§ Report to the least number of decimal places

· (e.g. 20.4 g - 1.322 g = 19.1 g).

· In multiple-step calculations always retain an extra significant figure until the end to prevent rounding errors.

1.6 Dimensional Analysis

· Method of calculation utilizing a knowledge of units.

· Given units can be multiplied and divided to give the desired units.

· Conversion factors are used to manipulate units:

§ Desired unit = given unit x (conversion factor)

· The conversion factors are simple ratios:

§ Conversion factor = (desired unit) / (given unit)

Using Two or More Conversion Factors

· We often need to use more than one conversion factor in order to complete a problem.

· When identical units are found in the numerator and denominator of a conversion. they will cancel. The final answer MUST have the correct units.

§ Example:

· Suppose that we want to convert length in meters to length in inches. We can do this conversion with the following conversion factors:

1 meter = 100 centimeters and 1 inch = 2.54 centimeters

· The calculation will involve both conversion factors; the units of the final answer will be inches:

Conversions Involving Volume

· We often will encounter conversions from one measure to a different measure.

§ Example:

1. Suppose that we wish to know the mass in grams of 2.00 cubic inches of gold given that the density of the gold is 19.3 g/cm .

2. We can do this conversion with the following conversion factors:

2.54 cm = 1 inch and 1 cm3 = 19.3 g gold

3. The calculation will involve both of these factors:

x g gold =

4. Note that the calculation will NOT be correct unless the centimeter to inch conversion is cubed! Both the units AND the number must be cubed.

Summary of Dimensional Analysis

· In dimensional analysis always ask three questions:

1. What data are we given?

2. What quantity do we need?

3. What conversion factors are available to take us from what we are given to what we need?

from CHEMISTRY The Central Science 8th Edition Brown, LeMay, Bursten Ch 1: Introduction: Matter and