Chapter 2. Measurement Chemistry is a physical science, one that depends heavily on measurement to...

Chapter 2.

Measurement

Chemistry is a physical science, one that depends heavily on measurement to obtain quantitative data.

Measurement is the determination of dimensions, capacity, quantity, or extent of something.

Precision, Accuracy, Error

Precision, Accuracy, Error

Precision refers to the reproducibility of a given measurement.

Accuracy refers to the agreement between a measurement (or average several measure-ments) to the true or accepted value.

Errors are deviations (differences) between the true value and a measurement. They can be random or systematic.

Uncertainty in Measurement

All measurements contain some error, or uncertainty.

Heisenberg's Uncertainty Principle is a natural law that states that the act of observing something changes it. In its mathematical form, it states the limits of accuracy in a measurement.

Uncertainty in Measurement

Generally, uncertainty is minimized (accuracy is maximized) by using equipment that is better designed, more expensive, and more demanding to use.

Beaker 1 milliliter $3.00

Cylinder 0.1 milliliter $6.00

Buret 0.01 milliliter $90.00

Uncertainty in Measurement

Uncertainty in Measurement

Numbers express uncertainty.

Exact numbers contain no uncertainty. They are obtained by counting objects (integers) or are defined, as in some conversion factors.

Inexact numbers contain uncertainty. They are obtained from measurements. The un-certainty in a measurement is shown by the number of digits recorded.

Uncertainty in Measurement

Significant figures are the digits in a measure-ment that are known with certainty, plus one digit that is uncertain. Last digit is uncertain.

14.3 gallons 0.1 gal (13 oz)

14.325 gallons 0.001 gal (¾ tsp)

Uncertainty in MeasurementSignificant figures allow us to estimate the

uncertainty of a value:

10 +/– 1 1 x 100% = 10% 10

10.0 +/– 0.1 0.1 x 100% = 1% 10.0

10.00 +/– 0.01 0.01 x 100% = 0.1% 10.00

Uncertainty in Measurement

Significant figures and measurements:

With digital instruments (reading is numbers) record all digits from the instrument. The last digit is uncertain and may fluctuate.

With analog instruments (reading is a scale) "read between the lines" for last digit. That digit is an estimate, and contains uncertainty.

Working with "Sig Figs"

How many significant figures are there in each of these numbers?

14.3 gallons

14.325 gallons

14.0 gallons

0.025 gallons

Working with "Sig Figs"

In a number from a measurement:

All nonzero digits are significant.

Internal zeroes are significant.

Leading zeroes are not significant.

Trailing zeroes are significant if there

is a decimal point in the number.

Working with "Sig Figs"

How many significant figures in each number?

1800

1800.

1800.00

Working with "Sig Figs"

Why does this matter? Because we usually do calculations on measurements.

I ride my bike to school. The distance is 4.1 miles. The other day it took 25.7 minutes. What was my average speed, in miles per minute, and in miles per hour?

4.1 mi = 0.1595440793 mi 25.7 min min

4.1 mi = 0.16 mi = 9.6 mi 25.7 min min hr

Working with "Sig Figs"

Why did I convert 0.1595440793 mi/min to

0.16 mi/min and not something else?

Rules for working with significant figures:

Rounding

Sig figs in calculations

Working with "Sig Figs"

Rounding numbers is disposing of nonsig-nificant digits

Only do this in calculations!

Do not round off measurements!

Working with "Sig Figs"

Rules for rounding:

1. Decide how many digits are significant.2. Underline them (till you catch on)3. Look at digit to right of last underlined digit a. If it's 1 4, drop it. b. If it's 5 9, add 1 to last sig. fig.

Working with "Sig Figs"

Examples:

Round the following values to 3 sig. figs.

27.428

39.572

0.01565

Calculating with "Sig Figs"

Rules for Calculations:

1. Multiplication and division:

The result has the same number of sig figs as the value with the fewest sig figs.

4.1 mi = 0.1595440793 mi 25.7 min min

4.1 mi = 0.16 mi = 9.6 mi 25.7 min min hr

Calculating with "Sig Figs"

Rules for Calculations:

2. Addition and Subtraction: The result has the same uncertainty as the

value with the greatest uncertainty. Keep digits as far to the right as all values have sig figs.

23.456 100.423 + 82.9 -100.312

106.356 ~ 106.4 0.111

Calculating with "Sig Figs"

Examples:

Calculate the quantities, and round to the correct number of significant figures:

15.86 x 2.34 12.356 =

907.369 – 15.5 =

(143.289 – 139.143) = 187.467

Calculating with "Sig Figs"

Rules for Calculations:

4. Working with exact and inexact numbers:

Exact numbers don't have sig figs because they don’t introduce uncertainty.

Just use sig figs in inexact numbers

0.16 mi x 60 min = 9.6 mi min hr hr

60 min/hour is an exact conversion

Calculating with "Sig Figs"

Rules for Calculations:

5. If your calculator gives you fewer sig figs

than the value should have, add zeroes

0.465 x 0.200 = 0.0930

4.389 – 2.589 = 1.800

Calculating with "Sig Figs"

Rules for Calculations:

6. If your calculator gives you more digits to

the left of the decimal than are significant,

use scientific notation.

75.3 x 24.8 x 675 = 1260522

= 1.26 x 106

Scientific Notation

Scientific notation is a numerical system in which a decimal number is expressed as the product of a number between 1 and 10, and 10 raised to a power.

1260522 = 1.260522 x 1,000,000 = 1.260522 x 106

= 1.26 x 106

1.26 is the coefficient 106 is the exponential term

Scientific Notation

Scientific notation is useful for expressing very large or very small numbers.

93,000,000 miles from earth to sun

9.3 x 107 miles

0.000 000 000 000 000 000 000 030 grams

3.0 x 10-23 grams is mass of 1 water molecule

Scientific Notation

When should scientific notation be used?

1. It MUST be used if you have more digits

than significant digits to the left of the

decimal point.

2. It CAN be used whenever it's convenient.

Often, this involves very small numbers.

Summary

It’s easy to keep track of significant figures if you remember WHY you’re doing it.

WHY are you doing it?

Significant figures reflect the accuracy of measured quantities and of results calculated from measured quantities.