1. Chapter 3. Units and Calculations
All measurements have three parts:
1. Number (value, quantity)
2. Uncertainty (error, shown by sig figs)
3. Unit (nature of quantity, label)
Units must always be shown with numbers!

2. The Metric System
The metric system is a decimal system of weights and measures based on the meter as a unit of length, the kilogram as a unit of mass, the second as a unit of time, and the kelvin as a unit of temperature.
Decimal: Unit conversions are factors of 10.

3. The Metric System Basic (fundamental, defined) Units:
Length; meter (m) (about 1 yard)
in lab, centimeter and millimeter
Mass; kilogram (kg) (about 2.2 pounds)
in lab, gram (g)
Time; second (s) (same as English)
minutes and hours, not decimal
Temperature; kelvin (K) (no negative values)
also Celcius, centigrade (C)

4. The Metric System

5. The Metric System
The fundamental units in the metric system are too large to be convenient in chemical labs. How do we get smaller units?
Some of the derived units in the metric system are very small. How do we get larger units?

6. The Metric System
We multiply the unit by some power of ten, for example 103 (1000) or 10-2 (0.01).
These multipliers relate to prefixes. The prefixes are combined with names of fundamental units to obtain larger or smaller units:
kilogram = grams
centimeter = meter

7. The Metric System Metric system prefixes (multipliers) to know:
My king died chewing M & M's
Prefix
Symbol
Meaning
Value
Exp.
mega m
million
1,000,000
106
kilo k
thousand
1,000
103
deci d
tenth
0.1
101
centi c
hundredth
0.01
102
milli
thousandth
0.001
103
micro

millionth
106

8. The Metric System How to use prefixes and multipliers:
Name of unit Value of unit
prefix unit multiplier x unit
milligram x 1 gram
one thousandth of a gram

9. The Metric System Examples: One centimeter = 1 cm = 0.01 meter
One kilogram = 1 kg = gram
One millisecond = 1 ms = second
One megahertz = 1 MHz = 1,000,000 Hz
One microfarad = 1F = F

10. The Metric System
Conversions within the metric system, e.g. convert meters to kilometers
1. Set up equality: prefix unit = multiplier x unit kilometer = meters
2. Convert to ratio with desired unit in numerator: km
1000 m
3. Multiply ratio by given units:
1 km x m = km

11. The Metric System Convert: 0.0285 kilograms to grams (kg to g)
27935 meters to kilometers (m to km)
53.8 milliseconds to seconds (ms to s)
0.084 meters to millimeters (m to mm)

12. The Metric System Convert: 0.000850 meters to micrometers (m to m)
250 micrograms to milligrams (g to mg)

13. The Metric System
Derived units are obtained by mathematical operations on one or more basic units.
Area = length squared
1 square meter = 1 m2
Volume = length cubed (space occupied)
1 cubic meter = 1 m3
The basic unit of volume in chemistry is
the liter (L). 1 L = 1 dm3 = 1000 cm3

14. The Metric System

15. The Metric System Other derived units: Speed = distance/time, m/s
Acceleration = distance/time2, m/s2
Force = mass x acceleration, kgm/s2 newton, N
Pressure = force/area, kg/ms2 pascal, P
Energy = force x distance, kgm2/s2 joule, J

16. Units in Math
Units can be multiplied, divided, squared, canceled, etc. -- just like numbers!
102 x 10 = m2 x m = m3
Pressure = force/area
= kgm x = kg
sec m msec2
Energy = force x distance
= kgm x m = kgm2
sec sec2

17. Units in Math
Conversion factors are ratios that specify how one unit of measurement is related to another unit of measurement. They can also be expressed as equalities.
2.54 cm = inch (exact)
1.00 in cm
2.54 cm in

18. Units in Math Example: How many centimeters are there in 12.0 inches?
2.54 cm x in = cm = cm
1.00 in

19. Units in Math
Dimensional Analysis is a method for setting up calculations in which the units associated with numbers are used as a guide.
Set up the calculation so that desired units remain in the answer, and all others cancel.
Dimensions are quantitative properties such as length, time, mass.
Units are defined measurements of dimensions, such as meters, seconds, and grams.

20. Dimensional Analysis How to do it:
1a. Figure out what quantity is to be deter-mined, and what are the desired units.
1b. Identify given quantities in the problem.

21. Dimensional Analysis
2a. Choose a given quantity or a conversion factor that has the desired units.
2b. Start an equation with this quantity. If it’s a ratio, the desired units should be in the numerator.
2c. Multiply this quantity by other given val-ues and conversion factors to make un-wanted units cancel and retain desired units.

22. Dimensional Analysis
3a. Perform mathematical operations as indicated in the equation you created.
3b. Reality check: Does the result make sense?
3c. Clean up: Round to correct number of sig figs.

23. Dimensional Analysis Example:
A premature infant weighs 1703 grams. What is its weight in pounds?
454 g = 1.00 lb (inexact)

24. Dimensional Analysis Example:
At room temperature, 1.00 L of water has a mass of 1.00 kilograms. What is its mass in grams?

25. Dimensional Analysis Example:
I can ride my bicycle at 9.6 miles per hour. How long will it take me to go 23 miles?

26. Dimensional Analysis Types of conversion factors:
Equality conversion factors are ratios that interconvert different units of the same dimension.
0.454 kg = 454 g = lb
1.00 lb kg
454 g lb

27. Dimensional Analysis Types of conversion factors:
Equivalence conversion factors are ratios that interconvert units of differ-ent dimensions.
Speed = distance miles
time hour
Density = mass grams
volume cm3

28. Dimensional Analysis Example:
An investigator found that 50.3 cm3 of bovine fat had a mass of 45.1 gram. What is the density of the fat?
The investigator also found that 49.8 cm3 of bovine lean muscle had a mass of 55.0 g. What is the density of the muscle?
Which is more dense?

29. Percentage Problems
Percent is the number of items of a specified type in a group of 100 total items.
Parts per hundred
Percent = number of items of interest x 100%
total items

30. Percentage Problems
A student answered 19 items correctly on a 23 point test. What was his score as a percentage?

31. Percentage Problems
Range as a percent of the average is a way to express precision.
% of average = (highest – lowest) x 100%
average
= (20.50 – 19.25) units x 100 % = 6.32% units

32. Percentage Problems
A technician measured the breaking strength of three samples of plastic. His results were:
Run 1: MPa
Run 2: MPa
Run 3: MPa
What was the range of his measurements as a percent of the average?
Note: 1 MPa = 145 pounds/in2

33. Percentage Problems Percent difference is a way to express accuracy.
% difference = (measured – actual) x 100%
actual
= (19.78 – 20.00) units x 100% = –1.1%
20.00 units

34. Percentage Problems
A student determined the density of aluminum metal to be 2.64 g/cm3. The accepted value is 2.70 g/cm3. What is the percent differ-ence between her result and the accepted value?
Did she do a good job?

35. Percentage Problems
A student did three experiments to determine the density of rubbing alcohol. Her results were: g/mL; g/mL; g/mL. What is her precision as % of average?
The true value is g/mL. What is her accuracy?