1. Mathematical Relationships in Chemistry

2. What You’ll Learn in this Unit
Significant Figures
Scientific Notation
Measurement
Dimensional Analysis
Error
Density
Graphical analysis

3. Review of Measurement Terms
Qualitative measurements - words
Quantitative measurements – involves numbers (quantities)
Depends on reliability of instrument
Depends on care with which it is read

4. Precision vs. Accuracy
Precision- the degree of agreement among several measurements of the same quantity.
Accuracy- the agreement of a particular value with the true value

5. Uncertainty Basis for significant figures
All measurements are uncertain to some degree
Random error - equal chance of being high or low- addressed by averaging measurements - expected

6. Significant Figures Meaningful digits in a measurement
The number of significant figures in your measurement will tell the reader how exact the instrumentation used
If it is measured or estimated, it has sig figs.
If not it is exact (e.g. 5 apples).

7. Significant Figures All numbers 1-9 are significant.
The problem are the ZEROS.
Which ones count and which don’t?
In between numbers 1-9 does
Example: ……… has 4 sig figs
Now let me tell you a story…

8. Left handed Archer
There once was a left handed archer who loved to shoot decimals. Zeros could not stop his arrow but numbers could.
If there is a decimal in the number begin on the left. Go through any zeros, come to the first number then all other numbers that follow are SIGNIFICANT!
→0.0040
→50.401

9. No decimals
If a number has no decimals you begin on the right hand side.
Go through any zeros , come to the first number.
Then all numbers after that count
5000←
405,000 ←

10. Doing the Math
Multiplication and division, same number of sig figs in answer as the least in the problem
Addition and subtraction, same number of decimal places in answer as least in problem.

11. Scientific Notation 100 = 1.0 x 102 0.001 = 1.0 x 10-3
-- This provides a way to show significant figures.

12. TOO QUICK FOR YOU! So here are the rules.. slowly!
Place decimal point after 1st real non-zero integer. (ex) 1.0 NOT 10.0
Raise 10 to the exponential which equals the number of places you moved.

13. Scientific Notation
The product of 2.3 x 10 x 10 x 10 equals 2300 (2.3 x 103)
Note:
Moving the decimal to the left will increase the power of 10
Moving the decimal to the right will decrease the power of 10

14. Sample Problems
2387

15. Answers
2.387 x 103
7.031 X 10-5
2.9 x 109
8.900 X 10-3
9.01 X 107
2.10 X 10-6

16. Scientific Notation Multiplication and Division
Use of a calculator is permitted
use it correctly
No calculator? Multiply the coefficients, and add the exponents
(3 x 104) x (2 x 102) =
(2.1 x 103) x (4.0 x 10-7) =
6 x 106
8.4 x 10-4

17. Scientific Notation Multiplication and Division
In division, divide the coefficients, and subtract the exponent in the denominator from the numerator
3.0 x 105
6.0 x 102
5 x 102 =

18. Scientific Notation 7 x 10-8 3.17 x 10-5 (6.6 x 10-8) + (4.0 x 10-9) =
Addition and Subtraction
Before numbers can be added or subtracted, the exponents must be the same
Calculators will take care of this
Doing it manually, you will have to make the exponents the same- it does not matter which one you change.
(6.6 x 10-8) + (4.0 x 10-9) =
(3.42 x 10-5) – (2.5 x 10-6) =
7 x 10-8
3.17 x 10-5

19. Measurement Every measurement has two parts
COMMON SI UNITS
Symbol  
Unit Name  
Quantity  
Definition
   m
meter
length
base unit
   kg
kilogram
mass
   s
second
time
   K
kelvin
temperature   
   °C
degree Celsius**  
temperature
   m3
cubic meter
volume m3
   L
liter**
dm3 = m3
   N
newton
force
kg·m/s2
   J
joule
energy
N·m
   W
watt
power
J/s
   Pa
pascal
pressure
N/m2
   Hz
hertz
frequency
1/s
Every measurement has two parts
Number with the correct sig - figs
Scale (unit)
We use the Systeme Internationale (SI).

20. Metric Base Units Mass - kilogram (kg) Length- meter (m) Volume- (L)
Time - second (s)
Temperature- Kelvin (K)
Electric current- ampere (amp, A)
Amount of substance- mole (mol)
Energy – joule (j)

21. Prefixes giga- G 1,000,000,000 109 mega - M 1,000,000 106
kilo - k 1,
deci- d
centi- c
milli- m
micro- m
nano- n

22. Using Units to solve problems
Dimensional Analysis
Using Units to solve problems

23. Dimensional Analysis Use conversion factors to change the units
1 foot = 12 inches (equivalence statement)
12 in = = 1 ft ft in
2 conversion factors
multiply by the one that will give you the correct units in your answer.

24. Example Problem
The speed of light is 3.00 x 108 m/s. How far will a beam of light travel in 1.00 ns?
Well, we know that 1.00 ns = 10-9 seconds
(3.00 x 108 m) X (10-9 s) = x m/ns
s (1.00 ns)

25. Example Problems 11 yards = 2 rod 40 rods = 1 furlong
8 furlongs = 1 mile
The Kentucky Derby race is 1.25 miles. How long is the race in rods, furlongs, meters, and kilometers?
A marathon race is 26 miles, 385 yards. What is this distance in rods, furlongs, meters, and kilometers?

26. Volume The space occupied by any sample of matter
Calculated for a solid by multiplying the length x width x height
SI unit = cubic meter (m3)
Everyday unit = Liter (L), which is non-SI

27. Units of Mass Mass is a measure of the quantity of matter
Weight is a force that measures the pull by gravity- it changes with location
Mass is constant, regardless of location.
The SI unit of mass is the kilogram (kg), even though a more convenient unit is the gram
Measuring instrument is the balance scale

28. Density
Which is heavier- lead or feathers?
It depends upon the amount of the material
A truckload of feathers is heavier than a small pellet of lead
The relationship here is between mass and volume- called Density

29. Density Common units are g/mL, or possibly g/cm3, (or g/L for gas)
Ratio of mass to volume
D = m/V
Common units are g/mL, or possibly g/cm3, (or g/L for gas)
Useful for identifying a compound
Useful for predicting weight
An intensive property- does not depend on what the material is

30. Things related to density
density of corn oil is 0.89 g/mL and water is 1.00 g/mL
What happens when corn oil and water are mixed?
Why?
Will lead float?

31. Example Problem
An empty container weighs g. Filled with carbon tetrachloride (density 1.53 g/cm3 ) the container weighs g. What is the volume of the container?

32. Density and Temperature
What happens to density as the temperature increases?
Mass remains the same
Most substances increase in volume as temperature increases
Thus, density generally decreases as the temperature increases

33. Density and water
Water is an important exception
Over certain temperatures, the volume of water increases as the temperature decreases
Does ice float in liquid
water?
Why?

34. Specific Gravity
A comparison of the density of an object to a reference standard (which is usually water) at the same temperature
Water density at 4 oC = 1 g/cm3

35. Formula Note there are no units left, since they cancel each other
D of substance (g/cm3)
D of water (g/cm3)
Note there are no units left, since they cancel each other
Measured with a hydrometer
Uses?
Gem purity
differentiating between different types of crude oils/gasoline
urine tests for concentration of all chemicals in your urine
Specific gravity =

36. Temperature
Heat moves from warmer object to the cooler object
Glass of iced tea gets colder?
Remember that most substances expand with a temperature increase?
Basis for thermometers

37. Temperature scales
Celsius scale- named after a Swedish astronomer
Uses the freezing point (0 oC) and boiling point (100 oC) of water as references
Divided into 100 equal intervals, or degrees Celsius

38. Temperature scales
Kelvin scale (or absolute scale)
Named after Lord Kelvin
K = oC + 273
A change of one degree Kelvin is the same as a change of one degree Celsius
No degree sign is used

39. Temperature scales
Water freezes at 273 K
Water boils at 373 K
0 K is called absolute zero, and equals –273 oC

40. Temperature A measure of the average kinetic energy
Different temperature scales, all are talking about the same height of mercury.
In lab take the reading in ºC then convert to our SI unit Kelvin
ºC = K

41. 100ºC = 212ºF
0ºC = 32ºF
100ºC = 180ºF
1ºC = (180/100)ºF
1ºC = 9/5ºF

42. Example problem
A 55.0 gal drum weighs 75.0 lbs. when empty. What will the total mass be when filled with ethanol? density g/cm gal = L
1 lb = 454 g

43. Error Calculations X 100 accepted value
Error = Experimental value- accepted value
% error = [error]
accepted value
X 100

44. Graphing
The relationship between two variables is often determined by graphing
A graph is a “picture” of the data

45. Graphing Rules – 10 items
Plot the independent variable on the x-axis (abscissa) – the horizontal axis. Generally controlled by the experimenter
Plot the dependent variable on the y-axis (ordinate) – the vertical axis.
3. Label the axis.
Quantities (temperature, length, etc.) and also the proper units (cm, oC, etc.)
4. Choose a range that includes all the results of the data

46. Graphing Rules
5. Calibrate the axis (all marks equal) 6. Enclose the dot in a circle (point protector) 7.Give the graph a title (telling what it is about) 8. Make the graph large – use the full piece of paper 9. Indent your graph from the left and bottom edges of the page 10. Use a best fit line, do not connect points