SCIENTIFIC NOTATION

  • Scientific Notation is a convenient way to express very large or very small quantities.
General form:

Using scientific notations:

  • Changing between conventional and scientific notation:
  • Addition and subtraction (NOT COVERED)
  • Multiplication and division :
  1. Change numbers to exponential form.
  2. Multiply or divide coefficients.
  3. Add exponents if multiplying, or subtract exponents if dividing.
  4. If needed, reconstruct answer in standard exponential notation.

Examples:

  1. Multiply
  1. Divide 60,000 by 0.003
Follow-up Problems:

ACCURACY & PRECISION

  • For measurements to be useful, it is important that they be precise and accurate.
  • Accuracy is closeness of a measurement to an external standard.
  • Precision is closeness of a measurement to another similarly obtained measurement.
Two types of error can affect measurements:
  • Systematic errors: those errors that are controllable, and cause measurements to be either higher or lower than the actual value.
  • Random errors: those errors that are uncontrollable, and cause measurements to be both higher and lower than the average value.
Evaluate the accuracy and precision of each set of data shown below:



ERRORS IN MEASUREMENTS

Two kinds of quantities are used in science:
  • Counted or Defined: exact numbers; no uncertainty (error)
  • Measured: are subject to error; have uncertainty (error)

Uncertainty in Measurements:

  • Every measurement has uncertainty because of instrument limitations, human error, and number of measurements.
  • The uncertainty in a measurement appears in the last recorded digit.
  • An uncertainty of one unit is assumed in all measurements, unless otherwise specified.
  • In reading a measurement scale, it is wrong to record more than one estimated digit.
  • The last digit is the estimated one.
    8.65 cm

RECORDING MEASUREMENTS TO THE PROPER NO. OF DIGITS

What is the correct value for each measurement shown above?
a) (1 certain, 1 uncertain)
b) (2 certain, 1 uncertain)
c) (3 certain, 1 uncertain)

SIGNIFICANT FIGURES

  • Scientists use significant figures to express the precision of a measurement.
  • Significant figures are the number of certain and uncertain digits

Examples:

Determine the number of significant figures in each of the following measurements:
0.05082 in 4 significant figures
3 significant figures
14.303 m 5 significant figures
0.00025 L 2 significant figures
150000 mg ambiguous (should be written in scientific notation)
2 significant figures
3 significant figures
4 significant figures

SIGNIFICANT FIGURES IN CALCULATIONS

Multiplication and Division:

  • The measurement with the least certainty limits the certainty of the results; or
  • The answer must contain the same number of significant figures as in the measurement with the least number of significant figures.

Examples:

Addition and Subtraction:

  • The answer must be rounded to the same number of decimal places as there are in the measurement with the fewest decimal places.

Examples:

83.5 5.74
+ 0.8233
106.78 (calculator answer) +2.651
106.8 (rounded answer)
9.214 (calculator answer)
9.21 (rounded answer)
4.8
-3.965
0.835 (calculator answer)
(rounded answer)

SIGNIFICANT FIGURES IN CALCULATIONS

Rounding Off Rules

When rounding to the correct number of significant figures:
  • round down if the rounded digit is 4 or less.
  • round up if the rounded digit is 5 or more
3 sig. figs 2 sig. figs.
8.4234 rounds off to 8.42 8.4
14.780 rounds off to 14.8 15
3256 rounds off to 3260 3300

Examples:

Perform the following operations to the correct number of significant figures:

MEASUREMENTS

  • Measurements are made by scientists to determine size, length and other properties of matter.
  • For measurements to be useful, a measurement standard must be used.
  • A standard is an exact quantity that people agree to use for comparison.
  • SI is the standard system of measurement used worldwide by scientists.
SI BASE UNITS
Measurement Units Symbol
Length meter m
Mass kilogram kg
Time seconds s
Temperature kelvin K
Amount of substance mole mol
SI PREFIXES
Prefix Symbol Meaning Multiplier
mega- M million
kilo- k thousand
hecto- h hundred
deca- da ten 10
1
deci- d tenth
centi- c hundredth
milli- m thousandth
micro- millionth
nano- n billionth
DERIVED UNITS
Measurement Units Symbol
Volume liters L
Density grams

VOLUME

  • Volume is a measure of the amount of space occupied by an object.
  • Volume is a derived quantity, with units of .
  • The SI base unit of volume is Liter (L) which is equal to cm .
  • Volume of various regular shapes can be calculated as follows:
Cube
Rectangular
solid
Cylinder
Sphere

DENSITY

  • Density is the ratio of mass of an object to its volume.
  • Density is an intensive property (i.e. independent of the amount of matter).
  • Mass and volume are examples of extensive properties (i.e. dependent on the amount of matter).
  • Density is a measure of how tightly packed an object's mass is.

Examples:

  1. A copper sample has a mass of 44.65 g and a volume of 5.0 mL . What is the density of copper?
  1. A silver bar with a volume of has a mass of 294 g . What is the density of this bar? m =

CONVERSION FACTORS

  • Many problems in chemistry and related fields require a change of units.
  • Any unit can be converted into another by use of the appropriate conversion factor.
  • Any equality in units can be written in the form of a fraction called a conversion factor.
For example:
Equality:
Conversion factors: or
Metric-Metric
Equality:
Conversion factors: or
Metric-English
  • Sometimes a conversion factor is given as a percentage. For example:
Percent quantity: body fat by mass
Conversion factors: or

CONVERSION OF UNITS

  • Problems involving conversion of units and other chemistry problems can be solved using the following step-wise method:
  1. Determine the intial unit given and the final unit needed.
  2. Plan a sequence of steps to convert the initial unit to the final unit.
  3. Write the conversion factor for each units change in your plan.
  4. Set up the problem by arranging cancelling units in the numerator and denominator of the steps involved.

Examples:

  1. Convert 164 lb to
Step 1: Given 164 lb Find kg
Step 2:
Step 3: or
Step 4:
2. Convert 5678 m to km .
Step 1: Given Find
Step 2 & 3:
Step 4:
  1. How many centimeters are in 2.0 ft ? ( )
Step 1: Given 2.0 ft Find cm
Step 2: ft \begin{tabular}{|l|}
\hline English-English

factor
in
\hline English-Metric

factor
\end{tabular} cm
Step 3: and
Step 4:

4. Bronze is by mass copper and by mass tin. A sculptor is preparing to cast a figure that requires 1.75 lb of bronze. How many grams of copper are needed for the brass figure?
Step 1:
Given
Find
Step 2:

Step 3:
Step 4:
1.75 lb bronze x _ x _
g copper

UNITS RAISED TO A POWER

  • When converting quantities with units raised to a power (e.g. ), the conversion factor must also be raised to that power. For example:

Examples:

  1. A large pizza has a surface area of . What is this area in ?
  1. How many cubic inches are there in ?
  2. A classroom has a volume of . What is this volume in ?

DENSITY AS A CONVERSION FACTOR

  • Density of a substance can be used as a conversion factor between mass and volume. Problems below show some examples of this.

Examples:

  1. If the density of gold is , how many grams does a nugget weigh?
  2. The gasoline in an automobile gas tank has a mass of 60.0 kg and a density of . What is the volume of the tank in ?
  3. If the density of milk is , what is the mass of 0.50 qt of milk? ( )