Chem. 1A Dr. Dan Evans

General Chemistry 1

Mathematical Operations

Multiplication/Division on Calculators

Calculate 22 on your calculator
45 * 12

The correct answer is 0.04074 (the wrong answer is 5.867)

Type it into the calculator this way:

22/(45 * 12) =

or

22/45/12 =

Rule: Multiply by everything in Numerator; Divide by everything in Denominator.

Exponential Notation

Chemistry deals with numbers that can be very large or very small. These numbers can require a large number of place holding zeros to write out. An easy way to represent and work with those numbers is with scientific notation. Scientific notation uses only the numbers that are significant and does not need any place holder zero’s

# x 10ⁿ

# is written using appropriate significant digits where the number is between 1 and 10

Any number can be represented using scientific notation.

The powers of ten replace the need for place holding zeros.

Example: 10⁶ = 1,000,000

To convert a number to Scientific notation.

Write the number.

Write the significant digits with the decimal point after the first digit. Multiply this number by 10 raised to a power. The number of places that the decimal place moved equals the power on ten.

Moving the decimal place to the left is a positive power.

Moving the decimal place to the right results in a negative power.

Example: 0.000004562
Write: 4.562
The decimal point moved 6 places to the right, so multiply this number by 10^-6
Scientific notation = 4.562 x 10^-6

Scientific Numbers on Calculators

To enter 4.562 x 10^-6 onto a calculator

Type 4.562 EE (–)6

Or possible 4.562 EE 6+

then, Enter, times, divide, etc. and continue with calculation

or

Type 4.562 Exp (-)6

Or possible 4.562 Exp 6+

Logarithms

A base 10 logarithm (log) returns the exponent on 10 that represents a number.

For example: 10,000 = 10⁴
So log(10,000) = 4

On a calculator: it is either Log # =
or # Log

The antilog takes the number and uses it as an exponent on 10.

Example: antilog 2.5 = 10^2.5

On a calculator:
It can be 2nd log #
or
10 ^ #

Quadratic Equations

A quadratic equation has the form of:

ax² + bx + c = 0

There are two solutions for x given by the quadratic formula:

x = -b ± Öb2 – 4ac)
2a

The two answers come from switching the sign ± between + and -.

Sometimes only one answer is valid.

Graphing

Graphs show relationships of data sets better than tables.

Graphs should contain certain components:
A Title at the top or bottom of graph to identify what is being graphed under what conditions;
Each axis should be numbered in a consistently linear fashion;
Each axis should have a label identifying what property is being graphed, and what units are being used.

Units with exponential notation:
If units have exponential notation, there are a couple ways to write then out;
If the units run from 1 x 10^-3 atm to 9 x 10^-3 atm,
Then the numbers can be written out whole through the axis, or
the axis can be numbered with 1 through 9 with a (x 10^-3) at the end of the line, or
the axis can be numbered with 1 through 9 with the label identifying "Pressure, atm (x 10³);
This last one means that the number 5 on the axis is 5 x 10^-3 (x 10³) = 5.

Adding a line to plotted points:
Do Not Connect the dots!
For a straight line, use a straight edge to draw the line and balance the points above and below the line through the length of the line,
For a curved line, draw a smooth curved line through the points balancing the points above and below the line.

Introduction: Matter and Measurement        

States of matter

Three states of matter: Solid, Liquid, Gas

Solid: atoms and molecules are close packed and in a rigid formation.

Liquid: atoms and molecules are close packed and freely moving

Gases: atoms and molecules are widely separated and freely moving

Physical States of Matter

Property	Solid	Liquid	Gas
Shape	Definite	Indefinite	Indefinite
Volume	Fixed	Fixed	Variable
Compressibility	Negligible	Negligible	Significant
Atom/Molecule spacing	Close packed	Close packed	Widely spaced
Atom/Molecule motion	Rigid location	Active motion	Active motion

Elements, Compounds and Mixtures

Element: have definite compositions and constant properties.
Elements cannot be broken down further by an ordinary chemical change.
No further separation is possible.

Examples: He, Fe, Ar, Al
With multiple atoms of one type: H₂, O₂, N₂, S₈, O₃

Compounds: have definite compositions and constant properties.
Compounds cannot be physically separated into more components.
However, compounds can be separated by chemical means into multiple elements.
Compounds generally are made up of 2 or more elements.

Pure substance: is material that has definite composition and constant properties. A pure substance can be an element or a compound.

Mixture: Material that can be separated using physical methods into two or more substances. Mixtures have variable composition and properties.

Homogeneous mixture: is a mixture that has uniform properties through out the material

Heterogeneous mixture: is a mixture whose properties vary through the material

Examples

Alloy: a solid homogeneous mixture of two or more metals.

Salt water: Homogeneous mixture

Salt and ice: Heterogeneous mixture (a small sample may be mostly ice or mostly salt)

Coffee: a homogeneous mixture of water (aqueous mixture), coffee bean extract, cream and sugar

Iced coffee: an heterogeneous mixture consisting of an aqueous coffee extract with cream and sugar along with small chunks of ice (frozen water)

Physical and Chemical Methods of Separation

Physical methods of separation: Does not change any compounds present; filtering, picking, distilling, evaporation, freezing, centrifuge, etc.

Chemical methods of separation: Changes one compound into another; combustion, reaction with other chemicals, decomposition, etc.

Physical and Chemical Properties

Physical Property:
Characteristics that can be observed without changing the composition of the compound.
(concentration changes in mixtures are not a composition change)

Examples:
Appearance
melting point
boiling point
density
specific heat
heat conductivity
electrical conductivity
solubility
physical state
crystallizes

Chemical Property:
Characteristics of chemical reactivity with other substances.
Produces new compounds.
(Does not include producing mixtures)

Examples:
Reacts with water
Reacts with air
combusts
decomposes
non-reactive

Physical and Chemical Changes

Physical Change:
Chemical compounds do not change

Examples:
Melting
freezing
boiling
crystallizing
evaporation
vaporizing

Chemical Change:
The composition of the material changes, a chemical reaction takes place producing new compounds or elements. Can be accompanied by release of gas, light or energy/heat.

Examples:
Electrolysis
combustion
decomposition

Metric Units and SI Units

The Metric System of units is the preferred system to use in science (and the world). Within the Metric System, the preferred set of units to use to allow ease of conversion between properties is the SI units (Systeme International d'Unites). We are not limited to using SI units, we can use whatever metric units are convenient.

Metric Units

Quantity	Unit	Symbol	SI?
Base Units (When SI)
Length	Meter	m	Yes
Mass	Kilogram	kg	Yes
	gram	g	No
	Unified atomic mass unit	u (sometimes amu)	No
Time	Second	s	Yes
Temperature, T	Kelvin	K	Yes
	Celsius	°C	No
Amount of Substance	Mole	mol	Yes
Electric Current	Ampere	A	Yes
Light Intensity	Candela	cd	Yes
Derived and Other Units
Heat/Energy	Joule	J [m^2 kg s^-2]	Yes
	calorie	cal	No
Volume	cubic meter	m^3	Yes
	Liter	L	No
Density	kilogram per cubic meter	kg/m^3	Yes
	gram per cubic centimeter	g/cc (g/cm^3)	No
	gram per milliliter	g/mL	No
Concentration	mole per cubic meter	mol/m^3	Yes
	Molarity (mole/Liter)	M [mol/L]	No
Pressure	pascal	Pa [N/m^2] [m^-1 kg s^-2]	Yes
	We will use several pressure units, the most common one will be: atmospheres	atm	No
Specific heat capacity	joule per kilogram Kelvin	J/(kg K) [m^2 s^-2 K^-1]	Yes
	We will most often use: Joule per gram degree Celsius; or calorie per gram degree Celsius	J/g°C cal/g°C	No

Prefixes

Prefixes are used to indicate decimal fractions of the base units.

Prefixes

Prefix	Symbol	Meaning	Example
exa	E	10^18
peta	P	10^15
tera	T	10^12
Giga	G	10^9	1 gigameter (Gm) = 1 x 10^9 meter
Mega	M	10^6	1 megameter (Mm) = 1 x 10^6 meter
Kilo	k	10^3	1 kilometer (km) = 1 x 10^3 meter
Deci	d	10^-1	1 decimeter (dm) = 1 x 10^-1 meter (0.1 meter)
Centi	c	10^-2	1 centimeter (cm) = 1 x 10^-2 meter (0.01 meter)
Milli	m	10^-3	1 millimeter (mm) = 1 x 10^-3 meter (0.001 meter)
Micro	m	10^-6	1 micrometer (mm) = 1 x 10^-6 meter
Nano	n	10^-9	1 nanometer (nm) = 1 x 10^-9 meter
Pico	p	10^-12	1 picometer (pm) = 1 x 10^-12 meter
femto	f	10^-15
atto	a	10^-18

Temperature and Heat

Temperature: a measure of the average energy of motion of the particles in a system. This energy of motion is related to how fast the atoms and molecules move, i.e. temperature measures average velocity.
Temperature is an intensive property (i.e. it does not depend on the quantity of substance)

Heat is a measure of the total energy of a material. Heat is also an extensive property that depends on the quantity of substance involved. If you double the amount of a material while retaining the same temperature, you have doubled the heat content of the system.

Note: There are other forms of energy [vibrational, rotational, etc.] that are not measured as temperature. These other forms of energy can be included in the related concept of heat and heat content.

Example: dissolving ammonia chloride in water results in a solution which is colder than the starting materials. Did the solution loose energy? No, but the energy was converted from motion to another form which does not affect temperature. So the temperature dropped, but the energy of the system, and heat content, did not change.

Note:

Temperature is an intensive property (i.e. it does not depend on the quantity of substance)

Heat is an extensive property which depends on the quantity of substance involved.

Temperature Scales

There are three common temperature scales:
Fahrenheit (°F), Celsius (°C), and Kelvin (K)

The zero temperature of each of these scales is based on:
Fahrenheit - salted ice water
Celsius - ice water
Kelvin - lowest possible temperature (Absolute zero)

Note: negative Kelvin temperature is impossible.

The divisions on the Celsius scale and the Kelvin scale are equal size.

Comparison of the Temperature Scales

	Fahrenheit	Celsius	Kelvin
Absolute zero	-460 °F	–273.15 °C	0 K
Freezing point of water	32 °F	0 °C	273.15 K
Average room temperature	70 °F	21 °C	294 K
Normal body temperature	98.6 °F	37 °C	310 K
Boiling point of water	212 °F	100 °C	373.15 K

We only need to know the bold temperatures

Absolute Zero:

The temperature at which theoretically the pressure and volume of a gas are zero.

The coldest theoretical temperature and corresponds to 0K or –273.15°C

An ideal gas would have no kinetic energy and no molecular motion at absolute zero.

Temperature Conversions

Fahrenheit to Celsius Conversions

(°F - 32°F) x 100°C/180°F = °C

(°F - 32°F) x 5°C/9°F = °C

(°F - 32°F)/1.8 = °C

This equation first aligns the temperature to the freezing point of water (the zero for the Celsius scale) then multiplies by the unit conversion factor.

Celsius to Fahrenheit Conversions

(°C x 180°F/100°C) + 32°F = °F

(°C x 9°F/5°C) + 32°F = °F

(°C x 1.8°F/°C) + 32°F = °F

Celsius to Kelvin Conversion

°C + 273.15° = K

Kelvin to Celsius Conversion

K – 273.15 = °C

Fahrenheit to Kelvin Conversions

(°F + 459.67°F)/1.8 = K

Example

What is the Celsius equivalent of oral body temperature, 98.6°F?

(98.6°F – 32.0°F)(5°C/9°F) = 37.0°C

Density

It is observed that different materials weigh more or less than other materials even though there size (volume) is the same. Some objects float while others sink. The property that describes this variability is called density (d).

Higher density objects sink in fluids

Lower density objects float in fluids

Density is the amount of mass in a unit volume.

Density = mass/volume (mass per volume)

d = m/V also
m = dv
v = m/d

Units

Solids and Liquids

Density = g/mL

or

Density = g/cm³ (g/cc)

Gases

Density = g/L

One Density to Remember

d(H₂O) @ 4°C = 1.00 g/mL

Density is an intensive property, it does not depend on the amount of material present a glass of water has the same density that a lake has.

Temperature Affect:
The density of liquids and solids vary slightly with temperature and the temperature effect can often be ignored.
The density of gases varies greatly with temperature and cannot be ignored.

Density Examples (Do Not Memorize)

Material	Density, g/mL	Material	Density, g/mL
Ethanol	0.789	Acetic Acid	1.05
Vegetable oil	0.79	Diamond	3.51
Methanol	0.791	Zinc	7.14
Ice	0.917	Copper	8.92
Water	1.00 @ 4°C	Gold	18.9
Air	1.29 g/L	O2	1.43 g/L @ 0°C

Density as unit Factor

The density of a substance can be used as an unit factor for the purpose of converting the units between mass and volume.

Volume and mass to Density

Question:
A block measuring 2 cm x 3 cm x 1.5 cm has a mass of 8.1 g.
What is the density of the block?

Formula:
d = m/v

Solution:
8.1 g /(2 cm x 3 cm x 1.5 cm)= 8.1 g/9 cm³ = 0.9 g/cm³ or 0.9 g/mL
In addition and subtraction: the units have to match prior to addition and subtraction.

Mass and density to volume

Question:
An irregular piece of ice is known to weigh 3.75 g. The density of ice is 0.917 g/cm³.
What is the volume of the ice?

Formula:
v = m/d

3.75 g / (0.917 g/cm³) = 3.75 g (1 cm³/0.917 g) = 4.09 cm³

Volume and density to mass

A piece of zinc was found to have a volume of 12.3 mL. The density of zinc is 7.14 g/mL.

What is the mass of the zinc?

Formula:
m = dv

Solution:
12.3 mL x 7.14 g/mL = 87.8 g

More Density Examples

Question:
Copper (density = 8.92 g/mL) with a mass of 52.7 g has what volume?

Formula:
d = m/V
or
V = m/d

Solution:
V = m/d = 52.7 g/8.92 g/mL = 52.7 g(1 mL/8.92 g) = 5.91 mL

Question:
12.3 mL of zinc (d = 7.14 g/mL) has what mass?

Formula:
d = m/V
or
m = dV

Solution:
m = dV =7.14 g/mL(12.3 mL) = 87.8 g

Precision (a prelude to Significant Figures)

High precision has low measurement error.

This means:
A lower range between individual measurements
or
A greater number of decimal places in the measured value

Example:

Low precision Higher precision
2.2 lbs/kg 2.204622 lbs/kg

Accuracy

How close the measured value is to the true or accepted value.

Measurement Uncertainty

Exact measurements are not possible, All measurements have uncertainty.

Counting objects can produce an exact number.

And Definitions produce exact numbers.

Measuring Devices

Measurement uncertainty is usually derived from the measurement device.
Some devices have more precision (less uncertainty) than others.

In analog devices, e.g. rulers and graduated cylinders, uncertainty in measurements is derived from the size of unit divisions of measuring device (we can estimate one significant figure more than unit divisions.)

Some measurement devices have stated precisions, such as digital devices.

Uncertainty is often represented by showing a range expressed as a ± number after the measured number.

Example with + range

10.0 ± 0.5 cm

This number has the possible range from
10.0 - 0.5 cm = 9.5 cm as the lowest possible value
to
10.0 + 0.5 cm = 10.5 cm as the highest possible value

Example using Significant Digits

The error is assumed to be ± 1 in the last significant digit if it is not otherwise stated.

A distance is measured to be 2340 m. What is the possible range for this measurement?

2340 m has 3 significant digits (the zero is not significant), 4 is the last significant digit, so the error on 4 is ± 1.

The error is 2340 m ± 10 m.

The range goes from the low of to the high of
2340 m - 10 m = 2330 m 2340 m + 10m = 2350 m

Measurement Uncertainty (an example)

If you weigh yourself, the scale may say 165.4 lbs. You get off and back on and the scale may read 164.7 lbs. Repeating the process can give; 164.4, 166.2, 165.7, 165.3, and 164.9 lbs. This variation is the result of the precision of the scale. In this series, the decimal place is obviously not valid. There is too much scatter in it. An average of these can give 165 lbs. The digit "5" has a minor amount of scatter from 4 to 6. This is considered the last significant digit. The decimal place is not included. So there are 3 significant digits: "1", "6", "5".

Averages

Averages can have less or more significant digits than the measured numbers that make them up.

Significant Digits

A representation of the precision of a measurement.

All measured values have an error associated with them, which is shown through the number of significant digits represented in the number.

Significant digits is the number of valid digits in a number (excluding place-holder zeros)

Significant digits include all certain digits and the first uncertain digit.

Some zero's are Place-holder zero's, which locate the decimal point.

Identifying Number of Significant Digits in a Measured Value

· Include all non-zero digits

· Zero's between nonzero digits are significant

· Zero's left of nonzero digits are NOT significant

· Zero's right of nonzero digits ARE significant ONLY when a decimal point is present

Exact Numbers

· Exact numbers have an unlimited number of significant digits.

· Exact Numbers do not affect # of significant digits in calculations

· Defined values are Exact Numbers.

· Counted values are Exact Numbers.

Exact number examples

1 ft = 12 in

1 kilometer = 1,000 meters

These are defined relationships, so both numbers are exact and have an unlimited number of significant digits

Significant Digit Examples

3,456 Four Significant digits

34,056 Five significant digits (count the zero)

34,560 Four significant digits (the zero is a place-holder zero to hold the decimal point which is not shown)

34,560. Five significant digits

74,000 Two significant digits (the zeros are place-holder zeros to hold the decimal point which is not shown, if any of the zero's were significant, scientific notation would be needed to show that; i.e. 7.40 x 10⁴ shows three significant digits for the same number)

0.00457 Three significant digits (the zero's are place-holder zeros)

0.004570 Four significant digits (left zero's are place-holder zero's, right zero is significant)

0.004070 Four significant digits (left zero's are place-holder zero's, center and right zeros are significant)

Working with Significant Digits

In addition or subtraction, significant digits are limited to the decimal place or column that is contained in all the numbers used. [Show all certain and first uncertain decimal place]

In multiplication and division, significant digits are limited to the smallest number of significant digits in each number used.

Rounding: <5 round down, >5 round up

, =5 exactly, round even

If 1st digit to cut is <5, then cut digit [replace with zero's if necessary to hold decimal place]

If 1st digit to cut >5, Round last retained digit up by 1 [add zero's to hold decimal place if necessary]

Significant Digits Examples

Rounding Examples

145,000 + 7,400 152,000 Not 152,400

0.001523 - 0.00045 0.00107 Not 0.001073

245 [3] x 0.47 [2] 120 [2] Not 115.15 or even 115

Round 14,756 to three significant digits yields 14,800 Not 148

Example

27.5 cm x 15 cm x 14.3 cm = 5898.75 cm³ = 5900 cm³

Answer is limited by number of significant digits in the value 15 cm

Example

Mass of 7 coins is 87.62 g. What is the average mass of each coin?

87.62 g/7 coins = 12.51714286 g/coin = 12.52 g/coin

The value "7" is a counted value and does not affect the number of significant figures of the answer.

Significant Figures, A reality Check example

0.5^2 = 0.5 x 0.5 = 0.25 = 0.3 (1) = (1) x (1) = (1) One Significant Digit

0.50^2 = 0.50 x 0.50 = 0.25 (2) = (2) x (2) = (2) Two significant digits

Check Limits

One Significant Digit	Two Significant Digits
0.6^2 = 0.6 x 0.6 = 0.36 = 0.4 (1) = (1) x (1) = (1)	0.51^2 = 0.51 x 0.51 = 0.2601 = 0.26 (2) = (2) x (2) = (2)
0.4^2 = 0.4 x 0.4 = 0.16 = 0.2 (1) = (1) x (1) = (1)	0.49^2 = 0.49 x 0.49 = 0.2401 = 0.24 (2) = (2) x (2) = (2)

By varying the last significant digit, you can see where the variability is and which digits are significant.

Doing this varying of the numbers and following the error through the equation is Sensitivity Analysis. It is used to determine which variables in a complex equation really affect the result.

Units in Calculations

In multiplication and division; the units are multiplied and divided along with the numbers

In addition and subtraction the units have to match first.

Example (Addition/Subtraction)

5 g + 0.025 kg	Convert one unit to match the other	5 g + 25 g
= ?		= 30. g

Example (Multiplication/Division)

4.0 g / 0.25 mole = 16 g/mole (16 g/ 1 mole)

Unit Conversions

Unit factors are used to convert between different units. This can be used to convert between metric units, between metric and English units, and between units representing different physical quantities.

· Find Unit Equation relating the two units to convert between.

· Turn unit equation into unit factor, such that units will cancel properly providing the desired unit(s).

· Multiply.

Example:

Convert 2,745 grams to kilograms.

The desired unit equation is: 1,000 grams = 1 kilogram.

Since grams is given and kilograms is desired,
we want the unit factor with kilograms in the numerator:
1 kilogram/1,000 grams

The conversion:
2,745 grams x (1 kilogram/1,000 grams) = 2.745 kilograms

Unit Conversions

Example 2:

Convert 213.245 pounds to kilograms.

The unit factor is 1 kg/2.205 pounds

The conversion:

213.245 lbs x (1 kg/2.205 lbs) = 96.70975 kg = 96.71 kg

Note: this unit conversion is not an exact equivalent so it influences significant digits. Since four significant digits are given for lbs, it implies four significant digits for 1.000 kg. Since it is not an exact equivalent, then additional significant digits are available. If, you do not want to loose significant digits in the conversion then you need six significant digits for unit factor.

213.245 lbs x (1 kg/2.20462 lbs) = 96.726419 kg = 96.7264 kg

Note that the last significant digit changed in the first part of this example.

Unit Analysis Problem Solving (more examples)

· Write down the units of the answer

· Write down the units given

· Multiply by appropriate unit factors to change units to that of the answer.

Why it Works

Any number can be multiplied by one without changing its value

Any equation can be multiplied by one without changing its value

Example

A tank is 1 meter by 2 meters by 4 meters.

How many liters does it hold?

Conversion needed: cubic meters to liters (1,000 L/1 m³)

1m x 2m x 4m x (1,000 L/m³) = 8 m³ x (1,000 L/m³) = 8,000 L

Example

175 in is how many cm?

Conversion needed: (2.54 cm/1 in)

175 in (2.54 cm/in) = 445 cm

Multiple unit factors may necessary to make the proper conversion.

Example: (km/hr m/s)

Convert 75 km/hr to m/s.

· Find Unit Factors.

We are given the units of km and we need meters, the unit factor is 1,000 m/1 km

We are given the units of hour and we need seconds, the unit factor is 3600 s/hr or 1 hr/3600 s.

· The conversion:

75 km/hr x (1,000 m/1 km) x (1 hr/3600 s) = 20.8 m/s (21 m/s)

Note: significant digits are not limited by 1 hr or 1 km since these are exact equivalents.

Another Example with multiple unit factors

65 mi/hr is how many meters/second (m/s)?

65 mi/hr (1760 yd/1 mi) (1 m/1.094 yd) (1 hr/60 min) (1 min/60 s) = 29 m/s

Metric-English Equivalents

Length	Mass	Volume
1 meter = 1.094 yards	1 kilogram = 2.205 pounds	1 liter = 1.06 quarts
2.54 centimeters = 1 inch	453.6 grams = 1 pound	1 gallon = 3.785 liters
1 foot = 0.3048 m	Ounce = 28.35 g	1 cup = 236.6 mL
1 mile = 1.609 km

Need to know one from each column.