Mid-Term Review

Please ask questions here, not email, so everyone can benefit.

Molarity

Molarity is a method of representing concentration.  In this case, it is the ratio of the moles of solute to the liter of solution. The abbreviation for molarity is a capital M (MUST be capitalized, "m" means molality).

Reminder- Solute is the substance that is dissolved or dispersed in a solution. Solvent is the substance that actively separates or disperses the solute. Water is the most common solvent.

There are 3 possible unknowns for simple problems dealing with molarity- M, g of solute, L of solution.

Example 1- What is the molarity of a solution made from completely dissolving 5.25 g of NaCl in 435 mL of water?

Example 2- How many grams of NaCl are needed to make 200.0 mL of 0.0150 M solution?

Example 3- How many mL of water are needed to make a 0.250 M solution using 10.00 g of NaCl?

Stoichiometry: Limiting Reactants

To put it in simplest terms, the limiting reactant is whatever runs out first. Once one of the essential components runs out, you can't make anymore of the final product.

Basic Example 1- A company builds little red wagons.  Each wagon must have a body, 4 wheels and 2 axels. If the company has 125 bodies, 400 wheels and 300 axels in stock, how many complete wagons can they produce?

While there are several ways of approaching this, we'll solve for the maximum number of complete wagons that can be made from the number of each component given.

While there are sufficient bodies and axels to make more wagons, once the company makes 100 wagons, it will run out of wheels. The wheels are the limiting reactant and therefore all other amounts are dependent on them.

Basic Example 1 con't- How many axels will the company have left over after all the wheels are used?

300 axels were available - 200 axels used = 100 axels left over

Now let's look at chemistry example-

Example 2 - 204.3 g of sodium hydroxide reacts with 79.4 g of aluminum chloride. How many grams of the base will be produced?

After 43.8 g of  aluminum hydroxide is produced, all of the aluminum chloride will be consumed (used up), so the reaction stops. Therefore the aluminum chloride is the limiting reactant (also called the limiting reagent). There will be sodium hydroxide left over- it is in excess.

Example 2 con't - What will be the mass of the excess reactant remaining?

To determine the amount remaining, we must know how much we started with (204.3 g NaOH) and how much will be used. To find how much is needed, we start with the LR (in this case, the 74.9 g of aluminum chloride).

Therefore, 204.3 g initially - 67.3 g needed = 137 g NaOH remaining

Mass-Mass Stoichiometry

In a mass-mass calculation you are given information in grams and asked for information in grams.  To complete stoichiometry problems you must know the ratio of particles.  The only way to determine this is by writing a balanced reaction.  The coefficients will provide you with the MOLE RATIO.

Remember that a reaction tells you the ratio of particles or moles, NOT MASS!  A gram of one substance will not have the same number of particles as a gram of another substance. That means you will need to convert grams to moles using molar mass.

We approach this type of problem using the same problem solving steps that we've been using all semester.

1. What are we given?
2. What are we looking for?
3. What additional information is needed?

Let's walk through an example problem:

     Example 1:  An excess amount of sodium chloride reacts with 25.0 grams of lead (II) nitrate in water.  How many grams of precipitate will be formed?

1. What are we given?
   excess NaCl, 25.0 g Pb(NO3)2
2. What are we looking for?
   grams of solid (?)
3. What additional information is needed?
   reacts means a reaction, so you need a balanced reaction
   grams means at some point molar mass will be needed
   solid, means you need to know what precipitates, so you need states

Example 2: How many grams of NaCl are actually needed to react all 25.0 g of lead (II) nitrate?

Ionic Reactions

In a molecular or complete reaction, all substances are shown in neutral (molecular) form.

In an ionic reaction, the substances that all always dissociate (strong) are broken into their ions.

Because lead (II) nitrate, sodium chloride and sodium nitrate all all soluble salt (strong electrolytes), they will all completely dissociate in water.  They will not exist as molecules, but as ions.  Remember you can't lose any mass in a reaction!  The Law of Conservation of Matter still applies!

 Notice that sodium and nitrate are EXACTLY the same on the reactant and product sides?  They are spectator ions.  Spectator ions do not change in a reaction.  When you cancel the spectator ions, you're left with the net ionic reaction.

This shows us what is really changing in a reaction- the driving force for the reaction to proceed.

Aqueous Solutions

Water is a very polar molecule, meaning it has a partial positive charge on the hydrogen end and a partial negative charge on the oxygen end.  This is caused by the unequal sharing of electrons by the hydrogen and oxygen atoms.

Because water is polar, it will dissolve most ionic compounds.  Since ionic compounds are composed of a positive ion (cation) and a negative ion (anion), the opposite charged end of a water molecule will be attracted and break a large crystal into smaller pieces.  This is called hydration. If the molecules are completely broken into their ions by water, it is called dissociation.

For instance, table salt (NaCl) will completely dissociate in water. Every single molecule will be broken apart into ions and kept apart by the water molecules.

NaCl (aq) --> Na+ (aq) + Cl- (aq)

In chemistry, STRONG means that every molecule will dissociate when dissolved in water.  WEAK means that it partially dissociates, or that only some of the particles will dissociate while others will remain in neutral/molecular form.

Strong acids, strong bases and strong electrolytes will always dissociate when dissolved in water.  Weak acids, weak bases and weak electrolytes may or may not dissociate.

Solubility Rules

Soluble means a substance will dissolve in water, whereas insoluble means it will not dissolve appreciably. If an insoluble product is created in an aqueous solution, it will precipitate out of solution as a solid.

Solubility Rules

Mainly soluble:

all nitrates & acetates

all halogens, except with silver, mercury and lead

all chlorates, except with silver, mercury and lead

all sulfates, except with calcium, strontium, barium, lead, mercury, and silver

all chromates, except with calcium, strontium, barium, lead, mercury, and silver

Mainly insoluble:

all sulfides, except with groups 1 & 2, and ammonium

all hydroxides, except with groups 1 & 2, and ammonium

all carbonates, except with group 1 and ammonium

all phosphates, except with group 1 and ammonium

Steps in Balancing Reactions by Inspection

Law of Conservation of Mass- Matter can neither be created nor destroyed, only rearranged in

a chemical reaction.

 Steps in balancing a chemical reaction:

1. predict the products in words

2. write the formula for all the reactants and products

(balance the charges and look for diatomic elements)

diatomic elements - H₂ , O₂ , N₂ , F₂ , Cl₂ , Br₂ , I₂

3. balance the polyatomic ions (nitrate. sulfate, hydroxide ...)

4. balance all other elements except hydrogen and oxygen

5. balance the hydrogen and oxygen  (don't forget to look for H and O in all compounds)

6. check every reaction again element by element 

Reactions: Basic Types

Synthesis (also called Addition, Composition and Combination)

     Atoms and or molecules are combined to make ONE more complex molecule.

A  +  B --> AB

Fe  +  S -->  FeS

metal oxide + water --> metal hydroxide

nonmetal oxide + water --> oxyacid

Decomposition (also called Analysis)

     A complex molecule is broken into simpler molecules and or atoms

AB --> A + B

KClO3 --> KCl  +  O2

metal chlorate --> metal chloride + oxygen

metal carbonate --> metal oxide + carbon dioxide

Single Replacement

     A single element replaces one component of a compound.  A metal must always replace a metal and a nonmetal must replace a nonmetal.  A single replacement replacement is a type of redox reaction where electrons are transfered from one atom to another.

A  + BX --> AX + B

K + CsCl --> KCl + Cs

X + AY --> AX  + Y

Cl2  +  MgBr2 --> MgCl2  +  Br2

Double Replacement (also called Ionic and Precipitation Reactions)

     Ionic compounds switch "partners."  Remember that a neutral compound must have a cation combined with an anion.

AX  +  BY --> AY  +  BX

CuCl  +  NaS --> CuS  +  NaCl

Determining an Empirical Formula

One way of determining the identity of an unknown in a lab is by analyzing its mass to determine its empirical formula (lowest whole number ratio of each element in a compound).  There are several types of problems, but all of them use the same concepts to start.

Example 1:  An unknown substance is composed of 24.7% potassium, 34.7% manganese and 40.5% oxygen.  Determine the empirical formula for this compound.

Problem!  You can't compare percentage by mass to determine the ratio of ATOMS!

1. The first step is to convert the percentages to MOLES using the MOLAR MASS for each element.
2. Once you have all the substances in moles, you can compare them to find the mole ratio.  There are several ways of doing this.  The easiest is to divide by the smallest value.  This usually works, but remember that an empirical formula is written in the LOWEST WHOLE NUMBER ratio, so if you a left with a fraction, you must multiply the entire ratio by a factor that will convert the fractions into WHOLE NUMBERS.
   
3. Therefore the ratio of K:Mn:O is 1:1:4, so the empirical formula is KMnO4.

Percent Composition

Percent always allows us to compare a part of something to the whole.
In general

% = part x 100

total

For percent composition

% = total mass of particles requested x 100

molar mass

Example: Determine the % oxygen in sulfuric acid.

% O =       4 oxygen          x 100

(2 H + 1 S + 4 O)

%O =               4(16.0) ___  _  x 100 = 48.9% O

                                                         2(1.0) + 32.1 + 4(16.0)

Example: Determine the % sulfate in sulfuric acid.

%SO4 =               32.1 + 4 (16.0)___  _  x 100 = 97.9% O

                                                            2(1.0) + 32.1 + 4(16.0)

Mole Conversions

While we tend to measure amounts in grams, the only way to compare amounts of atoms, molecules or ions is by using moles.  Unit analysis allows us convert one set of units to another.

To convert grams to moles, or visa-versa, we use molar mass that has the units grams/1 mole.

To convert number of particles to moles, or visa-versa, we use Avogadro's number (6.02 x 10^23) that has the units particles/1 mole.

Example: How many chlorine atoms are in 75.0 grams of sodium chloride?

Hydrates

Hydrates are ionic salts that trap water molecules in their crystal lattice.  This added mass must be used when making calculations therefore the ratio between molecules of ionic salt and water is given in the name.

For instance, calcium sulfate hexahydrate states that for every molecule of calcium sulfate there are 6 water molecules surround it.  We represent a hydrate with a large dot then the number of water molecules.  This dot is NOT a multiplication sign, it is actually a ratio.

To determine the molar mass of hydrate, determine the mass of the salt then add the mass of however many water molecules are attached to it.

1 Ca + 1 S + 4 O + 6(2 H + 1 O)

40.1 + 32.1 + 4(16.0) + 6(18.0)

244.2 g/mole

The Mole

Mole is a term used in chemistry to represent the number 6.02 x 10^23.  Just as we use the word "dozen" to mean 12 objects, "mole" represents 6.02 x 10^23.

Amedeo Avogadro studied molecular theory in the early 19th century and built on the ideas of Dalton and Guy Lussac.  The number of particles in a mole was actually discovered later in the century and named in his honor.

Atoms and molecules are VERY small. Remember we measure their mass in atomic mass units, amu.  An amu is equal to 1/12th the mass of a carbon-12 atom or approximately the mass of a proton or neutron. The wonderful thing about the very odd number is it allows to work with measurable quantities.

One mole of atoms of any element is equal to its atomic mass (average mass number) in grams.  This is called molar mass.

The molar mass of a compound is simply the sum of the masses of each of its atoms.

water is H20
there are 2 Hydrogen and 1 Oxygen
therefore its molar mass is 2(1.0) + 1(16.0) or 18.0 grams/mole

This means that if you have 18.0 grams of water, you will also have 6.02 x 10^23 molecules of water.

Naming Acids

The most fundamental definition of an acid is an anion combined with H+ to form a neutral compound.
All acids begin with a H⁺  and are combined with one of the three types of anions.

            -ide             to            hydro----ic acid                        H2S          hydrosulfuric acid

            -ate             to                        -ic acid                        H2SO4      sulfuric acid

            -ite              to                        -ous acid                     H2SO3      sulfurous acid

                        I -ate something and it made me s-ic.

                                        and you m-ite give it to -ous.

Naming Ionic Compounds

Ionic compounds are formed between oppositely charged particles called ions.  Ions are formed when electrons are lost, called cations, or when electrons are gained, called anions.

A molecule must be neutral.  The total amount of positive charges must cancel out the total amount negative charges to create an overall charge equal to zero.

Naming Covalent Compounds

Covalent bonds are formed by 2 or more atoms sharing electrons.  Typically this is between 2 non-metals since both atoms want to gain electrons to acquire a full outer shell.

To name covalent compounds, use the following prefixes to represent the subscripts and change the suffix of the more electronegative (always on the right) element to -ide.

1  mono

2  di

3  tri

4  tetra

5  pent

6  hex

7  hept

8  oct

9  non

10 dec

The Atom: Quantum Mechanical Model

Born, Planck, Heisenberg and Schrondinger built on the work of Rutherford to develop the Quantum Mechanical Model of the Atom.  Here are the primary points:

1. Energy is ultimately quantized into small discrete packet that we call photons.
2. Electrons can absorb (gain) or emit (lose) photons, but they can't use or lose a portion of a photon.
   - When an electron gains/absorbs a photon it will become excited and jump to a higher energy level.
   - When an electron loses/emits a photon it will has less energy and jump to a lower energy level.
   - Electrons DO NOT MOVE (Newtonian concept), they JUMP.  They cannot use a portion/fraction of a photon.  It's all or nothing.
   - When electrons jump from one energy level to another, it is called a Quantum Leap.  They have gained or lost a quantized amount of energy- a photon.
3. Heisenberg Uncertainty Principle- (basic) The exact velocity and location of an electron cannot be determined without changing its velocity and/or location.  We can only determine an electron's probable location in the electron cloud.  This probable position is explained by the 4 Quantum Numbers.
4. Quantum Numbers
• Principle Quantum Number describes the distance from the nucleus, or the energy level.
• Orbital Quantum Number describes the shape of the electron cloud.
• Magnetic Quantum Number describes the orientation of the electron cloud in space with respect to the 3 axis.
• Spin is a notation of up/down to represent that the electrons within a pair will always be as far apart as possible (Remember- like charges repel).
5. Pauli Exclusion Principle- No 2 electrons in the same atom can have the same 4 Quantum Numbers.  A pair of electrons in the same atom will be in the same energy level, shape and orientation, but they will always be as far apart as possible.

The Atom: Development of the Modern Theory

In 1897 British physicist Joseph John (J. J.) Thomson (1856–1940) discovered the electron in a series of experiments using a cathode ray tube at Cambridge University.  He said that there were "bodies much smaller than atoms" that had a negative charge.  He called the negative particles electrons. Thomson took the model of the atom one step further with his plum-pudding model.  While you might not be familiar with plum-pudding, you are probably very familiar with a chocolate cookie.  Thomson theorized that if there were negative particles in the atom, there must also be a positive charge to make an atom neutral. His model described the atom as being basically positive (the cookie) with areas of negative that can be broken off (the chips).

One of Thomson's students, Ernest Rutherford built on his research and further developed our modern concept of the atom. One of his first discoveries was that radiation was not a single substance but three. While he was at McGill University in Canada, he found that radiation could be influenced by an electrical charge.  The radiation that was attracted to a negative charged, he called alpha (positive charge).  The radiation that was attracted to a positive charge he called beta (negative charge).  Later gamma radiation was found to be unaffected by any charge (neutral).

After receiving the Nobel Prize, Rutherford moved to Manchester and attracted many young scientists to his research group. Under his direction, they continued to study the atoms and radiation.  The Gold Foil Experiment was to become one of the most famous.  In this experiment, alpha particles (+) were fired at a very thin sheet of gold foil.  It was expected that the particles would go straight through the foil like bowling balls through tissue paper.  Imagine their surprise when some of the particles seemed to be deflected, and even bounced back off.  This observation led to the conclusion that the atom is mostly empty space, with a small, hard positively charged center.  Rutherford called this the nucleus.  This group would go on to discover the proton in 1917.

Later Rutherford and a fellow Danish physicist, Neils Bohr, developed a new model of the atom.  They were bothered by the concept that if electrons were small and negative, and the nucleus was dense and positive, why don't the electrons simply fall into the nucleus?  They developed a model that described the atom as having a small positive nucleus with the electrons orbiting the nucleus in energy levels.  In this model, the electrons are held in the atom by the electrostatic attraction between the positive nucleus and negative electrons, but are constantly being pushed outward by their angular momentum.  This is the familiar planetary model that is commonly taught to young students.

During this same time, Max Planck was studying electromagnetic radiation in Berlin. Among his many discoveries, he found that EMR is quantized. That it is carried by small, discrete packets that he called photons. These small quantities cannot be divided, they are the smallest amount of energy.  This can be very difficult for us to conceptualize.  We live in a world where we can divide amounts easily into smaller amounts.  You don't have to use an entire gallon of gas at a time, you can just use a little, then a little more.  But when you start to look at the world on the subatomic level, you reach a point where it is all or nothing.  Think of money.  We normally think in terms of dollars.  That's the level we are comfortable with, that we use everyday. Dollars can be broken down into quarters, dimes, nickels and even pennies.  We don't think of paying for everything in pennies because they are very small in relation to what we are buying- but imagine buying something very small. Can you pay for something worth a portion of a penny?  No, you either use a whole penny or not.  Photons are the pennies of the electromagnetic spectrum.

Bohr and Rutherford incorporated this concept into their model. Now an electron could not move from one level to another, it had to jump or make a quantum leap. When an electron absorbs a photon, it will jump to a higher energy level, when an electron emits a photon, it will fall to a lower energy level.  Again, this is difficult for us to imagine.  We think in terms of something orbiting, but in reality electrons don't orbit, they jump. 

Werner Heisenberg, Max Born and Wolfgang Pauli took these ideas and formed the quantum mechanical model of the atom.

The Atom: Early Theories

The modern concept of the atom can be traced back to Democritus in 465 BC.  Unlike most Greeks of his time, he believed that everything was made up of small, indivisible shapes that he called atomos.

For the next 2 thousand years, most thinkers, philosophers and alchemists followed the basic idea set down by Aristotle.  He argued that everything was composed of some combination of the 4 elements- fire, earth, air and water.  This belief slowly began to break down during the Renaissance. This was when belief and debate began to be replaced with experimentation and observation.

Antoine Lavoisier (1743-1794) began to pull the ideas of the previous 200 years when he clearly stated the Law of Conservation of Matter.  From his experiments, he found that even if the appearance of a substance may change, the total mass of the materials before a change will always equal the total mass after the change. Thanks to Einstein, we now realize that we can make nuclear changes that will convert matter into energy, but Lavoisier's concept is still valid for chemical changes.
Law of Conservation of Matter- matter can neither be created nor destroyed by ordinary chemical means.

In the early nineteenth century, Dalton further defined matter by clearly stating that all matter is composed of atoms. He said an atom is the smallest unit of an element that can exist alone or in combination with other atoms of the same or different elements.

1. All matter is made up of very small particles called atoms.
2. Atoms of the same element are all chemically alike: atoms of different elements are chemically different.
3. The atoms of different elements have different average masses.
4. Atoms are not subdivided in chemical reactions, they unite in simple ratios to form compounds.

Over the next century, many others made contributions to our concept of the atom. Many different types of atoms (elements) were identified and investigated.  By the end of the nineteenth century, about 60 elements had been discovered and described.  Dmitri Mendeleev was the first to publicly attempt to organize the elements and look for patterns.  He organized the known elements by atomic weight and their known properties.  He was even able to predict the properties and atomic weights of elements that had not been discovered yet, but there were problems with his table.  There were several elements that seemed to be in the wrong place.  One of the most glaring was iodine.  Iodine clearly had properties similar to bromine, but it fell in the same row with oxygen.

Henry Gwyn Jeffreys Mosley (1887-1915):  Mosley began to work on the inconsistencies found in Mendeleev's table at Oxford. He arranged the elements in order of their atomic numbers horizontally and formed columns of elements with similar properties. Tragically for the development of science, Moseley was killed in action during WW1 at Gallipoli in 1915. 

The modern periodic law states that if the elements are arranged by increasing atomic number, their properties will reoccur at regular intervals.

The Atom: Isotopic Notation

Isotopes are atoms with the same number of protons but have a different number of neutrons.  They are basically different varieties of the same element.  Two of the most common isotopes that people hear about are Carbon-14 and Carbon-12.  Both isotopes are carbon because they each have 6 protons.  They are different because C-14 has 8 neutrons and C-12 only has 6.  While C-12 is the most abundant (common) isotope of carbon, both are called isotopes.

The number of protons in an atom is also called the atomic number.  The number of protons determines the identity of an atom, no matter how many neutrons or electrons are in the atom.  Atomic number is listed on the periodic table.  It is a whole number and is usually listed above the symbol for the element.  If you change the number of protons in an atom, the element has also changed.

Mass number is the total number of particles in the nucleus of an atom.  In other words, it is the number of protons plus the number of neutrons.  Isotopes have the same atomic number, but different mass numbers.  Mass numbers are NOT listed on the periodic table.  When you name a specific isotope, you MUST include its mass number, for instance Carbon-14 or Uranium-135.

The number below the symbol of the element on the periodic table is called the atomic mass.  It is the weighted average mass number of all the isotopes of a particular element.  Because it is an average, it has significant digits.

We can represent an isotope in an abbreviated form.  This is called the isotopic notation.

Here is the isotopic notation for carbon 14.

From looking at the isotopic notation, you can determine the number of protons, neutrons and electrons for a given atom.

In a neutral carbon 14 atom, determine the number of protons, neutrons and electrons?
Protons- The atomic number is 6, therefore there are 6 protons
Electrons- If the atoms is neutral the number of protons equals the number of electrons, therefore there are also 6 electrons.
Neutrons- The mass number equals the number of protons + neutrons, therefore 14-6 leaves 8 neutrons.

Mixtures

Pure substances are uniform throughout with a definite composition and properties, while mixtures are physical combinations of two or more pure substances. The properties of the substances in a mixture retain their own properties.

We can further divide pure substance in chemistry into elements and compounds.  An element is basically the name of a type of atom defined by the number of protons in the nucleus.  A compound in a chemical combination of two or more elements.  A compound is the name of a type of molecule (chemically bonded atoms).

Note:  Physical combination means that substances are just dispersed or close together. A chemical combination means that the atoms are chemical attached creating new molecules with new properties.  Chemical bonds cannot be separated by physical means.

Mixtures can be divided into homogeneous mixtures that appear to be a single substance and heterogeneous mixtures that are obviously two or more substances.

All mixtures are composed of a solute and a solvent. The solute is the substance that is dissolved or dispersed, while the solvent is the substance that does the dissolving.  The solvent separates and keeps the solute particles apart.

Mixture can be divided into three categories:

Solutions are homogeneous and clear.  The particles are so small they cannot be seen and do not reflect light.

     Ex:  windex, tap water, air

Colloids are homogeneous but appear cloudy.  Some of the particles are large enough to reflect light even though they can’t be seen with the naked eye.

     Ex:  milk, fog, mayonnaise

Suspensions are heterogeneous.  Given time gravity will separate a suspension with the most dense particles on the bottom, and the least dense (lightest) particles rising to the top.

     Ex:  Italian salad dressing, oil and water

Describing Matter

In lab we make both qualitative and quantitative measurements.  We can further describe matter in terms of extensive and intensive properties.  Extensive properties are quantity specific, such as mass and volume. The mass of a sample of water depends on how much water you have in a sample. Intensive properties are dependent only on the type of matter, not the quantity.  The density of water is 1.0 g/mL whether you have a drop or a swimming pool full.

We can further divide observations into physical and chemical properties.  Physical properties describe matter without changing its composition.  Chemical properties describe how matter interacts or changes with other matter.

Physical Properties                                Chemical Properties

Color                                                                Rusting

Odor                                                                 Burning

Density                                                            Tarnishing

Boiling Point

Malleability

The state of matter, and its boiling and melting points are all physical properties.  The state of matter is determined by the arrangement of its particles.

The change of state of a substance is a physical change.

                                    Boiling                                                Evaporating

                                    Melting                                               Condensing

                                    Freezing                                              Sublimation

                                    Solidifying

If the identity of a substance is changed, or a new substance is formed, it is a chemical change.

Unit Analysis: Density

You must use unit analysis to solve density problems in chemistry (NOT THE FORMULA & ALGEBRA).  Remember you are learning the process  of unit analysis, not simply finding the correct answer!

Use the technique you have learned in problem solving:  what are you looking for, what are you given, put the units together to cancel and solve.

Example #1

A sample of a known metal is dropped into a graduated cylinder with 25.0 mL of water. The level rises to 31.5 mL. If the given density of the metal is 2.56 kg/L, what is the mass of the sample in grams?

Possible Questions
Where did 6.5 mL come from? A substance displaces its volume when immersed in water.  The water level rose from 25.0 to 31.5 mL.
Why are there only 2 sig dig in the final answer?  When you multiply and divide, you use the least number of sig dig.  2.56 kg/1L has 3, 6.5 mL has 2, and the conversions for L and kg are exact values with an infinite number of sig dig.


Example #2

What is the volume of a sample of aluminum in L that weighs 56.7 g?  The density of aluminum is 2.71 g/mL.

Possible Questions

Why flip the density?  To make sure the units of volume will be in the numerator.

I got a different answer (0.153657), why? Remember that 2.71 is in the denominator.  You must divide, not multiply. You can either input 56.7 first, divided by 2.72, divided by 1000
OR start with 1 divided by 2.71, times 56.7, divided by 1000.

Unit Analysis: Solving Word Problems

Unit Analysis can make solving complex word problems much easier.  First, DON’T BE INTIMIDATED!  The problem is not going to jump off the page and bite you if you get it wrong!  Just TRY!  Follow these basic steps to simplify problem solving.

1. What are your looking for?  Read through the problem and determine the exact units requested.

          WRITE THAT DOWN!

2. What are you given?  Sometimes, there is so much information given it is a good idea to write it all down or underline it in the problem.  It also helps if you will label what type of given information it is.  For instance:  mass, distance, …

3. Is there any other information you need?  Conversions, molar mass, reactions, …

4. Put the units together in such as way that you cancel out the units you don’t want and end up with only the units requested.  If the units are reversed in your final answer, just flip your calculation.

Example #1

A farmer has 2 cows and he decides to change to chickens.  He can barter 4 emu for each cow, 3 emu for 5 pigs, 8 pigs for 3 llama,  a llama for 20 rabbits and 3 rabbits for 2 chickens.  How many chickens can he get for both his cows?

1. What are you looking for?  chickens

2. What are you given?  Many ridiculous ratios with animals.

3. Is there any other information needed?  Not for this problem, just watch out where you step.

4. Use UA to determine the units requested.

When you plug these values into the calculator, the screen reads 66.66666667, but we are looking for WHOLE, LIVE chicken, not parts.

Remember that in science we deal with objects and measurements, NUMBERS HAVE MEANING.  You must evaluate your answer based on what the units are, as well as significant digits.

Example #2

A machine produces 4.5 x 10³ m of spaghetti noodles each minute. A package of noodles contains 128 noodles that are each 12.5 inches long.  The company sells the noodles in cartons containing 20 packages for $75.50.  If the machine runs 12.0 hr a day, 5.00 days a week, 50.0 weeks a year, how much money can the company make each year from that one machine?

Unit Analysis: The Basics

Unit analysis or dimensional analysis is a method used to calculate values based on the units of each measurement.  We will start by using this method to simply convert one measurement in one unit to another unit. This technique may seem more complicated than necessary at this point, but remember you are learning how to use the units.  Later in the semester you will see that unit analysis will make problem solving so much easier!

Unit analysis is based on two very fundamental mathematical principles.

1. any number multiplied by one is equal to itself
2. a fraction equals one if the value of the numerator equals the value of the denominator

These two properties allow us to let the measurements determine how to do the calculation.  The final answer must have the units desired and all other units must be canceled.

Changing both the unit in the numerator and the denominator is just like changing only one set of units.  Just remember that all the units except the ones needed must cancel.

Change 3.40 m/sec to km/yr.

Notice that you must put the 60 sec in the numerator to cancel the sec you were given.  All units must cancel except the ones requested in the problem.  In this case, km/year.

Measurement: Precision & Accuracy

Accuracy and precision are terms used to explain the sources of error in a data set.  Accuracy describes how close a measurement is to the correct answer. Precision describes the spread of the data or how close the measurements are to each other.

To determine the accuracy of a measurement, the correct or accepted value must be known.  The most common calculation associated with accuracy is percent error.

percent error = |(accepted value - experimental value)|   x   100

                                   experimental value

The precision of a data set can be determined in a number of ways, including range, standard deviation and percent deviation. Range is determined by subtracting the smallest value from the largest value in a data set.

Deviation literally means difference, so we can calculate it using subtraction.  By finding the difference between an individual measurement and the average of all the measurements in a data set, we can find how "off" that single measurements is from all the others. A very basic way of looking at standard deviation is to think of it as the average of all the deviations of the individual measurements from the average of the data set.  

The problem with simply using standard deviation to determine precision is magnitude (the size of the numbers.) A standard deviation of 1.00 may sound large or small without some idea of the magnitude of the measurements in the data set.  If your measurements range from 1.20 to 3.56, it is huge!  But if the range of the data is 1000.0 to 1002.0, it would be much more acceptable.

Percent compares the part to the whole, so it takes away the uncertainty of magnitude.  Percent deviation allows us to compare the standard deviation to the average of the data set.  The lower the percentage that each individual measurement differs from the average of the data set, the better the precision.

percent deviation = standard deviation  x  100

                                average of the data set

Measurement: Significant Digits

There are two types of numbers in science: exact (counting) numbers and inexact (measurements and calculated quantities).  Exact or counting numbers represent objects.  For instance, a dozen eggs has exactly 12 eggs.  You can’t have 12.01 eggs.  Measurements and numbers based on calculations will always have some uncertainty. Significant digits are used to represent that uncertainty or the amount of confidence you have in a measurement.

Uncertainty occurs because we use equipment to make measurements.  You can only measure a length as exact as the increments on the ruler you are using.  Significant digits are the numbers we know with certainty plus one more that is estimated.

Basic Rules:

1. All non-zero digits are significant
2. All zeros between non-zero digits are significant
3. Zeros to the right of the decimal and to the right of a non-zero digit are significant
4. Zeros to the right of the decimal, but to the left of all non-zero digits are not significant
5. If there is no decimal, zeros to the right of the last non-zero digit are not significant

Rules for Calculations:

1. In addition and subtraction, use the LEAST number of DECIMALS.
2. In multiplication and division, use the LEAST number of SIGNIFICANT DIGITS.
3. Apply each rule using the order of operations.