Unit 5 The Mathematics of Chemistry
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Unit 5 The Mathematics of Chemistry. Relative Masses of Particles. What is the Mass Ratio of Golf Balls to Ping Pong Balls?. The Mass Ratio = 20 : 1. How much would 50 Ping Pong Balls weigh?. Since 5 weigh 10 grams, 50 would weigh 100 grams.

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Unit 5 The Mathematics of Chemistry

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Unit 5 the mathematics of chemistry

Unit 5 The Mathematics of Chemistry

Relative Masses of Particles

What is the Mass Ratio of Golf Balls to Ping Pong Balls?

The Mass Ratio = 20 : 1

How much would 50 Ping Pong Balls weigh?

Since 5 weigh 10 grams, 50 would weigh 100 grams.

How much would the same number of Golf Balls weigh?

2,000 grams

How many Ping Pong Balls?

5 ping pong balls = x ping pong balls10 grams 1,400 grams

X = 700 ping pong balls

How many Golf Balls?

5 golf balls = x golf balls200 grams 1,400 grams

X = 35 golf balls

1,400 grams in each

Unit 5 the mathematics of chemistry

Relative Masses and Numbers of Particles

How can chemists relate the number of atoms in a sample with the sample’s mass?

The Molar Mass (Gram Atomic Mass) of Carbon is 12.0 grams per mole (6.02 x 1023) of atoms.

What is a Mole?

The mass, in grams, of one mole of atoms.

A mole is the amount of a substance that contains the same number of particles as 12.0 grams of Carbon. That is, 6.02 x 1023 particles.

How many moles of Carbon are in a 60.0 gram sample?

12.0 grams = 60.0 grams 1.0 mole C x mole C

X = 5.0 moles of Carbon

How many moles of Carbon atoms are in a 3.0 gram sample?

12.0 grams = 3.0 grams 1.0 mole C x mole C

X = 0.25 moles of Carbon atoms

Unit 5 the mathematics of chemistry

Introduction to Dimensional Analysis

Would you agree that four quarters of an orange are equal to one whole orange?

4 = 4



Would you also agree that if you multiply any number by one, the product is equal to the original number?

So……. 17 x 1 x 1 x 1 = 17

If the top and bottom of a fraction are the same or are equal to each other, then one can multiply by that fraction and the answer is equal to the original quantity!!!!

Any conversion factor or equality can be written as a fraction and is equal to 1.

12 in. = 1 ft.


1 ft. = 12 in.


Unit 5 the mathematics of chemistry

Dimensional Analysis

What is a Dimension and what is Dimensional Analysis?

The Dimension in a measurement is the unit of measure: ft., mi., Lit., g. etc.

D-A is to study and manipulate the dimensions (units) of measurements in order to solve multi-step problems/conversions.

What are the steps in using Dimensional Analysis?

1. Dissect the question into “Desired” (what is…how many…etc.) and “Given” dimensions/units.

2. Write the “Given” quantity (number) and dimension (unit).

3. Analyze the conversion factors available and develop a plan to systematically change to the dimensions (units) that are “Desired.”

4. Set up “goal posts” to organize conversion factors. Flip the units of the “Given” value into the first “goal posts” (fraction). You can “cancel” any units on both the top and bottom of any fractions. Write conversion factors so that you can eliminate (cross off) all units except those that are “Desired.”

5. Complete the calculations: “Top times top times top times top….”

“Divided by the bottom divided by the bottom divided by the bottom…”

Unit 5 the mathematics of chemistry

Dimensional Analysis Sample Problem

1. What is the total cost ($) for socks for two leagues?

14 players

1 team

16 teams


6 pair socks

1 player


3 pair socks

2 leagues x



2 leagues = 2 x 16 x 14 x 6 x 9.50 ÷ 3 =

2. A sponsor has $500.00 for socks. How many teams* can be supplied with socks? *Include fractions

3 pair socks


1 player

6 pair socks

1 team

14 players

$500.00 x

= 1.88 teams


Unit 5 the mathematics of chemistry

Molar Mass or Gram Formula Mass

What is Gram Formula Mass?

GFM is the sum of the Gram Atomic Masses of the elements in the chemical formula.

What is Molar Mass?

Molar Mass is the mass (grams) of one mole of any kind of species: atoms, molecules, ionic compounds (formula units).

What is the molar mass or gram formula mass of CO2?

It is the mass of one mole of Carbon atoms + the mass of two moles of Oxygen atoms: 12.0 + (2 x 16.0) = 44 grams/mole

Determine the molar mass (GFM) of: H2O, CaCl2, NH3, H2SO4

18 gram/mole, 111 g/mol, 17 g/mol, 98 g/mol

How many moles of NaCl are in a 10.0 grams sample of salt?

number of moles = 10.0 grams 58.45 g/mole

= 0.171 moles of NaCl

Unit 5 the mathematics of chemistry

1 mole of any Gas @ STP = 22.4 Liters

1 mole

1 mole = 6.02 x 1023

Atoms, Molecules or Formula Units

Molar Mass = mass (g) of 1 mole

Unit 5 the mathematics of chemistry

22.4 Liters gas @ STP

1 mole

Molar Mass

6.02 x 1023 Particles

Unit 5 the mathematics of chemistry

Using the Mole Road Map

A mole map represents a simple method to convert between the mass of a sample, the number of moles of a sample and the number of representative particles (atoms, molecules or formula units) in a sample and the volume of any sample of gas at STP.

How many molecules of water are present in a 200.0 gram sample?

1 mole

18.0 g H2O

6.02 x 1023 molecules

1 mole

200.0 g H2O x

6.69 x 1024 molecules


What is the mass of 0.350 moles of Na2CO3?

106 g Na2CO3

1 mole Na2CO3

0.350 mole Na2CO3 x

37.1 g Na2CO3

Link to Video

Link to another video

Unit 5 the mathematics of chemistry

Chemical Equations

What does a balanced chemical equation indicate?

It Indicates the Ratio by which the Reactants combine and the Products are produced in a chemical reaction. The Net Energy Change may also be indicated.

What do the Coefficients indicate?

They indicate the simplest Mole Ratio of Reactants and Products.

C3H8 + O2 CO2 + H2O + 2,219.2 kJ





Every atom in the reactants must be accounted for in the products! It’s the Law of Conservation of Mass (or Matter).

It may be helpful to think of the balanced equation in terms of individual molecules or formula units.

Unit 5 the mathematics of chemistry

Reaction Stoichiometry

What is Reaction Stoichiometry?

It is the study of Quantitative Relationships between Reactants, Products and Energy in Chemical Reactions.

1 C3H8 + 5 O2 3 CO2 + 4 H2O + 2,219.2 kJ

Each component in the balanced reaction can form a MolarConversion Factor with any other component.

Every 1 mole of propane, C3H8, that reacts requires 5 moles of O2 and produces 3 moles of CO2, 4 moles of H2O and 2,219.2 kJ of Energy.

Ex. If 13.5 grams of propane react completely, then how many grams of water will be produced?

18 g H2O

1 mole H2O

4 mole H2O

1 mole C3H8

1 mole C3H8

44 g C3H8

13.5 grams C3H8

= 22.1 g H2O

Unit 5 the mathematics of chemistry

Types of Chemical Formulas

Empirical Formula: Gives Simplest Whole Number Ratio of each element in the compound. Ex. C6H12O6CH2O “Carbohydrates” All Ionic Formulas are Empirical Formulas.

Molecular Formula: Tells the Exact Number of Each Atom in one Molecule. Only for Molecular compounds. Ex. C8H18 C6H12O6 CO2

Structural Formula: Shows “which atoms are bonded to which” in a molecule. Uses straight lines to indicate covalent bonds. Also only for molecular compounds. Sometimes called “stick diagrams”.

Formula of a Hydrate: Gives ratio of water molecules to salt. ex. CuSO4 . 5H2O, MgSO4. 7H2O, BaCl2. 2H2O Calculate the Molar Mass of each Hydrate.

249.5 g/mol 246.3 g/mol244.3 g/mol

Unit 5 the mathematics of chemistry

Practice with Chemical Formulas

Give the Empirical Formula of each: C4H10C2H4O2 N2H4 CH3CH2CH2COOH

C2H5 CH2O NH2 C4H8O2 C2H4O

Given The Empirical Formula CH2O and the Molar Mass of 150.0 g/mole, find the Molecular Formula.

CH2O = 30 g/mol 30 x ? = 150  Molecular Formula is C5H10O5

Given the Empirical Formula CH2 and the Gram Formula Mass of 98.0 g/mole, find the Molecular Formula.


Find the % Composition (by mass) of Copper in CuSO4


Find the % by mass of Oxygen in K2CrO4


Unit 5 the mathematics of chemistry

Four Types of Chemical Reactions

1. CO2 C + O2

2. NaCl + AgNO3 NaNO3 + AgCl

3. S + Cl2 SCl2

4. BaCl2 + 2 NaOH  2 NaCl + Ba(OH)2

5. Zn + CuSO4 ZnSO4 + Cu

6. CH4 C + 2H2

7. Pb(NO3)2 + Mg  Pb + Mg(NO3)2

8. Mg + 2HCl  MgCl2 + H2

9. H2SO4 H2 + S + 2O2

10. 2 O2 + N2 N2O4

11. 3 CaBr2 + 2 Na3P  Ca3P2 + 6 NaBr

12. 2 KI + Br2 2 KBr + I2

13. 2 NaF  2 Na + F2

14. Si + O2  SiO2

15. 2 NaI + Pb(NO3)2 2 NaNO3 + PbI2

16. NaI + Cs  CsI + Na

17. H2 + CO + O2 H2CO3

18. Li3PO4 3 Li + P + 2 O2

19. CS2 + 2 F2 CF4 + 2 S

20. Cl2 + 2 HBr  2 HCl + Br2

Examine these reactions and Identify four different types:

(1) Write a General Formula for each type listed below. (2) Apply one of these four names to each of the sample reactions.



Single Replacement

Double Replacement

A + B  AB

3, 10, 14, 17,

AB  A + B

1, 6, 9, 13, 18

5, 7, 8, 12, 16, 19 20

AB + C  CB + A

2, 4, 11, 15,

AB + CD  AD + CB*

Unit 5 the mathematics of chemistry

Concentration of Solutions

What does the Concentration of a Solution indicate?

Concentration describes the Ratio of Solute to Solution.

Common methods of defining concentration:

part x 100whole

% by mass or % by volume

percent concentration (%)

parts per million (ppm)

moles liters

Molarity (M)

1,000,000 ppm = 100% 1 mg/kg = 1 ppm For aqueous solutions only: Density = 1 g/mL 1 liter of solution = 1 kilogram

Unit 5 the mathematics of chemistry

Practice with Solutions and Concentrations

The EPA recommends treatment of water with lead concentrations of 0.015 ppm or higher. How many grams of lead are present in a 2.50 liter sample of water with this concentration?

parts per million = grams of solute x 1,000,000 grams of solution

0.015 ppm = x grams of lead x 1,000,000 1 2,500 grams of water

x = 3.8 x 10-5 grams

How many grams of salt are required to prepare 250.0 mL of a 0.750 Molar aqueous solution of NaCl?

Convert mL to liters

molarity = moles of solute liters of solution

Solve for moles Convert to grams

X = .1875 mol x 58.5 grams 1 mole NaCl

0.750 M = x moles NaCl

1 0.250 liters

= 11.0 g

250.0 mL x 1 liter x 0.750 moles x 58.5 grams 1,000 mL 1 liter 1 mole NaCl

= 11.0 g

Unit 5 the mathematics of chemistry

How to Make a Solution

Watch the linked Video and list the steps in making an aqueous solution with a specific concentration.

  • Calculate the mass of solute required. Measure the mass of solute required.

  • Add ALL of the solute to the proper Volumetric Flask.

  • Add distilled water; approximately 2/3 of the total. Cover the top and swirl the solution to dissolve the solute.

  • Add distilled water exactly to the only mark on the neck of the flask.

  • Cover the top and invert the flask several times to complete the mixing.

How to Dilute a Solution

C = concentration V = volume i = initial f = final

Ci Vi = Cf Vf

Muriatic acid is typically a 20% (by volume) solution of HCl (aq). How would you dilute the solution to create 75 mL at 12% ?

Ci Vi = Cf Vf

X = 45 mL  “Use 45 mL of the 20% solution and add water to make 75 mL.”

20% · X mL = 12% · 75 mL

Unit 5 the mathematics of chemistry

Characteristics of Solutions

Name and describe the two main components of all solutions.

The Solute is what is dissolved and the Solvent is what the solute is dissolved in.

What are the Characteristics of Solutions?

Solutions are Homogeneous Mixtures with very small Solute Particles.

What effects can Solutes have on the properties of Solvents?

The addition of nonvolatile (not likely to vaporize) solute particles to a Solvent will: 1. Raise the Boiling Point, 2. Lower the Freezing Point, 3. Increase Osmotic Pressure, and 4. Lower the Vapor Pressure of the Solvent.

Describe some examples of these effects.

Winter Ice: Salt lowers the freezing point of water. Radiator Antifreeze also raises the boiling point of coolant water. Salt causes cells to dry out. Cooking Pasta? One tablespoon of salt will raise the boiling pt. of 5 qt. of water by about 0.05 oC!!!

These effects are Colligative Properties: They depend only on the Concentration of Solute Particles. Explain the different effects of: one mole of CaCl2 vs. one mole of C6H12O6 in same volume of water.

1 CaCl2 (s) 1 Ca2+ (aq) + 2 Cl- (aq) vs. 1 C6H12O6 (s)  1 C6H12O6 (aq) Ionic three moles of solute particles Molecular one mole of solute particles

Unit 5 the mathematics of chemistry

Temperature and Solubility

What does the Solubility of a given solute in a given solvent describe?

Solubility describes the normal, Maximum possible concentration of a solute at a given temperature. Usually some solute is undissolved.

What is a Saturated Solution?

The Solution described above is Saturated.

What is an Unsaturated Solution?

An Unsaturated Solution has a concentration of solute below the maximum. The solute is completely dissolved.

What is a Supersaturated Solution?

A Supersaturated Solution has more solute dissolved than should be dissolved!

How does Increasing Temperature affect Solubility?

Most Solids Increase solubility, but Gases Decrease solubility as Temp. Increases.

What is the Solubility of KCl at 64oC?


What concentration of NaNO3 would be Supersaturated at 40oC?

Any concentration Greater than 106g/100gH2O is Supersaturated.

Unit 5 the mathematics of chemistry

Kinetic Molecular Theory and Gases

The concept of an Ideal Gas can help us understand Real Gas behavior. Kinetic Molecular Theory describes an Ideal Gas as follows:

  • Gases are composed of individual particles in continuous, random, straight-line motion until they collide with other particles or the walls of the container.

  • Particle collisions involve a transfer of Kinetic Energy, but the total energy of an isolated system remains unchanged.

  • The Volume of the gas particles is negligibly small compared to the volume the gas occupies in its container. Why?

  • There are NO Attractive Forces (IFA) between gas particles. True?

  • The Average Kinetic Energy of the particles is proportional to the Kelvin Temperature.

What conditions of Temp. and Pressure would cause Real Gases to behave most like Ideal Gases? Why?

High Temp. and Low Pressure would reduce the effect of IFA’s as molecules pass each other at high speed and greater distance.

What Real Gases are most like Ideal Gases?

Small, nonpolar molecules: He or H2

Unit 5 the mathematics of chemistry

Characteristics of Gases

Gas Pressure is a measure of Force per unit Area. What is a typical recommended car tire pressure?

32 psi = 32 pounds of force on every square inch of surface (psi)

What is typical air pressure at sea level?

14.7 psi or 1 atmosphere (atm) or 101.3 kiloPascals (kPa) see Ref. Table A

Gas Pressure depends on How Hard and How Often the gas molecules hit a surface.

See Hyperlink

Explain how changes in Temperature affect the pressure of gas in a sealed container.

Higher Temp  Faster moving gas particles hit harder and more often. 2X the Kelvin Temp  2X Pressure

Explain how changes in Volume affect the pressure of gas in a sealed container.

If the container Volume becomes Half as large, the gas particles will hit twice as often. ½ Volume  2X Pressure

To Predict the effect of changing Temp. Pressure or Volume:

Temp. Must be Kelvin.

All units Must Match.

Make an Inventory of information given in the problem. Substitute and Solve.

Unit 5 the mathematics of chemistry

Solving Gas Law Problems







101.3 kPa

2.0 Liter

273 K

90. kPa

X Liter

303 K

101.3 x 2.0 = 90. x X 273 303

X = 2.5 Liter

At the same temp. and pressure, equal volumes of any gases have the same number of moles of molecules. Why?

The Volume of the gas particles is negligibly small compared to the volume the gas occupies in its container.

Unit 5 the mathematics of chemistry

The Ideal Gas Law

How can one relate the amount of gas (moles or grams) to the volume, pressure and temperature of a sample?

P = pressure (atm)

V = volume (mL)

n = number of moles* of molecules

R = 82.1 (atm)(mL)(mol-1)(K-1)**

T = temperature (K)

*grams and moles are easily interconverted **all units must match those of R

PV = nRT

Recall from the mole map that the volume of one mole of any gas at STP = 22.4 Liters

Ex. What is the pressure of 0.0981 grams of water vapor at 155 oC if the volume is 0.287 mL?

0.0981 g H2O 1 mole H2O = 18 grams H2O

0.00545 moles H2O

P = x atm.

V = 0.287 mL

n = 0.00545 mol

R = 82.1

T = 428K

P = nRT V

P = 667 atm

Unit 5 the mathematics of chemistry

Gas Temperature, Pressure and Volume

Sketch a graph of the following Properties of Gases:

Click the Hyperlinks below:

Temperature vs. Pressure (constant volume)


Direct Relationship

Inverse relationship

Pressure vs. Volume (constant temperature)




Temperature vs. Volume (constant pressure)

Direct Relationship

Volume vs. Pressure (constant temperature)

Inverse Relationship

Volume vs. 1/Pressure (constant temperature)

Inverse of an Inverse = Direct Relationship

Number of Molecules vs. Pressure(constant volume)

Direct Relationship

Number of Molecules vs. Volume(constant pressure)

Direct Relationship

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