 Shane Plunkett
 plunkes@tcd.ie
 Some Mathematical Functions
 What is a mole?
 Avogadro’s Number
 Converting between moles and mass
 Calculating mass % from a chemical formula
 Determining empirical and molecular formulae from mass
 Recommended reading
 T.R. Dickson, Introduction to Chemistry, 8th Ed., Wiley, Chapters 2 & 4
 M.S. Silberberg, Chemistry, The Molecular Nature of Matter and Change,
 3rd Ed., Chapter 3
 P. Atkins & L. Jones, Molecules, Matter and Change, 3rd Ed., Chapter 2
 Multiple choice tests: http://www.mhhe.com/silberberg3
Carrying out Calculations  In chemistry, must deal with several mathematical functions.
 Scientific Notation
 Makes it easier to deal with large numbers, especially concentrations
 Written as A ×10b, where A is a decimal number and b is a whole number
 Example: Avogadro’s number
 602 213 670 000 000 000 000 000
 It is very inconvenient to write this. Instead, use scientific notation:
 6.022 × 1023
 Calculators:
 Sharp & Casio Type in 6.022
 Press the exponential function [EXP]
 Key in 23

Questions  (a) 784000000
 (b) 0.00023
 (c) 9220000
 (d) 0.000000015
 How would you write the following:
 7.84 × 108

 2.3 × 104
 9.22 × 106
 1.5 × 108
 (a) (1.38 × 104) × (8.21 × 106)
 (b) (8.56 × 108) × (2.39 × 104)
2. Logarithms  Makes dealing with a wide range of numbers more convenient, especially pH
 Two types: common logarithms and natural logarithms
 Common Logarithms
 Common log of x is denoted log x
 gives the power to which 10 must be raised to equal x
 10n = x
 written as: log10x = n (base 10 is not always specified)
 Example:
 The common log of 1000 is 3, i.e. 10 must be raised to the power of 3 to get 1000
 Written as: log101000 = 3
Calculators  Sharp: Press the [LOG] function
 Type the number
 Hit answer
 Casio: Key in the number
 Calculate the common logarithms of the following:
 log 10
 log 1000000
 log 0.001
 log 853
Natural logarithms  Natural log of x is denoted ln x
 the difference here is, instead of base 10, we have base e (where e = 2.71828)
 Gives the power to which e must be raised to equal x
 lnx or logex = n or en = x
 Example
 The natural log of 10 is 2.303, i.e. e must be raised to the power of 2.303 to get 10
 Calculators
 Sharp: Press the [ln] function
 Enter the number and hit answer
 Casio: Enter the number
 Press the [ln] function
Questions  What is the natural log of:
 50
 1.25 × 105
 2.36 × 103
 8.98 × 1013
 ln 50
 ln 1.25x105
 ln 2.36x103
 ln 8.98x1013
3. Graphs  Experimental data often represented in graph form, especially in straight lines
 Equation of straight line given by
 y = mx + c
 where x and y are the axes values
 m is the slope of the graph
 c is the intercept of the plot
Sign of slope tells you the direction of the line  Sign of slope tells you the direction of the line
 Magnitude of slope tells you steepness of line
 Slope found by taking two x values and the two corresponding y values and substituting these into the following relation:
 Example
 Given the (x, y) coordinates (2, 4) and (5, 9), find the slope of the line containing these two points.
 Sub into above relation: m = 9 – 4 = 5 = 1.67
 5 – 2 3
4. Quadratic Equations  May be encountered when dealing with concentrations
 Involve x2 (xsquared terms)
 Take the form ax2 + bx + c = 0
 Can be solved by:
Find the roots of the equation x2 – 6x + 8 = 0 Question  You have been asked to calculate the concentration of [H3O+] ions in a
 chemical reaction.
 The following quadratic equation has been given Solve for x.
 2.4x2 + 1.5x – 3.6 = 0
 Because we are dealing with concentrations, a negative value will not make sense. Therefore, we report the positive x value, 0.95, as our answer. Never round up this number!
Important formulae so far… Calculations: The Mole  Stoichiometry is the study of quantitative aspects of chemical
 formulas and reactions
 Mole: SI unit of the amount of a substance
 Definition: A mole is the number of atoms in exactly 12g of the
 carbon12 isotope
 This number is called Avogadro’s number and is given by 6.022 ×1023
 The mole is NOT just a counting unit, like the dozen, which specifies only the number of objects. The definition of a mole specifies the number of objects in a fixed mass of substance.
 Mass spectrometry tells us that the mass of a carbon12 atom is
 1.9926×1023g.
 No. of carbon12 atoms = atomic mass (g)
 mass of one atom (g)
Other definitions of the Mole  One mole contains Avogadro’s Number (6.022 x 1023)
 A mole is the amount of a substance of a system which contains as many elementary entities as there are atoms in 0.012kg (or12g) of Carbon12
 A mole is that quantity of a substance whose mass in grams is the same as its formula weight
 E.g. Fe55.85
 Iron has an atomic mass or 55.85g mol1, so one mole of iron has a mass or 55.85g
One mole of any object always means 6.022 × 1023 units of those objects. For example, 1 mol of H2O contains 6.022 × 1023 molecules 1 mol of NaCl contains 6.022 × 1023 formula units  Avogadro’s number is used to convert between the number of
 moles and the number of atoms, ions or molecules.
 Example
 0.450mol of iron contains how many atoms?
 Number of atoms = number of moles × Avogadro’s number (NA)
 Therefore No. of atoms = (0.450mol) × (6.022 × 1023)
 = 2.7 × 1023 atoms
 Calculating the number of particles
Example  How many molecules are there in 4 moles of hydrogen peroxide (H2O2)?
 No. of molecules = no. of moles × Avogadro’s number (NA)
 = 4mol × (6.022 × 1023 mol1)
 How many atoms are there in 7.2 moles of gold (Au)?
 The visible universe is estimated to contain 1022 stars. How many moles of stars are there?
 Answer: 1022 stars = 1022 = 0.17 mol.
 6.022×1023
Calculating the mass of one molecule  Example: What is the mass of one molecule of water?
 Step 1: Calculate the molar mass of water
 Molar mass of water = (2 × atomic mass H) + (1 × atomic mass O)
 Molar mass H2O = (2 × 1.008g mol1) + (1 ×16.000g mol1)
 = 18.00 g mol1
 Step 2: Employ Avogadro’s number
 Mass of one molecule = Molar mass
 Avogadro’s no.
 = 18.00g mol1 6.022×1023mol1
 Note: Always check the units you have in your answer to ensure you are correct
Example  Calculate the mass of one molecule of ammonium carbonate [(NH4)2CO3]
 2 Nitrogen atoms
 8 Hydrogen atoms
 1 Carbon atom
 3 Oxygen atoms
 2 × 14.01gmol1
 8 × 1.008 gmol1
 1 ×12.01gmol1
 3 × 16.00 gmol1
 = 28.02 gmol1
 = 8.064 gmol1
 = 12.01 gmol1
 = 48.00 gmol1
 Step 2: Employ Avogadro’s Number, NA
 96.09 gmol1 .
 6.022×1023mol1
 Calculate the mass of one molecule of:
 Ethanoic acid (CH3COOH)
 Methane (CH4)
 Potassium dichromate (K2Cr2O7)
Converting between mass and moles  In the lab, we measure the mass of our reactants in grams using a
 balance. However, when these react they do so in a ratio of moles.
 Therefore, we need to convert between the mass we measure and the
 number of moles we require.
 The expression relating mass and number of moles is:
 Mass of sample (g) = no. of moles (mol) × molar mass (gmol1)
 Step 1: Find the molar mass of the compound
 Step 2: Substitute into the above expression
Questions  Calculate the mass in grams present in:
 (a) 0.57mol of potassium permanganate (KMnO4)
 Answer: Molar mass KMnO4 = 158.03 gmol1
 Mass in grams = 0.57mol × 158.03 gmol1
 = 90.07 g
 (b) 1.16mol of oxalic acid (H2C2O4)
 Answer: Molar mass H2C2O4 = 90.04 gmol1
 Mass in grams = 1.16mol × 90.04 gmol1
 = 104.44 g
 (c) 2.36mol of calcium hydroxide (Ca(OH)2)
 Answer: Molar mass Ca(OH)2 = 74.1 gmol1
 Mass in grams = 2.36mol × 74.1 gmol1
 = 174.87 g
Converting between moles and mass  Number of moles = mass of sample (g)
 molar mass (gmol1)
 Convert 25.0g of KMnO4 to moles
 Step 1: Calculate the molar mass
 1 × 39.10 gmol1
 1 × 54.93 gmol1
 4 × 16.00 gmol1
 39.10 gmol1
 54.93 gmol1
 64.00 gmol1
 Step 2: Substitute into above expression
Questions  Calculate the number of moles in:
 Answer: Molar mass water = 18.02 gmol1
 (b) 3.0g of carbon dioxide (CO2)
 Answer: Molar mass carbon dioxide = 44 gmol1
 (c) 500g of sucrose (C12H22O11)
 Answer: Molar mass sucrose = 342.30 gmol1
 (d) 2.00g of silver chloride (AgCl)
 Answer: Molar mass silver chloride = 143.38 gmol1
Important formulae so far….  No. of carbon12 atoms = atomic mass (g)
 mass of one atom (g)
 No. of atoms = No. of moles × Avogadro’s number (NA)
 No. of molecules = No. of moles × Avogadro’s number (NA)
 Mass of one molecule = Molar mass
 Avogadro’s no.
 Mass of sample (g) = no. of moles (mol) × molar mass (gmol1)
 Number of moles = mass of sample (g)
 molar mass (gmol1)
 Calculating the number of atoms or molecules, given the number of moles:
 Calculating the mass of an individual molecule:
Calculating mass percentage from a chemical formula  Many of the elements in the periodic table of the elements occur in
 combination with other elements to form compounds.
 A chemical formula of a compound tells you the composition of that compound in terms of the number of atoms of each element present.
 The mass percentage composition allows you to determine the fraction of the total mass each element contributes to the compound.
 Ammonium nitrate (NH4NO3) is an important compound in the fertiliser industry. What is the mass % composition of ammonium nitrate?
 Step 1: Calculate the molar mass of ammonium nitrate
 Molar mass NH4NO3 = 80.05 gmol1
 Two N atoms: 28.016 gmol1
 Four H atoms: 4.032 gmol1
 Three O atoms: 48.00 gmol1
Step 2: Determine the mass % composition for each element  Nitrogen: 28.016g N in one mol of ammonium nitrate
 Mass fraction of N = 28.016g
 80.05g
 Mass % composition of N = 28.016g × 100%
 80.05g
 = 34.99% ≈ 35%
 Hydrogen: 4.032g H in one mol of ammonium nitrate
 Mass fraction of N = 4.032g
 80.05g
 Mass % composition of H = 4.032g × 100%
 80.05g
 = 5.04% ≈ 5%
 Oxygen: 48.00g O in one mol of ammonium nitrate
 As above, the mass % composition of O is found to be 60%
Therefore, the mass % composition of ammonium nitrate (NH4NO3) is:  % Nitrogen: 35%
 % Hydrogen: 5%
 % Oxygen: 60%
 To check your answer, make sure it adds up to 100%
 What is the mass % composition of C12H22O11?
 Answer: % Carbon: 42.1%
 % Hydrogen: 6.5%
 % Oxygen: 51.4%
 The empirical formula of a compound tells you the relative number of
 atoms of each element present in that compound. It gives you the
 simplest ratio of the elements in the compound.
 For example, the empirical formula of glucose (C6H12O6) is CH2O, giving the C:H:O ratio of 1:2:1
 If you know the mass % composition and the molar mass of elements present in a compound, you can work out the empirical formula
 What is the empirical formula of a compound which has a mass % composition of 50.05% S and 49.95% O?
 Step 1: Find the atomic masses of the elements present
 Sulfur (S) : 32.066 gmol1
 Oxygen (O) : 16.000 gmol1
Step 2: Determine the number of moles of each element present  Since we are dealing with percentages, we can express the mass % as
 grams if we assume we have 100g of the compound.
 Therefore, 100g of our compound contains 50.05g of sulfur and 49.95g of oxygen.
 Convert number of grams to number of moles
 Number of mol Sulfur = mass of sulfur in sample (g)
 atomic mass of sulfur (gmol1)
 Similarly, the no. of mol of Oxygen is found to be 3.12mol
 Step 3: Determining the ratios of elements
 Sulfur: 1.56mol
 Oxygen: 3.12mol
 Ratio must be in whole numbers. Here we must divide across by 1.56
 Therefore, we have a ratio of 1:2 giving an empirical formula of SO2
Question  Determine the empirical formula of a compound that contains 27.3
 mass% Carbon and 72.7 mass% Oxygen.
 Answer: No. of mol Carbon = 2.27mol
 No. of mol Oxygen = 4.54mol
 Ratio 1:2 Empirical formula CO2
 Monosodium glutamate (MSG) has the following mass percentage composition: 35.51% C, 4.77 % H, 37.85% O, 8.29% N, and 13.60% Na. What is its molecular formula if its molar mass is 169 gmol1?
Important calculations  Calculating mass percentage from a chemical formula
 Step 1: Calculate the molar mass
 Step 2: Determine the mass % composition for each element
 Determining empirical formula from mass
 Step 1: Find the atomic masses of the elements present
 Step 2: Determine the number of moles of each element present
 Step 3: Determining the ratios of elements
Molarity  The concentration of a solution is the amount of solute present in a given quantity of solvent or solution
 This concentration may be expressed in terms of molarity (M) or molar concentration:
 M = Molarity = no. of moles
 volume in Litres
 Molarity is the number of moles of solute in 1 Litre (L) of solution
What is molarity of an 85.0mL ethanol (C2H5OH) solution containing 1.77g of ethanol?  Step 1: Determine the number of moles of ethanol
 Molar mass of ethanol, C2H5OH:
 No. of moles = mass in g
 molar mass
 No. of moles ethanol = 1.77g .
 46.07 gmol1
Step 2: Convert to molarity  Have 0.085 L of ethanol
 Molarity = no. of moles
 volume in L
 Calculate the molarities of each of the following solutions:
 (a) 2.357g of sodium chloride (NaCl) in 75mL solution
 (b) 1.567mol of silver nitrate (AgNO3) in 250mL solution
Example  An antacid tablet is not pure CaCO3; it contains starch, flavouring, etc.
 If it takes 41.3mL of 0.206 M HCl to react with all the CaCO3 in one
 tablet, how many grams of CaCO3 are in the tablet. You are given the
 following balanced equation:
 2HCl(aq) + CaCO3(s) CaCl2(aq) + H2O(l) + CO2(g)
 Step 1: Determine the no. of moles of HCl that react
 Have 0.206 M HCl solution have 0.206 mol in one litre
 Have 41.3 mL of HCl solution have 0.0413 L of HCl solution
 Molarity = no. of moles
 volume in L
 no. of moles = Molarity × volume in L
Step 2: Determine no. of moles of CaCO3 used in the reaction  2HCl(aq) + CaCO3(s) CaCl2(aq) + H2O(l) + CO2(g)
 From the balanced equation, we can see that 2 moles of HCl are required to react with one mole of CaCO3
 Therefore, if 8.5 × 103 mol of HCl are present in the reaction, we must have 4.25 × 103 mol of CaCO3 present.
 No. of mols = mass in g
 molar mass
 Mass in g = no. of mols × molar mass
 = (4.25 × 103 mol) × (100 gmol1)
 = 0.425 g CaCO3 present in tablet
Questions  (a) How many moles of NaCl are present in 25.00mL of 1.85M NaCl(aq)?
 Answer: 4.62 × 102 mol NaCl
 (b) What volume of a 1.25 × 103 M solution of C6H12O6(aq) contains
 1.44 × 106 mol of glucose?
 (c) If stomach acid, given as 0.1 M HCl, reacts completely with an antacid tablet containing 500mg of CaCO3, what volume of acid in millilitres will be consumed? The balanced equation is:
 CaCO3(s) + 2HCl(aq) CaCl2(aq) + CO2(g) + H2O(l)
Important formulae…  No. of moles = mass in g
 molar mass
 Molarity = no. of moles
 volume in L
 Calculating the number of moles:
 Calculating the molarity or concentration:
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