Measurements and Significant Figures

• There are many devices & units that are used to quantify properties of matter

• All measuring devices are limited in their accuracy

• To communicate the accuracy of a measurement, we used significant figures (or digits)

• All counted values are significant (in essence, meaningful) figures (or sig fig for short)

• Significant figures, when using a measuring device, are all of the values that can be read off of the device, as well as one that is estimated

• Even with electronic measurement devices, the last digit is an estimate

Sig Fig Rules

• All non-zero digits are significant

• 321 has 3 sig fig because none of the digits are 0

• Zeros in between non-zero digits are significant

• 301 has 3 sig fig because the 0 is between 3 and 1, which are both non-zero

• Leading zeros are not significant

• 0.0321 has 3 sig fig - the two zeros are placeholders and aren't significant

• Trailing zeros are only significant when there is a decimal

• 3210 has 3 sig fig because the zero at the end is a placeholder

• 321**.**0 has 4 sig fig because the zero at the end is measured and is not a placeholder as it tells us more about the quantity

A and P Method:

• If the decimal is Absent, start counting sig figs from the first non-zero digit starting from the right side (AKA the Atlantic side)

• 40200 ⇐ right side, start at the first non-zero digit –– 3 sig fig

• If the decimal is Present, start counting sig fig from the first non-zero digit starting from the left side (AKA the Pacific side)

• Left side, start at the first non-zero digit ⇒ 0.0420 –– 3 sig fig

• All digits written in scientific notation are significant!!

• Scientific notation is a way of expressing values with exponents - very big or very small numbers are expressed this way

• Always move the decimal point so that it is one digit in from the side with the non-zero digits

• Eg. 34000000 ⇒ 3.4 x 10⁶

• Eg. 0.0000034 ⇒ 3.4 x 10⁻⁶

Adding and subtracting with significant figures

• Cannot be more accurate than the least accurate measurement

• When adding or subtracting values, the answer will have the same number of digits after the decimal as the least accurate measurement

• Chop off what is more accurate than the least accurate measurement

Chop it off, you can't calculate a number you don't have!!!

Multiplying and dividing with significant figures

• The sig fig in the answer from multiplying and dividing measured values have the same number of total sig figs as the value with the least amount of sig figs in the question

• Chop off digits that have more than the least amount of sig figs in the question

Rounding Rules

• Values between 1 and 4, round down

• Eg. round 52.36 to 2 sig fig ⇒ 52

• Values between 6 and 9, round up

• Eg. round 52.74 to 2 sig fig ⇒ 53

• Values with a 5 will round to the nearest EVEN value

• Round 52.51 to 2 sig fig ⇒ 52 (because it's nearer than 54)

• Round 53.57 to 2 sig fig ⇒ 54 (because it's nearer than 52)

Factor Label Method

• Many calculations involve converting, from one unit to another, for many purposes

• These require conversion factors & can be organized using the factor label method, which is also called dimensional analysis

• Conversion factor: a relationship between one unit and another unit

Factor label method can be broken into 3 steps:

1. Write the name or unit of the quantity that you're looking for & an equal sign; this is your target

2. Write the given value that you have to begin with

3. Multiply the given value by conversion factors arranged so that the given units are cancelled (divided out) and the target unit remains

• Conversion factors can be multiplied together in the same equation, if more than one is needed to reach your target unit!

• You can add as many conversion factors “blocks” if it helps you “build a road” to your target unit

Benefits of using this method:

• You don't have to remember as many formulas

• You keep your units organized & make fewer mistakes as a result

• Can be applied to many subjects like chemistry, physics, math, phone bill, and more

The Mole

• In chemistry, quantities like mass & volume are measured in units like grams and litres

• Tiny particles, like atoms & molecules are measured relative to each other, in atomic mass units, or amu, using a mass spectrometer

• On paper, we balance equations using particles, but in the lab we can only measure large units like masses & volumes

• THE MOLE IS A BRIDGE BETWEEN THESE TWO KINDS OF MEASUREMENTS

Quantities of invisible tiny particles                             Quantities of measurable amounts

like molecules and atoms                   ⟺                    like mass and volume

Mole logic

If you know the relative mass of 2 items, then you can always determine when you have equal quantities of both

• For instance: a helium atom has 4 times the mass of a hydrogen atom. What mass of helium has the same number of atoms as 5.0 g of hydrogen?

• If each helium is 4x the mass of hydrogen, and we have 5 grams of it, 4x5=20, so 20.0g helium will have the same number of atoms

Molar mass

• There are the same number of atoms in the mass of an element represented by the average atomic mass of an element measured in grams

• This is one mole of the element

For compounds: just add the mass of each element in the compound together

Eg. what is the mass of 15.5 moles of carbon dioxide (C = 12, 0 = 16 (x2) – 44 g)

Mass = 15.5 mol =    44 g    = 682g                   Use factor label method!

1 mol

Avogadro's number

• The quantity of particles in a mole is named after Avogrado, who discovered that equal volumes of gases must contain the same number of particles

• This quantity of particles in the mass elements on the periodic table turns out to be:

6.02 x 1023 particles/mol

• Avogrado determined that equal quantities of gas occupy the same volume under standard temperature & pressure

• Under standard temperature & pressure, one mole of any gas occupies 22.4 L

Mole map

Note: not all substances are made of molecules! The particles that make up most elements are atoms. Ionic compounds are a repeating pattern of ions, the smallest of which can be called a formula unit.

Percent Composition and Empirical Formula

• The formula of a compound gives the ratio (proportion of particles) of atoms present

• Using the mole concept, a formula can also give you the percentage of mass of each element in a compound

Eg. One mole of water (H2O -- 18g/mol) contains 2 moles of Hydrogen (1g/mol + 1g/mol = 2g/mol) and 1 mole of Oxygen (16g/mol)

General formula to find the percent mass of something

Grams of element you're looking for            x 100%             =  % of element you were looking for

Grams of the compound you have

• An empirical formula is the lowest whole number ratio of elements in a compound

• The empirical formula can be determined by working backwards from the percentage mass data to find the ratio of element moles

Steps

1. Find moles of each element

2. Convert to lowest whole number ratio

MASS → MOLES → LOWEST WHOLE NUMBER RATIO

• Always assume you have 100g if grams aren't given so that you have an easy starting point

Eg. a sample was found to contain 50% sulfur and 50% oxygen by mass - what is the empirical formula?

Then, you can see that the formula created is SO2 because the lowest wholes represent the quantities of the elements in the formula

• Sometimes...the moles of each element do not give you a nice even whole number, and you'll need to multiply to find the best ratio

• For a ratio of 1:1.5 ⇒ multiply by 2 to get 2:3

• For a ratio of 1:1.333 ⇒ multiply by 3 to get 3:4

• For a ratio of 1:1.25 ⇒ multiply by 4 to get 4:5

• Sometimes, the percentages are not given - masses need to be found in another way….

Eg. a 50.51 g sample of a compound made from iron and oxygen is decomposed, and 35.36 g of the iron remains. Find the empirical formula

Mass                                                Moles                                                   Lowest whole

Fe: 35.36                                   x 1 mol Fe       = 0.6331 mol                  ⇒ 1        x 2

                                                    55.85 g            0.6331

O: 50.51-35.36 = 15.15            x 1 mol O         = 0.9469 mol                   ⇒ 1.5     x2

(subtract whole sample                  16 g                0.6331

from Fe which was also

given)

• A molecular formula is not always the lowest whole number ratio

• In order to determine the molecular formula, you can divide the empirical molar mass into the molecular molar mass and then multiply the empirical formula by the same factor

Eg. If the empirical formula of a molecule is CH3 and the molar mass of the compound is 30 g/mol, what is the molecular formula?

• The empirical formula & formula of a molecule are different, & the empirical formula may not be the same formula for the compound

Stoichiometry

• Balanced equations make sure that the number of atoms of each element are the same on each side of an equation by adding coefficients

• Since a mole is simply a large number of particles, we can use moles in the same way as we would use particles in a balanced equation

Eg.

2H2    +   O2   →     2H2O

2 particles H2 + 1 particle O2   →  2 particles H2O

2 moles H2    +   1 mole O2     →     2 moles H2O

• Mole ratio: the coefficients from the balanced equation can be used to convert between moles consumed and moles produced in a reaction

• Use factor label method

Eg. If 5.0 moles of Hydrogen are consumed, how many moles of oxygen are also consumed?

Moles O2 = 5.0 mol H2 x   1 mol O2  =  2.5 mol O2

2 mol H2

Eg. if 5.0 moles of oxygen are consumed, how many moles of water are produced?

Moles H2O = 5.0 mol O2 x 2 mol H2O    = 10.0 mol H2O

1 mol O2

• Use the quantities from the balanced equation (equivalent in moles) to calculate what you need

• Stoichiometry: involves 3 steps to determine the quantity involved in a chemical reaction

1. Convert given quantity to moles

2. Convert given moles to moles of the quantity that you want (target, whatever you’re asked for)

3. Convert the target moles to the unit you want

Eg. Quantity consumed: what mass of oxygen is consumed when reacting with 5.0 g of Hydrogen?

2H2 + O2 → 2H2O

Steps

Step 1 - convert into moles

Moles H2 = 5.0 g H2 x 1 mol H2  = 2.5 mol H2

2g

Step 2 - convert the given moles to the moles of the quantity you want (target)

Moles O2 = 2.5 mol H2 x 1 mol O2    = 1.25 mol H2

2 mol H2

Step 3 - convert target moles to the unit you want

Mass O2 = 1.25 mol O2, x   32g        = 40 g O2

1 mol

Putting it all together:

Mass O2  = 5.0 H2 x 1 mol H2  x 1 mol O2    x     32g         = 40 g O2

2g          2 mol H2      1 mol O2

Generalized formula

[target unit] [target element/compound] = [given amount of given e/c] x [1 mol/molar mass of given e/c] x [quantity of target e/c in balanced equation/quantity of given e/c in balanced equation] x [conversion factor to get into target units]

• Always 3 steps!

Expanded map

Limiting Reagent

• A chemical reaction is like a factory with input components (reactants) and output (products)

• If one of the reactants runs out, the reaction will then stop

• Therefore, that reactant then controls how much product can be made

• The reactant that runs out = the limiting reagent

• The leftover reactant = the excess reagent

• Whenever you have more than one quantity of a reactant, you will need to determine the limiting reagent before any other quantities can be found

Finding the limiting reagent:

• Convert the reactant quantities into moles (given amt x 1mol/molar mass of given)

• Divide what you get by the coefficient in the balanced equation

The smaller value is the limiting reagent

Eg) if 3.0g of H2 reacts with 6.0g of O2

2H2 + O2 → 2H2O

H2 :                                                                                O2 :

3.0g H2 x 1 mol   = 1.5mol H = 0.75                              6.0g O2 x 1 mol = 0.25   = 0.25

2g           2                                                                   32g       1

                   0.25 is smaller than 0.75 ∴ O2 is the Limiting Reagent

Product Yield

Yield = the quantity of a product that can be expected to be made

Eg) What mass of water can be formed from 3.0g of H2 reacting with 6.0g of O2?

*********you need to use the limiting reagent to solve this question!!!

MassH2O = 6.0g O2 x   1 mol O2  x   2 mol H2O   x 18g H2O  = 6.8g H2O

(yield)           ↑                  32 g           1 mol O2         1 mol

Given quantity LR             ↑                     ↑                  ↑

1 mol/molar mass        target/given       target molar mass

of LR                     equation             to get into grams

coefficients

Excess Reagent: what mass of excess reactant remains from 3.0g of H2 reacting with 6.0g of O2?

*******you also need to use the LR here too

MassH2  = 6.0g O2  x 1 mol O2  x 2 mol H2  x  2gH2    = 0.75g H2

(used)          ↑               32g         1 mol O2        1 mol

Given quantity LR           ↑                  ↑               ↑

1 mol/molar mass         target/given         target molar mass

of LR                  coefficients          to get into g

Remaining H2 = 3.00g - 0.75g = 2.25g

↑              ↑

Given               Used

amount            amount

Percentage Yield, Purity, and Error

• Yield = the amount of product that is produced in a chemical reaction

• Theoretical yield = maximum amount of product that is possible from a calculation

• Actual yield = the amount of product that was actually collected from the real, actual experiment

Percentage yield    =       Actual yield         x 100%

Theoretical yield

Why isn't the yield always 100%

• Incomplete reactions (like incomplete combustion)

• Side reactions (where more than one reaction occurs)

• Experimental error (your experiment wasn't set up optimally)

Purity

• Purity = how much of a given sample is actually one substance

• It can be expressed as a percentage

Percentage purity    =     Mass of a pure substance      x 100%

Total mass of a sample

Percentage Error

• Error percentage = a measure of the difference between a measured value and a known/accepted value

• This percentage is often used in labs to quantify the closeness that an experiment comes to producing the expected value

Percent error    =      |Experimental value - Accepted value|        x 100%

Accepted value