1.2MEASUREMENTS AND

CALCULATIONS

The Scientific Method

Scientists follow a series of steps known as the scientific method in order to answer questions and solve problems.

They can use all steps, or just some steps.

Ultimate goal is to come up with reliable answers and solutions.

Steps include:

Make observations Ask question Form hypothesis Test hypothesis Analyze results Draw conclusions Communicate results

The Metric System

The International System (SI) of Units is the modern form of the metric system, based around the number ten. ▪ French derived “Le Systeme International d’Unites”

Density

Density is the ratio of mass to volume, or mass divided by volume. An object made of cork feels lighter than a lead object of the same size.

Sample Problem 1

A sample of aluminum metal has a mass of 8.4 g. The volume of the sample is 3.1 cm3. Calculate the density of aluminum.

2.7 g/cm3

Sample Problem 2

What is the density of a block of marble occupying 310. cm3 and having a mass of 853 g?

2.8 g/cm3

Sample Problem 3

Diamond has a density of 3.26 g/cm3. What is the mass of a diamond having a volume of 0.351 cm3?

1.14 g

Sample Problem 4

What is the volume of a sample of liquid mercury having a mass of 76.2 g, given the density of mercury is 13.6 g/mL?

5.60 mL

Dimensional Analysis

A mathematical technique allowing you to convert units to solve problems is called dimensional analysis. When you want to use a conversion factor to

change a unit in a problem, you can set up the problem in the following way:

▪ Quantity sought (?) = quantity given x conversion factor

▪ Example: How many quarters are in 12 dollars?

? quarters = 12 dollars x conversion factor

? quarters = 12 dollars x 4 quarters 1 dollar

48 quarters in 12 dollars

Sample Problem 5

Express a mass of 5.712 grams in milligrams and in kilograms.

5712 mg

0.005712 kg

Sample Problem 6

Express a length of 16.45 m in centimeters and in kilometers.

1645 cm

0.01645 km

Sample Problem 7

Express a mass of 0.014 mg in grams.

14 g

Sample Problem 8

I am having a party this weekend and inviting 15 people, anticipating each person will eat 4 slices of pizza. Knowing each pizza has 12 slices, how many total pizzas will I need to order to have enough for everyone at the party?

5 pizzas

HOMEWORK

A sample of calcium nitrate, Ca(NO3)2, has a formula weight of 164 g/mol and contains 5.00 x 1025 atoms of oxygen. Knowing there are 6.02 x 1023 molecules of whatever in a mole, what is the mass in kg of this sample?

Accuracy vs. Precision

Another important detail when dealing with numbers is to understand values.

Do the words accuracy and precision sound familiar? What’s the difference?

Accuracy refers to how closely a measured value agrees with the correct value.

Precision refers to how closely individual measurements agree with each other.

Percent Error

Percent error is calculated by subtracting the accepted value from the observed value, dividing the difference by the accepted value, and then multiplying by 100. Negative value if the accepted value is greater than the

observed value.

Positive value is the accepted value is less than the observed value.

𝑃𝑒𝑟𝑐𝑒𝑛𝑡 𝐸𝑟𝑟𝑜𝑟=𝑂𝑏𝑠𝑒𝑟𝑣𝑒𝑑− 𝐴𝑐𝑐𝑒𝑝𝑡𝑒𝑑

𝐴𝑐𝑐𝑒𝑝𝑡𝑒𝑑∗100

Sample Problem 9

A student measures the mass and volume of a substance and calculates its density as 1.40 g/mL. The correct, or accepted, value of the density is 1.30 g/mL. What is the percent error of the student’s measurement?

7.69 % error

Sample Problem 10

What is the percent error for a mass measurement of 17.7 g, given the accepted value is 21.2 g?

-16.5 % error

Sample Problem 11

A volume is measured experimentally as 4.26 mL. What is the percent error, given the accepted value is 4.15 mL?

2.65 % error

Significant Figures

When dealing with numerical values, it is imperative to focus on something probably familiar to many of you. The number of significant

figures (“sig figs”) is the number of digits believed to be correct by the person doing the measuring.

Significant Figures

Rules for calculating significant figures:

- Digits from 1-9 are always significant.

- Leading zeros are never significant.

- Imbedded zeros are always significant.

- Trailing zeros are only significant if the decimal point is specified.

Significant Figures

Further rules for calculating significant figures when performing mathematical operations:

Addition and SubtractionThe answer may only show as many significant decimal places as the measurement having the least number of significant decimal places.

Multiplication and DivisionThe answer may only show as many sig figs as the

measurement having the least number of sig figs.

Sample Problem 12

Using the correct number of sig figs and decimal places, solve the following:

5.26 + 1 + 29 – 3.74

2.3 x 4.28 x 6 x 1.05

32Least significant decimal place in measurements is located in

the “ones” position

Least number of sig figs in measurements is 1

60

Scientific Notation

Often times in Chemistry, numbers are either so large or small it is inconvenient to write them out.

Rather than writing them out, it is easier to use scientific notation, a system used to express very large or very small numbers without using a lot of zeros. Example: 384,000 becomes …

3.84 x 105

Proportions

Two quantities are directly proportional to each other if dividing one by the other gives a constant value. y ∞ x

Example: as one goes up, the other goes up. As one goes down, the other goes down.

Straight-line graph results from direct proportions.

Proportions

Two quantities are inversely proportional to each other if their product is constant. y ∞ 1/x

Example: as one goes up, the other goes down. As one goes down, the other goes up.

Hyperbola graph results from direct proportions.