Scientific Method: review chapter 2, section 1

Nature of Measurement

• Measurement - quantitative observation consisting of 2 parts• Part 1 – number• Part 2 – scale (unit)

• Examples:

• 20 grams

• 6.63 6.63 × 10× 1044 meters per second (m/s)

Can you think of other examples?

What quantity is this a measurement of?

What quantity is this a measurement of?

A base unit is not made up of other units.

A derived unit is made up of two or more other units.

gram is an example of a base unit.

m/s is an example of a derived unit.

A number without a unit is NOT a measurement!

The 7 Fundamental SI Units (Base Units)Quantity (symbol) Name of Unit Abbreviation

Mass (symbol is “m”) kilogram kg

Length (symbol is “l”) meter m

Time (symbol is “t”) second s

Temperature (symbol is “T”) kelvin K

Amount of (symbol is “n”) substance

mole mol

Electric current (symbol is “I”) ampere A

Luminous (symbol is “IV”)

intensity

candela cd

Why is volume not listed as a quantity?

Volume is a derived unit (length x length x length = volume)!

Any unit not listed above is a derived unit.

Derived Units

Derived units are units that are defined by a combination of two or more base units.

Volume = (length x width x height)

If distances are in meters, what would the units for volume be? (m)x(m)x(m) = m3 (cubic meters)

Density = mass ÷ volume = m/V

If mass is in grams and volume is in mL, what would the units for density be?

(g)÷(mL) = g/mL (gram per milliliter)

A definition: 1 cm3 = 1 mL

(one cubic centimeter is one milliliter)

Definitions of SI Prefixes:

terra (T) means 1X1012 1000 000 000 000giga (G) means 1X109 1000 000 000

Big mega (M) means 1X106 1000 000kilo (k) means 1X103 1000hecto (h) means 1X102 100deka (da) means 1X101 10

Middle 1X100 1

deci (d) means 1X10-1 0.1centi (c) means 1X10-2 0.01milli (m) means 1X10-3 0.001

Small micro () means 1X10-6 0.000001nano (n) means 1X10-9 0.000000001pico (p) means 1X10-12 0.000000000001

  femto (f) means 1X10-15 0.000000000000001atto (a) means 1X10-18 0.000000000000000001

1 TL = 1X1012 L1 Gs = 1X109 s1 Mg = 1X106 g1 km = 1X103 m

 

 1 g = 1X103 mg1 L = 1X106 L1 m = 1X109 nm1 s = 1X1012 ps

 

Notice that if we write 1 large unit on the left side of the equal sign, then there must be a larger number of the smaller units on the right side to be equal.

Notice the 3, 6, 9, 12 pattern!

SI Prefixes written in Equivalence Statement format

Problem Solving Strategy Illustration

A solid object is found to have a mass of 84.241 g and a volume of 28.53 mL. What is the density of the object?

First Step: Highlight key concepts or quantities in the word problem

Second Step: Assign an appropriate symbol for all key quantities

mass = m = 84.241 g volume = V = 28.53 mL density = d = ?

Third Step: Use the list of symbols to identify any useful equations

d = mV

Fourth and Fifth Steps: Arrange the symbols in the equation so that the unknown variable is by itself on one side and then substitute quantities into the mathematical equation and complete the indicated mathematics

m = 84.241 g V = 28.53 mL d = ?

d = mV

Sixth Step: Check significant figures and units and write the correct answer

d = 84.241 g 28.53 mL

You have 14.3 mL of an object that has a density of 7.932 g/mL.What is the mass of the object?

You have 435.3 g of a liquid that has a density of 0.8325 g/mL.What is the volume of the liquid?

What if you could only remember one of the two temperature conversion equations? Can you change one into the other?

oC = (oF 32.00)59

oF = (oC) + 32.00 95

To change from oC to K: K = oC + 237.15

Convert 15 oF into oC

Convert 45 oC into K

Convert 245 K into oF

Practice

Mathematics with Scientific Notation:

Let your calculator handle the exponents!

Let’s do an example: 3.4X106 + 2.8X105

On your calculator (Texas Instruments), type the following in order:

3.4 2nd EE 6 + 2.8 2nd EE 5 =

On your calculator (Casio), type the following in order:

3.4 exp 6 + 2.8 exp 5 =

How many Gm are in 1.5X1013 meters?

How many g are in 3.42X104 g?

1.5X1013 m( )( )

1 Gm = 1X109 mWe need the following equivalence statement:

Now we use the equivalence statement so that the “m” units cancel out and are replaced by the units “Gm”.

1 Gm

1X109 m= 1.5X104 Gm

( )( )3.42X104 g

Equivalence Statements and Conversion Factors

Any statement that says that one quantity is equal to another.

12 things = 1 dozen 1 inch = 2.54 cm 1 km = 1000 m

Each of these equivalence statements can be used to create conversion factors.

Example:

1 inch 1 inch

1 inch 2.54 cm=

This is a conversion factor that converts from “in” into “cm”

2.54 cm 2.54 cm

2.54 cm 1 inch =

This is a conversion factor that converts from “cm” into “in”

How many dozen apples do you have if you have 270 apples?

270 apples( ) 12 apples

1 dozen apples= 22.5 dozen apples

We need the following equivalence statement: 12 apples = 1 dozen apples

( )Conversion Factor (created from the equivalence statement)

We could have tried to remember that 1/12 is 0.0833 and then used the value 0.0833 as a conversion factor. However, in the long run it is more efficient to learn the equivalence statements and then use them to create conversion factors as needed.

How many apples do you have if you have 13.5 dozen apples?

If you know that your car has a mileage rating of 23.5 miles per gallon and you assume that gas costs $3.60 per gallon, how much will it cost you to travel 545 miles?

Dimensional Analysis

Using units to guide your use of conversion factors to solve problems.

Important Equivalence statements: 1 gal = 23.5 miles $3.60 = 1 gal

What were you given?

What were you asked to find? Cost for traveling 545 miles

Travel 545 miles

(545 miles)

Start with what you were given and convert the units into what you were asked to find (using the equivalence statements you know).

23.5 miles

1 gal( ) 1 gal

$ 3.60( )= $ 83.5

a) Not Precise and not Accurate

b) Not Accurate but Precise

c) Accurate and Precise

The Difference Between Precision and Accuracy can be more difficult to see when numbers are given instead of the darts.

TrialVolume

(mL)

1 3.6

2 3.5

3 3.7

Average 3.6

The values are only changing in the last decimal place, so they are precise.

If the TRUE value for the volume is 3.6, then the data is also accurate.

However, if the TRUE value for the volume was 4.2, then the data would only be precise and would not be accurate.

The Difference Between Precision and Accuracy can be more difficult to see when numbers are given instead of the darts.

TrialVolume

(mL)

1 3.8

2 2.8

3 4.2

Average 3.6

The values are changing both decimal place, so they are not precise.

If the TRUE value for the volume was 4.2, then the data would not be precise and would not be accurate.

However, if the TRUE value for the volume is 3.6, then the data is accurate by accident.

Uncertainty in Measurement

• A digit that must be estimated is called an uncertain digit.

• All measurements include all the digits we are certain of plus one guess digit.

• A measurement always has some degree of uncertainty because we can always make a guess about the last digit.

Generally, the more digits a measurement has, the more precise it is considered to be. Between two numbers, the number with uncertainty in the smallest decimal place is the more precise number.

3.28 g

3.2764 g

Since the nail is longer than 6.3 cm but is not longer than 6.4 cm, we are certain of the digits 6.3

All non-digital devices have precisions that are one place smaller than the smallest marking on the device. In the case of the ruler, the smallest marks are at the 0.1 cm scale. Therefore, the precision would be at the 0.01 cm scale. We would say the measurement was 6.36 cm +/ 0.01 cm (or +/ 0.05 cm depending upon how well we can estimate our guess). This measurement would have 3 significant figures!

See figure 3.2 on page 44

Since there are no marks between 6.3 cm and 6.4 cm, we must guess how far between the marks we think the length is-this guess is an uncertain digit.

Significant Figures are those digits in a measurement that we are certain of plus one guess digit at the end.

However, if you are not making the measurement, and the measurement is given to you, you must use different rules to determine significant figures.

Rules for Significant Figures in Measurements given to you by an outside source

• Nonzero integers always count as significant figures.

– 3456 has 4 sig figs.

• Leading zeros do not count as significant figures.

– 0.048 has 2 sig figs.

• Captive zeros always count as significant figures.

– 16.07 has 4 sig figs.

• Trailing zeros are significant only if the number contains a decimal point.

– 9.300 has 4 sig figs

– 150 has 2 sig figs.

• Exact numbers have an infinite number of significant figures.

– 1 inch = 2.54 cm, exactly

Exact numbers are definitions or simple counting: 12 is 1 dozen and 4 cars

How many significant figures are in each of the following numbers?

450 g 0.029 m 20.3 s

0.00300 g $45,700,000 13 people

6.2X10-2 mL 1.300X108 m

These measurements are given to you by someone else-they are not numbers that you obtained from a measurement. This means that we must apply the arbitrary rules for significant figures.

Significant Figure Rules for Mathematical Operations:

Multiplication and Division: the number with the fewest significant figures in the calculation determines how many significant figures the answer will have.

Examples: (4.53 m)*(0.28 m)*(1.342 m) = 1.7021928 m3

(from the calculator)

1.7 m3

(correct answer)

(678.3 m)÷(18.4 s) = 36.86413043 m/s(from the calculator)

36.9 m/s(correct)

Significant Figure Rules for Mathematical Operations:

Addition and Subtraction: The largest position “guess” number determines the position of the last significant figure in the answer.

4,300 m298 m+

4,598 m (calculator answer)

4,600 m (correct answer)

the “3” in 4,300 is a guess number and is in the hundred’s position

the “8” in 298 is a guess number and is in the one’s position

Since the hundred’s position is larger than the one’s position, the answer must have its guess number in the hundreds position.

321.4 m298 m-23.4 m (calculator answer)

23 m (correct answer)

a)

b)

Error (or absolute error) is the difference between the accepted value for a measurement and the experimental value for a measurement.

Example: The accepted density for chloroform is 2.97 g/mL;In an experiment, a student obtained a value of 2.85 g/mL.

The Error in her measurement is:

Percent Error is the error expressed as a percentage!

Error = 2.85 g/mL – 2.97 g/mL = 0.12 g/mL

Error = Experimental Value – Accepted Value

% Error = [(2.85 – 2.97)/2.97]*100 = 4.0%

% Error =( ) *100

Notice that the error was negative, but the percent error was not. Percent error is always positive.

Circle Graph

Seniors

Juniors

Sophomors

Freshmen

0

5

10

15

20

25

Seniors Juniors Sophomors Freshmen

Bar Graph

Examples of Two Types of Graphs

0

1

2

3

4

5

6

7

0 1 2 3 4 5 6 7

X values

Y v

alu

es

A third type of graph: X-Y scatter plot

Line will have the form: y = mx + b

where “m” is slope and “b” is y intercept

What happens to “y” as “x” increases?

Since “y” gets smaller as “x” gets larger, the slope will be negative.

This data represents directly proportional data (y/x = constant).

This data represents inversely proportional data (y*x = constant).