Chapter 3 Scientific Measurement

Measurement In chemistry, #’s are either very small or very large 1 gram of hydrogen = 602,000,000,000,000,000,000,000 atoms Mass of an atom of gold = gram

Scientific Notation Condensed form of writing large or small numbers When a given number is written as the product of 2 numbers M x 10 n  M must be: greater than or equal to 1 less than 10  n must be: whole number positive or negative

 Find M by moving the decimal point over in the original number to the left or right so that only one non-zero number is to the left of the decimal.

 Find n by counting the number of places you moved the decimal: To the left (+) or To the right (-)

Scientific Notation Examples  20 =  200 =  501 =  2000 = 2.0 x x x x 10 3

More examples…  0.3 =  0.21 =  0.06 =  =  = Rule: If a number starts out as < 1, the exponent is always negative. 3 x x x x x 10 -4

Scientific Notation  Adding & Subtracting: if they have the same n, just add or subtract the M values and keep same n if they don’t have the same n, change them so they do

Scientific Notation  Multiplying: the M values are multiplied the n values are added

Scientific Notation  Division: the M values are divided the n values are subtracted

Accuracy & Precision ‘How close you are really counts!’

Accuracy Accuracy – a measure of how close a measurement comes to the actual or true value of what is measured To evaluate… the measured value must be compared to the correct value

Precision Precision – a measure of how close a series of measurements are to one another To evaluate… you must compare the values of 2 or more repeated measurements

Accuracy vs. Precision

Errors are Unavoidable Measuring instruments have limitations Hence, there will always be errors in measurement.

Not All Errors are Equal Consider the following two errors: You fly from NY to San Francisco Your plane is blown off course by 3cm You are an eye surgeon Your scalpel misses the mark by 3cm The errors sound equal… but are they?

Absolute Error The error in each of the previous examples is 3cm But the error in each is not equivalent! This type of error is the absolute error. Absolute error = | measured value – accepted value | Accepted value is the most probable value or the value based on references Only the size of the error matters, not the sign

Significance of an Error The absolute error tells you how far you are from the accepted value It does not tell you how significant the error is. o Being 3cm off course on a trip to San Francisco is insignificant because the city of San Francisco is very large. o Being 3cm off if you are an eye surgeon means your operating on the wrong eye! It is necessary to compare the size of the error to the size of what is being measured to understand the significance of the error.

Percentage Error The percentage error compares the absolute error to the size of what is being measured. % error = |measured value – accepted value| x 100% accepted value

Example: Measuring the boiling point of H 2 O Thermometer reads – 99.1 O C You know it should read – 100 O C Error = measured value – accepted value % error = |error| x 100% accepted value Sample Problem

% error = |99.1 o C – o C| x 100% 100 o C = 0.9 o x 100% 100 o = x 100% = 0.9% C C

Significant Figures Used as a way to express which numbers are known with certainty and which are estimated

What are significant figures? Significant Figures – all the digits that are known, plus a last digit that is estimated

Rules 1) All digits 1-9 are significant Example: 129 2) Embedded zeros between significant digits are always significant Example: 5,007 3) Trailing zeros in a number are significant only if the number contains a decimal point Example: sig figs 4 sig figs 2 sig figs

4) Leading zeros at the beginning of a number are never significant Example: ) Zeros following a decimal significant figure are always significant Example: ) Exceptions to the rule are numbers with an unlimited number of sig figs Example = Counting – 25 students Exact quantities – 1hr = 60min, 100cm = 1m 2 sig figs 3 sig figs 5 sig figs

Significant Figure Examples  123m =  x 10 4 m =  =  40, 506 =  22 meter sticks =  98, 000 =  143 grams =  m =  x g = unlimited

Calculations Using Significant Figures Rounding 1 st determine the number of sig figs Then, count from the left, & round If the digit < 5, the value remains the same. If the digit is ≥ 5, the value of the last sig fig is increased by 1.

Try your hand at rounding…  Round each measurement to 3 sig figs.  meters =  x 10 8 meters =  meter =  9009 meters =  x meter =  meters = 87.1m m or 1.55 x m 4.36 x 10 8 m 9010m 1.78 x m 630. m or 6.30 x 10 2 m

Multiplying and Dividing Limit and round to the least number of significant figures in any of the factors. 23.0cm x 432cm x 19cm = Answer = Because 19 only has 2 sig figs 190,000cm 3 or 1.9 x 10 3 cm 3 188,784cm 3

Addition and Subtraction Limit and round your answer to least number of decimal places in any of the numbers that make up your answer mL mL mL = Answer = Because 46.0 has only 1 decimal place 255.5mL mL

The International System of Units

Based on the #10 Makes conversions easier Old name = metric system

Units and Quantities Length – the distance between 2 points or objects Base unit = meter Volume – the space occupied by any sample of matter V = length x width x height Base unit = liter Based on a 10cm cube (10cm x 10cm x 10cm = 1000cm 3 ) 1 liter = 1000cm 3

Mass – the amount of matter contained in an object Base unit = gram Different than weight… Weight - a force that measures the pull of gravity

Temperature – a measure of the energy of motion How fast are the molecules moving? When 2 objects are at different temperatures heat is always transferred from the warmer → the colder object

Temperature Scales Celsius scale – Freezing point of H 2 O = 0 o C Boiling point of H 2 O = 100 o C Kelvin scale – Freezing point of H 2 O = K Boiling point of H 2 O = K K = C C = K - 273

Temperature Scale Conversions

Conversion Factors and Unit Cancellation

A physical quantity must include: Number + Unit + Unit

1 foot = 12 inches 1 foot = 12 inches

1 foot = 12 inches 1 foot 12 inches = 1

1 foot = 12 inches 1 foot 12 inches = 1 12 inches 1 foot = 1

1 foot 12 inches 1 foot “Conversion factors”

1 foot 12 inches 1 foot “Conversion factors” 3 feet 3 feet 12 inches 12 inches 1 foot 1 foot = 36 inches ( ( ) ) ( ( ) ) How many inches are in 3 feet?

How many cm are in 1.32 meters? conversion factors: equality: or X cm = 1.32 m= 1 m = 100 cm ______1 m 100 cm We use the idea of unit cancellation to decide upon which one of the two conversion factors we choose. ______ 1 m 100 cm () ______ 1 m 100 cm 132 cm

How many meters is 8.72 cm? conversion factors: equality: or X m = 8.72 cm= 1 m = 100 cm ______1 m 100 cm Again, the units must cancel. ______ 1 m 100 cm () ______ m 1 m 100 cm

How many feet is inches? conversion factors: equality: or X ft = in= 1 ft = 12 in ______1 ft 12 in Again, the units must cancel. () ____ 3.28 ft 1 ft 12 in ______ 1 ft 12 in

How many kilometers is 15,000 decimeters? X km = 15,000 dm= 1.5 km () ____ 1,000 m 1 km 10 dm 1 m () ______

How many seconds is 4.38 days? = 1 h 60 min24 h 1 d1 min 60 s ____ ()() () _____ X s = 4.38 d 378,432 s3.78 x 10 5 s If we are accounting for significant figures, we would change this to…

Why do some objects float in water while others sink? Need to know the ratio of the mass of an object to it’s volume Pure H 2 O at 4 o C = 1.000g/cm 3 If an object has a lower ratio it will float If an object has a greater ratio it will sink

Density The ratio of an object’s mass to it’s volume Density = mass volume Example: A 10.0cm 3 piece of lead has a mass of 114g. What is the density of lead? 114g = 10.0cm g/cm 3

 Recall… What type of property is density? Does the density of a material change in relation to the sample size? NO… density is an Intensive property it depends only on the composition of the material

What might affect a substance’s density? Temperature  The volume of most substances ↑ withan ↑ in temperature  the mass remains the same If the volume increases… what affect does it have on a substance’s density? The density decreases *Exception – Water’s volume ↑ with a ↓ in temperature Its density decreases & ice floats H2OH2O

Calculating Density What is the volume of a pure silver coin that has a mass of 14g, and a density of 10.5g/cm 3 ? D = 10.5g/cm 3 M = 14g V = ? Rearrange the density formula to solve for V D M V

V = M D V = 14g = 14 x 1 cm 3 = 10.5g/cm g g 1.3cm 3

What is the mass of mercury that has a density of 13.5g/cm3 and a volume of 0.324cm 3 ? Once again, rearrange the density formula… and solve for M. D M V M = D x V M = 13.5g x cm 3 = 4.62g