2 - 1 Measurement Uncertainty in Measurement Significant Figures

2 - 2 Measurement Observation can be both QUALITATIVE and QUANTITIVE A qualitative observation is a description in words. is a description in words. A quantitative observation is a description with numbers and units. A measurement is a comparison to a standard.

2 - 3 Units are important has little meaning, just a number $45,000 has some meaning - money $45,000/yr more meaning - person’s salary

2 - 4 Uncertainty in Measurement Use of Significant Figures It is important to realize that a measurement always has some degree of uncertainty, which depends on the precision of the measuring device. Therefore, it is important to indicate the uncertainty in any measurement. This is done by using significant figures.

2 - 5 Uncertainty in Measurement Every measurement has an uncertainty associated with it, unless it is an exact, counted integer, such as the count of trials performed or a definition.

2 - 6 Uncertainty in Measurement Every calculated result also has an uncertainty, related to the uncertainty in the measured data used to calculate it. This uncertainty should be reported either as an explicit ± value or as an implicit uncertainty, by using the appropriate number of significant figures.

2 - 7 Uncertainty in Measurement The numerical value of a "plus or minus" (±) uncertainty value tells you the range of the result. When significant figures are used as an implicit way of indicating uncertainty, the last digit is considered uncertain.

2 - 8 Uncertainty in Measurement. A significant figure is one that has been measured with certainty or has been 'properly' estimated. The significant figures in a number includes all certain digits as read from the instrument plus one estimate digit.

2 - 9 Uncertainty in Measurement Significant digits or significant figures - are digits read from the measuring instrument plus one doubtful digit estimated by the observer. This doubtful estimate will be a fractional part of the least count of the instrument.

Uncertainty in Measurement All measurements contain some uncertainty. Limit of the skill and carefulness of person measuring Limit of the measuring tool/equipment being used Uncertainty is measured with Accuracy AccuracyHow close to the true value Precision PrecisionHow close to each other

Precision Here the numbers are close together so we have good precision. Poor accuracy. Large systematic error. How well our values agree with each other. xx x

Accuracy Here the average value would give a accurate number but the numbers don’t agree, are not precise. Large random error How close our values agree with the true value. x

Accuracy and precision Our goal! Good precision and accuracy. These are values we can trust. xx x

Accuracy and precision Predict the effect on accuracy and precision. Instrument not ‘zeroed’ properly Reagents made at wrong concentration Temperature in room varies ‘wildly’ Person running test is not properly trained

Types of errors Instrument not ‘zeroed’ properly Reagents made at wrong concentration Temperature in room varies ‘wildly’ Person running test is not properly trained Random Systematic

Errors Systematic Errors in a single direction (high or low). Can be corrected by proper calibration or running controls and blanks.Random Errors in any direction. Can’t be corrected. Can only be accounted for by using statistics.

Errors Systematic: ACCURACY Errors in a single direction (high or low). Can be corrected by proper calibration or running controls and blanks. Random: PRECISION Errors in any direction. Can’t be corrected. Can only be accounted for by using statistics.

Significant figures Method used to express precision. You can’t report numbers better than the method used to measure them units = three significant figures ONLY ONE UNCERTAIN DIGIT IS REPORTED Certain Digits Uncertain Digit

Significant figures The number of significant digits is independent of the decimal point These numbers All have three significant figures!

Significant figures: Rules for zeros are not Leading zeros are not significant three significant figures Leading zero are Captive zeros are significant four significant figures are Trailing zeros are significant five significant figures Captive zero Trailing zero

Significant figures Zeros are what will give you a headache! They are used/misused all of the time.Example The press might report that the federal deficit is three trillion dollars. What did they mean? $3 x meaning +/- a trillion dollars or $3,000,000,000, meaning +/- a penny

Significant figures In science, all of our numbers are either measured or exact. Exact Exact - Infinite number of significant figures. Measured Measured - the tool used will tell you the level of significance. Varies based on the tool.Example Ruler with lines at 1/16” intervals. A balance might be able to measure to the nearest 0.1 grams.

Significant figures: Rules for zeros Scientific notation Scientific notation - can be used to clearly express significant figures. A properly written number in scientific notation always has the the proper number of significant figures = 3.21 x Three Significant Figures Three Significant Figures

Scientific notation Method to express really big or small numbers. Format isMantissa x Base Power Decimal part of original number Decimals you moved We just move the decimal point around.

Scientific notation If a number is larger than 1 The original decimal point is moved X places to the left. The resulting number is multiplied by 10 X. The exponent is the number of places you moved the decimal point = 1.23 x 10 8

Scientific notation If a number is smaller than 1 The original decimal point is moved X places to the right. The resulting number is multiplied by 10 -X. The exponent is the number of places you moved the decimal point = 1.23 x 10 -7

Most calculators use scientific notation when the numbers get very large or small. How scientific notation is displayed can vary. It may use x10 n or may be displayed using an E. They usually have an Exp or EE This is to enter in the exponent. Scientific notation E-2

Examples x x x x

Significant figures and calculations An answer can’t have more significant figures than the quantities used to produce it.Example How fast did you run if you went 1.0 km in 3.0 minutes? speed = 1.0 km / 3.0 min = 0.33 km / min

Significant figures and calculations Addition and subtraction Report your answer with the same number of digits to the right of the decimal point as the number having the fewest to start with g g g g g 83.7 g

Significant figures and calculations Multiplication and division. Report your answer with the same number of digits as the quantity have the smallest number of significant figures. Example. Density of a rectangular solid kg / [ (18.5 m) ( m) (2.1m) ] = 2.8 kg / m 3 (2.1 m - only has two significant figures)

Example 257 mg \__ 3 significant figures 102 miles \__ 3 significant figures kg \__ 3 significant figures 23, $/yr \__ 7 significant figures

Rounding off numbers After calculations, you may need to round off.

If a set of calculations gave you the following numbers and you knew each was supposed to have four significant figures then becomes becomes Rounding off 1st uncertain digit

Converting units Factor label method Regardless of conversion, keeping track of units makes thing come out right Must use conversion factors - The relationship between two units Canceling out units is a way of checking that your calculation is set up right!

Common conversion factors Factor SomeEnglish/ Metric conversions Factor 1 liter= quarts1.057 qt/L 1 kilogram= 2.2 pounds2.2 lb/kg 1 meter= yards1.094 yd/m 1 inch= 2.54 cm2.54 cm/inch

Example A nerve impulse in the body can travel as fast as 400 feet/second. What is its speed in meters/min ? Conversions Needed 1 meter = 3.3 feet 1 minute= 60 seconds

m 400 ft 1 m 60 sec min 1 sec 3.3 ft 1 min Example m 400 ft 1 m 60 sec min 1 sec 3.3 ft 1 min ? ? =xx ? ? =xx m min....Fast 7273

Extensive and intensive properties Extensive properties Depend on the quantity of sample measured. Example Example - mass and volume of a sample. Intensive properties Independent of the sample size. Properties that are often characteristic of the substance being measured. Examples Examples - density, melting and boiling points.

Density Density is an intensive property of a substance based on two extensive properties. Common units are g / cm 3 or g / mL. g / cm 3 Air Bone Water 1.0Urine Gold19.3Gasoline Density = Mass Volume cm 3 = mL

Example. Density calculation What is the density of 5.00 mL of a fluid if it has a mass of 5.23 grams? d = mass / volume d = 5.23 g / 5.00 mL d = 1.05 g / mL What would be the mass of 1.00 liters of this sample?

Example. Density calculation What would be the mass of 1.00 liters of the fluid sample? The density was 1.05 g/mL. density = mass / volume somass = volume x density mass = 1.00 L x 1000 x 1.05 = 1.05 x 10 3 g ml L g mL

Specific gravity The density of a substance compared to a reference substance. Specific Gravity = Specific Gravity is unitless. Reference is commonly water at 4 o C. At 4 o C, density = specific gravity. Commonly used to test urine. density of substance density of reference

Specific gravity measurement Hydrometer Float height will be based on Specific Gravity.

Measuring time The SI unit is the second (s). For longer time periods, we can use SI prefixes or units such as minutes (min), hours (h), days (day) and years. Months are never used - they vary in size.

The mole Number of atoms in grams of 12 C 1 mol = x atoms mol = grams / formula weight u Atoms, ions and molecules are too small to directly measure - measured in u. Using moles gives us a practical unit. thegram We can then relate atoms, ions and molecules, using an easy to measure unit - the gram.

The mole If we had one mole of water and one mole of hydrogen, we would have the name number of molecules of each. 1 mol H 2 O = x molecules 1 mol H 2 = x molecules We can’t weigh out moles -- we use grams. We would need to weigh out a different number of grams to have the same number of molecules

Moles and masses Atoms come in different sizes and masses. A mole of atoms of one type would have a different mass than a mole of atoms of another type. H u or grams/mol O u or grams/mol Mo u or grams/mol Pb u or grams/mol We rely on a straight forward system to relate mass and moles.

Masses of atoms and molecules Atomic mass The average, relative mass of an atom in an element. Atomic mass unit (u) Arbitrary mass unit used for atoms. Relative to one type of carbon. Molecular or formula mass The total mass for all atoms in a compound.

Molar masses Once you know the mass of an atom, ion, or molecule, just remember: Mass of one unit - use u Mass of one mole of units - use grams/mole DON’T The numbers DON’T change -- just the units.

Masses of atoms and molecules H 2 O H 2 O - water 2 hydrogen 2 x1.008 u 1 oxygen1 x u mass of molecule u g/mol Rounded off based on significant figures Rounded off based on significant figures

Another example CH 3 CH 2 OH CH 3 CH 2 OH - ethyl alcohol 2 carbon2 x12.01 u 6 hydrogen6 x1.008 u 1 oxygen1 x16.00 u mass of molecule46.07 u g/mol

Molecular mass vs. formula mass Formula mass Add the masses of all the atoms in formula - for molecular and ionic compounds. Molecular mass Calculated the same as formula mass - only valid for molecules. Both have units of either u or grams/mole.

Formula mass, FM The sum of the atomic masses of all elements in a compound based on the chemical formula. You must use the atomic masses of the elements listed in the periodic table. CO 2 1 atom of C and 2 atoms of O 1 atom C x u = u 2 atoms O x u = u Formula mass = u Formula mass = u or g/mol or g/mol

Example - (NH 4 ) 2 SO 4 OK, this example is a little more complicated. The formula is in a format to show you how the various atoms are hooked up. ( N H 4 ) 2 S O 4 ( N H 4 ) 2 S O 4 We have two (NH 4 + ) units and one SO 4 2- unit. Now we can determine the number of atoms.

Example - (NH 4 ) 2 SO 4 Ammonium sulfate contains 2 nitrogen, 8 hydrogen, 1 sulfur & 4 oxygen. 2 Nx14.01 = Hx1.008 = Sx32.06 = Ox16.00=64.00 Formula mass= The units are either u or grams / mol.

Example - (NH 4 ) 2 SO 4 How many atoms are in 20.0 grams of ammonium sulfate? Formula weight = grams/mol Atoms in formula= 15 atoms / unit moles = 20.0 g x = mol 1 mol g atoms = mol x 15 x 6.02 x10 23 atoms unit units mol atoms = 1.36 x10 24