Chapter 2 Measurement & Problem Solving

Uncertainty There is a certain amount of doubt in every measurement – It is important to know the uncertainty when measurements are recorded or read – Why is this important in science? This helps to communicate the amount of accuracy and precision of the measurement

Accuracy The closeness between a measurement and its true value. – The true value is rarely known, so accuracy is based on how close independent studies agree – The closer they agree, the more confidence scientists have in the accuracy of their results

Accuracy Accuracy is affected by determinate errors – Determinate errors are errors due to 1)poor technique 2)incorrectly calibrated instruments

Which graduated cylinder is more accurate? 50 mL up by 1 mL 25 mL up by 0.5 mL 10 mL up by 0.1 mL **Has a smaller scale so measurement will be closer to true value

Precision The reproducibility of an experiment (how close are the repeated measurements to each other) Precision is affected by indeterminate errors – Indeterminate errors - errors due to estimating the last uncertain digit of a measurement Random Cannot be eliminated

Graduated Cylinder 5.72 mL 3.0 mL 0.33 mL

Precision What are other examples of when precision is important to you?

Which graduated cylinder is more precise? 50 mL up by 1 mL 25 mL up by 0.5 mL 10 mL up by 0.1 mL **Has a smaller scale so measurement will be more precise

Is it possible to be precisely inaccurate? YES!! Image from:

What is the volume? Up front, two of the graduated cylinders have water in them 10 students to “read” the volumes – Write down your measurements on a scratch piece of paper: Volumes of water in 50mL grad. cylinder = 10mL grad. cylinder= 1 student to write measurements on the board and tally them if there are multiples

Is the variation in those measurements due to determinate or indeterminate errors? BOTH! Determinate errors- may have had poor technique (didn’t read from bottom of meniscus, didn’t lower eyes to the level of the meniscus) Indeterminate errors- estimating the last digit (when the level of the meniscus is between two distinct lines) **This illustrates that there is always uncertainty in measurements!

Uncertainty Instruments are not always calibrated perfectly (remember, humans do the calibrating and many times the measuring!) It is very important that in science we can communicate how accurate or precise our measurements are

Communicating Uncertainty In science, it is understood that the last digit of a number is “uncertain” or the estimated digit (if another person were to make the same measurement, the number may be different) *The estimate is made between the smallest divisions on the scale of the instrument (i.e. ruler, graduated cylinder, thermometer) **Generally, the more numbers after the decimal point, the more precise the instrument is

General Rule for Measuring Uncertainty Unless otherwise indicated, assume that the uncertainty in a measurement is one-half the smallest division Examples: ________________ _________________ _______________

Which is more precise? 1002 cmor cm This number is measured to more decimal places which indicates smaller increments of measurement In both numbers, which numbers are uncertain? Instrument measures every 0.1 cm, so estimated to the hundredths place Instrument measures every 10 cm, so estimated to the ones place

Things to keep in mind… When using an electric device, the last digit reported will be considered the uncertain digit “2” and “2.0” are not the same number! – By saying “2” we are saying the number is not 1 or 3 – By saying “2.0” we are saying the number is not 1.9 or 2.1 **It is the responsibility of the experimenter to write numbers in a way to reflect the precision of the instrument being measured

Standard Deviation Values reported as results from experiments are in the form of the “mean plus-or-minus one standard deviation” ( X ± s.d.) – This gives a range over which we have confidence that the true value will fall Used in scientific journals Requires a sample of at least three Most calculators are able to compute the mean and standard deviation – The standard deviation indicates how much each value differs from the average – The size of the deviation would indicate precision in running a sequence of experiments (the smaller the deviation, the more precise the work was)

Scientific Notation Scientific Notation is often used to write very large or very small numbers – The numbers are expressed as a coefficient number multiplied by an exponent of 10 coefficient exponent 4.32 x 10 5

Scientific Notation The coefficient is a number greater than 1 but less than 10 The exponent is: – Positive if the number is greater than 1 Ex: 1,200 = 1.2 x 10 3 – Negative if the number is less than 1 Ex: = 2.3 x – Zero if the number is equal to 1 Ex: 1 = 10 0

Addition & Subtraction with Scientific Notation 1)Exponents must be made equal before you can perform the operation Ex: 5.4x x10 2 = 5.4x x10 3 2) Then add or subtract the coefficients & the exponent will stay the same Ex: 5.4 x x10 3 = 6.0 x10 3

Multiplication and Division with Scientific Notation 1) Multiply the coefficients and add the exponents Ex: ( 3.0 x10 4 ) x ( 5.0 x10 2 ) = 15.0 x10 6 = 1.50x10 7 2) Divide the coefficients and subtract the exponents – be careful when subtracting a negative exponent!! Ex: ( 8.0 x10 4 ) ÷ ( 2.0 x10 -2 ) = 4.0 x10 6

Significant Figures In science, numbers are only significant if they were measured. The last number reported is the “estimated” number and therefore is the uncertain number

Significant Figures All non-zero digits are significant What to do with the zeros: – Zeros between significant digits are significant (Sandwich Rule) Ex: 307cm (3 sig figs) – Zeros at the end of a number and to the RIGHT of a decimal are significant Ex: 5.20cm (3 sig figs)

Significant Figures Zeros at the end of a number and to the LEFT of an assumed decimal may or may not be significant **The presence of a decimal indicates significance, while the absence of the decimal point makes it insignificant Ex: 750.cm (3 sig figs) 750cm (2 sigs figs)

Significant Figures “Cosmetic” zeros written to the LEFT of the first non-zero digit are NOT significant Ex: 0.146cm (3 sig figs) 09 cm (1 sig fig) Zeros that are place holders are NOT significant Ex:.006cm (1 sig fig) 0.014cm (2 sig figs)

Significant Figures The difference between zeros used for accuracy and those being used as place holders can only be addressed using scientific notation – 1,000,000 – 1.00x10 6 – x10 6 All zeros are placeholders (no decimal) 1 sig fig The first two zeros are significant 3 sig figs All the zeros are significant 6 sig figs

Sig Fig Practice mL s g x10 23 molecules mol m J x10 -3 cm mg 4 sig figs 2 sig figs 6 sig figs 3 sig figs 2 sig figs 4 sig figs 6 sig figs 3 sig figs

Exact Values Some numbers are exact values which involve no uncertainty. For example, there is an exact number of people in the classroom. If there are 20 people, the zero is automatically significant. Chemistry examples: – There are joules in a calorie – There are exactly 2 hydrogen atoms in a water molecule – Stoichiometric coefficients and subscripts 6CO 2 – Defined quantities (i.e. If we know 1 liter = 1000 mL, the number 1000 actually has an infinite number of significant digits)

Significant Figures & Rounding Round down if last (or leftmost) digit dropped is 4 or less Ex: cm 56.32cm Round up if the last digit dropped is 5 or more Ex: cm 4.53cm ** Make sure you only use the last digit being dropped to decide which direction to round (ignore all digits to the right of it)

Addition & Subtraction of Significant Figures The number of places to the right of the decimal point (degree of accuracy), determines the number of sig figs reported cm 4.1 cm cm cm **This number has the least number of digits to the right of the decimal – Answer should be rounded to the tenth place Add the decimals as you would normally (add in placement zeros if it helps you, but do not consider them when you decide on the # of sig figs Correct Answer: 43.0 cm

Multiplication & Division of Significant Figures The number or reported digits is determined by the lease accurate value (the one with the fewest significant figures) 23.5 cm x cm x 2. cm = 2, cm This number has the least amount of sig figs, so the answer will need to have only 1 sig fig Correct answer: 3000 cm or 3 x 10 3 cm

Keep in mind… For calculations involving multiple steps, round only the final answer In calculations involving both multiplication/division and addition/subtraction, do the steps in parenthesis first

Determining Sig Figs in Measurements The last significant figure in your measurement needs to be one that you estimate The digits preceding this last digit, are considered “certain” because the instrument will have specific “marks” that should be exact measurements. The final “estimated” digit will be located between the specific “marks” (depending on the location of the measurement, but you may also decide that the estimated digit is a zero)

How many centimeters is this pencil? Your answer should have 3 sig figs! The ruler measures 0.1cm (1mm) so your measurement should have an estimated digit in the hundredths position ~6.12 cm

How many mL are in this graduated cylinder? Your answer should have 4 sig figs. The graduated cylinder measures to every 0.1 mL so your estimated digit will be in the hundreths position ~40.30 mL

How long is this pen? The tip of the pen is between 4.7 cm and 4.8 cm. You will need to estimate the digit between the two points