Measurement in Scientific Study and Uncertainty in Measurement Lecture #3 Measurement in Scientific Study and Uncertainty in Measurement Chemistry 142 B James B. Callis, Instructor Autumn Quarter, 2004

Precision and Accuracy Errors in Scientific Measurements Precision - Refers to reproducibility or how close the measurements are to each other. Accuracy - Refers to how close a measurement is to the ‘true’ value. Systematic Error - produces values that are either all higher or all lower than the actual value. Random Error - in the absence of systematic error, produces some values that are higher and some that are lower than the actual value.

Rules for Determining Which Digits Are Significant All digits are significant, except zeros that are used only to position the decimal point. 1. Make sure that the measured quantity has a decimal point. 2. Start at the left of the number and move right until you reach the first nonzero digit. 3. Count that digit and every digit to its right as significant. Zeros that end a number and lie either after or before the decimal point are significant; thus 1.030 mL has four significant figures, and 5300. L has four significant figures also. Numbers such as 5300 L have 2 sig. figs., but 5.30x103 L has 3. A terminal decimal point is often used to clarify the situation, but scientific notation is clearer (best).

Examples of Significant Digits in Numbers Number - Sig digits Number - Sig digits 0.0050 L 1.34000 x 107 nm six 18.00 g four 5600 ng 0.00012 kg two 87,000 L two 83.0001 L six 78,002.3 ng six 0.006002 g four 0.000007800 g four 875,000 oz 1.089 x 10–6 L 30,000 kg one 0.0000010048 oz five 5.0000 m3 five 6.67000 kg six 23001.00 lbs seven 2.70879000 mL nine 0.000108 g 1.0008000 kg eight 1,470,000 L three 1,000,000,000 g

Examples of Significant Digits in Numbers Number - Sig digits Number - Sig digits 0.0050 L two 1.34000 x 107 nm six 18.00 g four 5600 ng two 0.00012 kg two 87,000 L two 83.0001 L five 78,002.3 ng six 0.006002 g four 0.000007800 g four 875,000 oz three 1.089 x 10 -6L four 30,000 kg one 0.0000010048 oz five 5.0000 m3 five 6.67000 kg six 23,001.00 lbs seven 2.70879000 mL nine 0.000108 g three 1.0008000 kg eight 1,470,000 L three 1,000,000,000 g one

Rules for Significant Figures in answers 1. For multiplication and division. The number with the least certainty limits the certainty of the result. therefore, the answer contains the same number of significant figures as there are in the measurement with the fewest significant figures. Multiply the following numbers: 9.2 cm x 6.8 cm x 0.3744 cm = 2. For addition and subtraction. The answer has the same number of decimal places as there are in the measurement with the fewest decimal places. Example, adding two volumes 83.5 mL + 23.28 mL = Example subtracting two volumes: 865.9 mL - 2.8121393 mL =

Rules for Significant Figures in answers 1. For multiplication and division. The number with the least certainty limits the certainty of the result. therefore, the answer contains the same number of significant figures as there are in the measurement with the fewest significant figures. Multiply the following numbers: 9.2 cm x 6.8 cm x 0.3744 cm = 23.4225 cm3 = 23 cm3 2. For addition and subtraction. The answer has the same number of decimal places as there are in the measurement with the fewest decimal places. Example, adding two volumes 83.5 mL + 23.28 mL = 106.78 mL = 106.8 mL Example subtracting two volumes: 865.9 mL - 2.8121393 mL = 863.0878607 mL = 863.1 mL

Rules for Rounding Off Numbers (1) In a series of calculations*, carry the extra digits through to the final result, then round off. ** (2) If the digit to be removed is less than 5, the preceding digit stays the same. For example, 1.33 rounds to 1.3. is equal to or greater than 5, the preceding digit is increased by one. For example, 1.36 rounds to 1.4. (3) When rounding, use only the first number to the right of the last significant figure. Do not round off sequentially. For example, the number 4.348 when rounded to two significant figures is 4.3, not 4.4. Notes: * Your TI-93 calculator has the round function which you can use to get the correct result. Find round by pressing the math key and moving to NUM. Its use is round(num, no of decimal places desired), e.g. round(2.746,1) =2.7. ** Your book will show intermediate results rounded off. Don’t use these rounded results to get the final answer.

Rounding Off Numbers – Problems (3-1a) Round 5.379 to three significant figures Ans: (3-1b) Round 5.379 to two significant figures We used the rule: If the digit removed is greater than or equal to 5, the preceding number increases by 1. (3-2a) Round 0.2413 to three significant figures (3-2b) Round 0.2413 to two significant figures We used the rule: If the digit removed is less than 5, the preceding number is unchanged

Sample Problem – 3-3 Lithium (Li) is a soft, gray solid that has the lowest density of any metal. If a slab of Li weighs 1.49 x 103 mg and has sides that measure 20.9 mm by 11.1 mm by 12.0 mm, what is the density of Li in g/ cm3 ? Lengths (mm) of sides Mass (mg) of Li Lengths (cm) of sides Mass (g) of Li Volume (cm3) Density (g/cm3) of Li

Sample Problem – 3-3(cont.) Mass (g) of Li = 1.49 x 103 mg Length (cm) of one side = 20.9 mm Similarly, the other side lengths are: Volume (cm3) = Density = mass/volume Density of Li =

Problem 3-4: Volume by Displacement Problem: Calculate the density of an irregularly shaped metal object that has a mass of 567.85 g if, when it is placed into a 2.00 liter graduated cylinder containing 900.00 mL of water, the final volume of the water in the cylinder is 1277.56 mL ? Plan: Calculate the volume from the different volume readings, and calculate the density using the mass that was given. Solution: Volume = mass Density = volume

Definitions - Mass & Weight Mass - The quantity of matter an object contains kilogram - ( kg ) - the SI base unit of mass, is a platinum - iridium cylinder kept in Paris as a standard! Weight - depends upon an object’s mass and the strength of the gravitational field pulling on it, i.e. w = f = ma.

Problem 3-5: Computer Chips Future computers might use memory bits which require an area of a square with 0.25 mm sides. (a) How many bits could be put on a 1 in x 1 in computer chip? (b) If each bit required that 25 % of its area to be coated with a gold film 10 nm thick, what mass of gold would be needed to make one chip? Approach: use Achip = (b) use r = m/V

Solution to Chip Problem (3-7)

Solution to Chip Problem (3-7)

Temperature Scales and Interconversions Kelvin ( K ) - The “Absolute temperature scale” begins at absolute zero and only has positive values. Celsius ( oC ) - The temperature scale used by science, formally called centigrade and most commonly used scale around the world, water freezes at 0oC, and boils at 100oC. Fahrenheit ( oF ) - Commonly used scale in America for our weather reports, water freezes at 32oF, and boils at 212oF. T (in K) = T (in oC) + 273.15 T (in oC) = T (in K) - 273.15 T (in oF) = 9/5 T (in oC) + 32 T (in oC) = [ T (in oF) - 32 ] 5/9

Problem 3-6:Temperature Conversions (a) The boiling point of Liquid Nitrogen is -195.8 oC, what is the temperature in Kelvin and degrees Fahrenheit? T (in K) = T (in oC) + 273.15 T (in K) = T (in oF) = 9/5 T (in oC) + 32 T (in oF) = (b)The normal body temperature is 98.6oF, what is it in Kelvin and degrees Celsius? T (in oC) = [ T (in oF) - 32] 5/9 T (in oC) = T (in K) = T (in oC) + 273.15 T (in K) =

Answers to Problems in Lecture #3 (a)5.38; (b) 5.4 (a) 0.241; (b) 0.24 0.536 g/cm3 1.5040 g / mL 31 mg gold (a) 77.4 K; -320.4 oF; (b) 37.0 oC; 310.2 K