Weights measured for a person using a home balance (100, 130, 110, 150, 120 lbs). What number do you believe?

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1 Lecture 2 Measurements, Errors, and Uncertainties Anytime one does a measurement there will be some error or uncertainty in the measurement. Examples: Weights measured for a person using a home balance (100, 130, 110, 150, 120 lbs). What number do you believe? What if the weights were for a cat? Would you believe it? What the measured weight was listed as lbs? Do you see any problem? QUALITATIVE EXPRESSION OF UNCERTAINTY We describe these uncertainties qualitatively using two basic terms: (1) precision - closeness of the set of values from identical measurements (2) accuracy - closeness of a single measurement to the true value These two terms are distinct. Example 1: You weigh a crucible ten times and get the same number every time. Do you have high precision? high accuracy? What if your balanced wasn't zeroed properly? Example 2: The Hubble Telescope QUANTITATIVE EXPRESSION OF UNCERTAINTY (1) The inaccuracy of a measurement can be expressed as an exact value. Example One weighs a 10.0 lb standard and obtains a weight of 12.5 lbs. The error in the measurement is = 2.5 lbs

2 (2) The precision of a set of data is generally expressed by using a standard deviation something you ll learn in more advanced classes. A related type of uncertainty results from the maximum precision inherent to the measuring instrument. This uncertainty addressed by expressing measured data using significant digits. Significant digits The significant digits in a measured number are all of digits that are certain, plus a final one having some uncertainty. Example: What would you report if you measured a penny on a balance three times and got g g g The number of significant figures is the number of digits reported for a measured quantity. Example: 3.07 is reported to 3 significant figures. In order to determine which digits are significant, use the following rules: (1). All digits are significant except zeros at the beginning of the number and possibly terminal zeros (one or more zeros at the end of the number), (2) Terminal zeros at the right of the decimal point are significant ) Terminal zeros ending to the left of the decimal point may or may not be significant ,000 Examples: How many significant digits do the following numbers have?

3 Mathematical Operations and Significant Digits (+,-) - keep the same number as the number with the least number of decimal place = = = Examples: How many significant digits are in the following numbers or expressions? = = = = (*,/) - keep the same number as the value with the least number of significant digits 1 * = 1.0 * = 1.00 * = Example 12.0 * 0.1 = 12 / 0.1 = 12* /0.10 =

4 One exception to these rules is when one has exact number. An exact number is a number that arises when you count items or sometimes when you define a unit. (e.g. 10 cars) Here you keep the number of significant digits of the measured value. Example 1: One penny weighs g. How much does 10 pennies weigh? Example 2: A rod weighs grams per meter of length. How much does a 10 meter rod weigh? Significant Digits in the Real World Example: Suppose the density of water is being determined by measuring the volume and mass of 100 ml of water. Volume is measured using a volumetric cylinder (100.0 ± 0.2 ml). Mass is measured using a balance ( ± g). The density is given by mass volume = g 100.0mL = g/ml One of the benefits of using significant figures is that it helps you determine where you need to be careful (volume) (weight) Which do you need to measure more carefully? Scientific Notation Numbers like 200 or 5000 create problems for scientific notation. One way to overcome this problem is to use scientific notation representing numbers in the form A x 10 n.

5 Examples: x ? 800 (three significant figures)? 800 (one significant figure)? Rounding Rounding is the procedure of dropping non-significant digits in a calculation result and adjusting the last digit reported. Example Write to four significant figures? Is it 1234 or 1235? Rules for rounding: (1) If its greater than XXXX.5, round up. (2) If its less than XXXX.5 round down. (3) If its equal to XXXX.5, round to even. The best way to understand these rules is through examples. Example 1 Round the following numbers to 3 significant digits. (What does 3 significant digits mean?)

6 Example 2 What do the following numbers round to using 3 significant digits: Example 3 Report the following to the correct number of significant digits * (5 x 10 2 ) = 1.0 * = Dimensional Analysis IMPORTANT TOPIC Dimensional analysis is a method of calculation in which one carries along the units with the numerical quantities. Typically dimensional analysis is used when doing conversions between units. Example 1: How many eggs are in 3 dozen? 3 x 12 eggs = 36 eggs This is easy Example 2: A crate contains 10 cartons of eggs. Each carton contains a dozen eggs. Each egg weighs 10.0 grams. An egg is 80% water by weight. How much water (by weight) is there in 20 cartons of eggs? This problem is easily solved using dimensional analysis.

7 To solve a problem using dimensional analysis, follow the following steps. 1) Write the units you want in the final answer. 2) Write the numbers and units of the values you have. 3) Fill in the conversion factors that relate the units you have to the units you want. Conversion factors Conversion factors relate equal amounts. Both 12 eggs and 1 dozen are identical amounts. They have different numerical values, but this is due to the different units. Hence, if we multiply a number by a conversion factor, it is equivalent to multiplying it by the scalar number 1. Example: How many dozen is 30 eggs? The critical thing when using conversion factors is to end up with the correct units. Here, the conversion factor must have units of dozen/eggs to be able to cancel the inches. What happens if the conversion factor is used upside down? You get the wrong units... Example 1: How many eggs are in a crate of 20 egg cartons?

8 Example 2: Convert 20 g/l to g/ml Example 3: One particularly difficult type of the problem for students are those with squared or cubed terms such as m 3. If 1 cm 3 = 1 ml, how many ml would a box of dimensions 10 cm x 10 cm x 10 cm hold? How about a 1 m x 1 m x 1 m box? At least one problem related to dimensional analysis will be on the quiz - probably two. Thought Problem: You can t use dimensional analysis to do temperature conversions. Why?

9 The Scientific Method The scientific method is an approach by which one can come to valid theories. In chemistry, this involves interative experiments designed to test the hypotheses and refine it. EXPERIMENT (explanation) HYPOTHESIS Once a hypothesis has been confirmed by many experiments it can be considered a theory. Example: All rubber balls bounce. If a theory is basic enough, it can be a law. Three basic laws relevant to chemistry 1. Law of conservation of mass 2. Law of conservation of energy 3. Law of definite proportions Law of Conservation of Mass (Lavoisier): The total mass remains constant during a chemical change (or chemical reaction). Example: If you take a sealed flask containing mercury and oxygen, then heat it, it will form mercury oxide. The weight of the vial will remain the same. This is not true however if you burn coal in the open air. Why? Law of Conservation of Energy: Energy cannot be created or destroyed in a chemical reaction or physical change. It can only be converted from one form to another. Example: Pendulum Law of Definite Proportions (or law of constant composition) (Proust). A pure compound whatever its source always contains definite or constant proportions of elements by mass. Example: 1 g of NaCl will always contain: g of sodium g of chloride