Measurements, Significant Figures, Scientific Notation

"If you have only one watch, you always know exactly what time it is "If you have only one watch, you always know exactly what time it is. If you have two watches, you are never quite sure..." The fact is, when we take a single reading, whether from a thermometer, a wristwatch, or a fancy laboratory instrument, we tend to accept the readout without really thinking about its validity. People seem to do this even when they know the reading is inaccurate. Your fancy digital watch is probably off by a minute or two right now! Try as you might, with the most expensive instruments, under the most ideal conditions, every measurement is subject to errors and inaccuracies. But what is worse, modern digital instruments convey such an aura of accuracy and reliability, that we forget this basic rule... There is no such thing as a Perfect Measurement

Measurements 1. What is precision? The precision of an instrument reflects the number of significant digits in a reading or the closeness of a group of measurements to each other (both are related to how small of a measurement it is capable of making and how consistent it is)

Measurements The accuracy of an instrument reflects how close the reading is to the 'true' value measured.

Accuracy When can you trust a measurement (is it inaccurate)? Consider when you would need to measure temperature—you would have 3 options: Glass Bulb Type - The expanding liquid is constrained to a thin glass tube, and the height of the liquid is a function of its temperature. Dial Type - Working on the principle that different metals expand at different rates when heated. A metallic coil unwinds with heat and a pointer can be attached to make the thermometer. If the pointer is bent or forced beyond its normal range of movement, inaccurate readings may result.

Accuracy Electronic Type - Working on the principle that the electrical characteristics of certain components (resistors, diodes, transistors) varies with temperature, this circuit essentially measures those varying characteristics and converts the result to a digital readout. Because the characteristics do not vary uniformly, the exact relationship between, (for example) apparent resistance and temperature, is not easily translated. Outside of the operating range, very unexpected results can occur and it may not be obvious at all that the operating range has been exceeded.

Accuracy vs. Precision Note that an accurate instrument is not necessarily precise, and instruments are often precise but far from accurate. For example, you might read out time right down to the second, even though you know your watch is one minute slow. This reading is precise, but not accurate. It makes little sense to quote values to high precision beyond the expected accuracy of the measurement. Without stating the estimated accuracy, such a reading cannot be used in serious computations. Worse, even by quoting the time down to the second, you have implied some accuracy which you cannot justify.

Precision Any measurement must be recorded in such a way as to show the degree of precision to which it was made-- no more, no less. Calculations based on the measured quantities can have no more (or no less) precision than the measurements themselves. The answers to the calculations must be recorded to the proper number of significant figures. To do otherwise is misleading and improper. 

Significant Figures Determining the Number of Significant Figures The number of significant figures in a measurement, such as 2.531, is equal to the number of digits that are known with some degree of confidence (2, 5, and 3) plus the last digit (1), which is an estimate or approximation. As we improve the sensitivity of the equipment used to make a measurement, the number of significant figures increases. Postage Scale: 3 1 g 1 significant figure Two-pan balance: 2.53 0.01 g 3 significant figs Analytical balance:2.531 0.001 g 4 significant figs *Note that the accuracy is the same for all three scales, just the precision is different

Significant Figures Rules for counting zeros as significant figures are summarized below: Zeros within a number are always significant. Both 4308 and 40.05 contain four significant figures. Zeros that do nothing but set the decimal point are not significant. Thus, 470,000 has two significant figures. Trailing zeros that aren't needed to hold the decimal point are significant. For example, 4.00 has three significant figures. If you are not sure whether a digit is significant, assume that it isn't. For example, if the directions for an experiment read: "Add the sample to 400 mL of water," assume the volume of water is known to one significant figure. (Unless it’s written as 400. mL of water)

Significant Figures A simpler set of rules will also work: Zeros that are found on the left of all non-zero digits are never significant Zeros that are found in between any non-zero digits are always significant Zeros that are found on the right of all non-zero digits are significant if there is a decimal point present.

Significant Figures Practice determining the number of sig figs in the following numbers: 2.3004 0.00004 0.00400 0.40040 230 230. 345,000 345,000.00

Calculating with Significant Figures Multiplication and Division With Significant Figures The same principle governs the use of significant figures in multiplication and division: the final result can be no more precise than the least precise measurement. So we count the significant figures in each measurement: When measurements are multiplied or divided, the answer can contain no more significant figures than the least precise measurement.

Multiplication and Division With Significant Figures Practice multiplying or dividing the following using appropriate sig fig rules: 230 x 12 = ? 0.4058 / 0.003 = ? 5482.3 / 25 = ? 74.077 x 2.100 x 16.0037 = ?

Calculating with Significant Figures Addition and Subtraction with Significant Figures When combining measurements with different degrees of precision, the precision of the final answer can be no greater than the least precise measurement. This principle can be translated into a simple rule for addition and subtraction: When measurements are added or subtracted, the answer can include no more values on the right side of the number than the least precise measurement. 150.0 g H2O (using significant figures) + 0.507 g salt 150.5 g solution

Addition and Subtraction with Significant Figures Practice adding or subtracting the following using appropriate sig fig rules: 230 + 12 = ? 0.4058 – 0.003 = ? 5482.3 + 25 = ? 74,077 + 2,100 + 16,003.7 = ?

Rounding Rounding Off When the answer to a calculation contains too many significant figures, it must be rounded off. There are 10 digits that can occur in the last decimal place in a calculation. One way of rounding off involves underestimating the answer for five of these digits (0, 1, 2, 3, and 4) and overestimating the answer for the other five (5, 6, 7, 8, and 9). This approach to rounding off is summarized as follows. If the digit is smaller than 5, drop this digit and leave the remaining number unchanged. Thus, 1.684 becomes 1.68. If the digit is 5 or larger, drop this digit and add 1 to the preceding digit. Thus, 1.247 becomes 1.25.

Scientific Notation To write a number in scientific notation: Put the decimal after the first digit and drop the zeroes on the right. In the number 123,000,000,000 The coefficient will be 1.23 To find the exponent count the number of places from the decimal to the end of the number.

Scientific Notation In 123,000,000,000 there are 11 places. Therefore we write 123,000,000,000 as:

Scientific Notation In scientific notation, the digit term indicates the number of significant figures in the number. The exponential term only places the decimal point. As an example, 46600000 = 4.66 x 107 This number only has 3 significant figures. The zeros are not significant; they are only holding a place. As another example, 0.00053 = 5.3 x 10-4 This number has 2 significant figures. The zeros are only place holders.

Scientific Notation Addition and Subtraction: All numbers are converted to the same power of 10, and the digit terms are added or subtracted. Example: (4.215 x 10-2) + (3.2 x 10-4) = (4.215 x 10-2) + (0.032 x 10-2) = 4.247 x 10-2 Example: (8.97 x 104) - (2.62 x 103) = (8.97 x 104) - (0.262 x 104) = 8.71 x 104

Scientific Notation Multiplication: The digit terms are multiplied in the normal way and the exponents are added. The end result is changed so that there is only one nonzero digit to the left of the decimal. Example: (3.4 x 106)(4.2 x 103) = (3.4)(4.2) x 10(6+3) = 14.28 x 109 = 1.4 x 1010 (to 2 significant figures) Example: (6.73 x 10-5)(2.91 x 102) = (6.73)(2.91) x 10(-5+2) = 19.58 x 10-3 = 1.96 x 10-2 (to 3 significant figures)

Scientific Notation Division: The digit terms are divided in the normal way and the exponents are subtracted. The quotient is changed (if necessary) so that there is only one nonzero digit to the left of the decimal. Example: (6.4 x 106)/(8.9 x 102) = (6.4)/(8.9) x 10(6-2) = 0.719 x 104 = 7.2 x 103 (to 2 significant figures) Example: (3.2 x 103)/(5.7 x 10-2) = (3.2)/(5.7) x 103-(-2) = 0.561 x 105 = 5.6 x 104 (to 2 significant figures)

Scientific Notation Powers of Exponentials: The digit term is raised to the indicated power and the exponent is multiplied by the number that indicates the power. Example: (2.4 x 104)3 = (2.4)3 x 10(4x3) = 13.824 x 1012 = 1.4 x 1013 (to 2 significant figures) Example: (6.53 x 10-3)2 = (6.53)2 x 10(-3)x2 = 42.64 x 10-6 = 4.26 x 10-5 (to 3 significant figures)