Chapter 3 - Experimental Error - WH Freeman

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$Math for Life - Life: The Science of Biology, Seventh ... - WH Freeman$

54 3 <strong>Experimental</strong> <strong>Error</strong> The real rule: The first uncertain figure is the last significant figure. The rationale for finding the uncertainty in the molecular mass of NH 3 is explained at the end of this chapter: N: 14.006 74 0.000 07 3H: 3(1.007 94 0.000 07) N: 14.006 74 0.000 07 3H: 3.023 82 0.000 21 NH 3 : 17.030 56 20.000 07 2 0.000 21 2 17.030 5 6 0.000 2 2 17.030 6 0.000 2 Convert absolute uncertainty to % relative uncertainty for multiplication. The Real Rule for Significant Figures The first uncertain figure of the answer is the last significant figure. For example, in the quotient the uncertainty (0.000 2) occurs in the fourth decimal place. Therefore, the answer is properly expressed with three significant figures, even though the original data have four figures. The first uncertain figure of the answer is the last significant figure. The quotient is expressed with four significant figures because the uncertainty occurs in the fourth place. The quotient 0.821 (0.002) 1.022 (0.004) 0.803 (0.002) is expressed with four figures even though the dividend and divisor each have three figures. Example Significant Figures in Laboratory Work You prepared a 0.250 M NH 3 solution by diluting 8.45 (0.04) mL of 28.0 (0.5) wt % NH 3 [density 0.899 (0.003) g/mL] up to 500.0 (0.2) mL. Find the uncertainty in 0.250 M. The molecular mass of NH 3 , 17.030 6 g/mol, has negligible uncertainty relative to other uncertainties in this problem. SOLUTION To find the uncertainty in molarity, we need to find the uncertainty in moles delivered to the 500-mL flask. The concentrated reagent contains 0.899 (0.003) g of solution per milliliter. The weight percent tells us that the reagent contains 0.280 (0.005) g of NH 3 per gram of solution. In the following calculations, we retain extra insignificant digits and round off only at the end. Grams of NH 3 per mL in concentrated reagent 0.002 364 (0.000 003) 0.025 00 (0.000 05) 0.002 664 (0.000 003) 0.025 00 (0.000 05) 0.899 (0.003) g solution mL 0.899 (0.334%) g solution mL 0.251 7 (1.82%) g NH 3 mL 0.094 6 (0.000 2) 0.106 6 (0.000 2) g NH 3 0.280 (0.005) g solution g NH 3 0.280 (1.79%) g solution because 2(0.334%) 2 (1.79%) 2 1.82%. Next, we find the moles of ammonia contained in 8.45 (0.04) mL of concentrated reagent. The relative uncertainty in volume is 0.04/8.45 0.473%. 0.251 7 (1.82%) g NH 3 mL mol NH 3 17.030 6 (0%) g NH 3 mol 0.124 9 (1.88%) mol 8.45 (0.473%) mL because 2(1.82%) 2 (0.473%) 2 (0%) 2 1.88%. This much ammonia was diluted to 0.500 0 (0.000 2) L. The relative uncertainty in the final volume is 0.000 2/0.500 0 0.04%. The molarity is
mol NH 3 L 0.124 9 (1.88%) mol 0.500 0 (0.04%) L 0.249 8 (1.88%) M because 2(1.88%) 2 (0.04%) 2 1.88%. The absolute uncertainty is 1.88% of 0.249 8 M 0.004 7 M. The uncertainty in molarity is in the third decimal place, so our final, rounded answer is 55 3-5 Propagation of Uncertainty [NH 3 ] 0.250 (0.005) M Exponents and Logarithms For the function y x a , the percent relative uncertainty in y(%e y ) is equal to a times the percent relative uncertainty in x(%e x ): Uncertainty for powers and roots: (3-7) For example, if y 2x x 1/2 , a 2% uncertainty in x will yield a (2%) 1% uncertainty in y. If y x 2 , a 3% uncertainty in x leads to a (2)(3%) 6% uncertainty in y (Box 3-2). If y is the base 10 logarithm of x, then the absolute uncertainty in y (e y ) is proportional to the relative uncertainty in x (e x /x): Uncertainty for logarithm: y x a 1 %e y a(%e x ) y log x1 e y 1 ln 10 e x x 0.434 29 e x x (3-8) You should not work with percent relative uncertainty [100 (e x /x)] in calculations with logs and antilogs, because one side of Equation 3-8 has relative uncertainty and the other has absolute uncertainty. 1 1 22 To calculate a power or root on your calculator, use the y x button. For example, to find a cube root (y 1/3 ), raise y to the 0.333 333 333 ...power with the y x button. Use relative uncertainty (e x /x), not percent relative uncertainty [100 (e x /x)], in calculations involving log x, ln x, 10 x , and e x . Box 3-2 Propagation of Uncertainty in the Product xx Table 3-1 says that the uncertainty in the function y x a is %e y a(%e x ). If y x 2 , then %e y 2(%e x ). A 3% uncertainty in x leads to a (2)(3%) 6% uncertainty in y. But what if we just apply the multiplication formula 3-6 to the product xx? x(e 1 ) x (e 2 ) x 2 (e 3 ) % e 3 2(%e 1 ) 2 (%e 2 ) 2 2(3%) 2 (3%) 2 4. 2 % Which uncertainty is correct, 6% from Table 3-1 or 4. 2 % from Equation 3-6? Table 3-1 (6%) is correct. In the formula y x 2 , the error in a measured value of x is always positive or always negative. If the true value of x is 1.00 and the measured value is 1.01, the computed value of x 2 is (1.01) 2 1.02. That is, if the measured x is high by 1%, the computed value of x 2 is high by 2% because we are multiplying the high value by the high value. Equation 3-6 presumes that the uncertainty in each factor of the product x z is random and independent of the other. In the product x z, the measured value of x could be high sometimes and the measured value of z could be low sometimes. In the majority of cases, uncertainty in the product x z is not as great as the uncertainty in x 2 . Example. The distance traveled by a falling object in time 1 t is 2gt 2 , where g is the acceleration of gravity. If t has an uncertainty of 1%, the uncertainty in t 2 is 2(%e t ) 2(1%) 2%. 1 The uncertainty in distance computed from 2gt 2 will also be 2%. If you (incorrectly) used Equation 3-6, you would compute an uncertainty in distance of 21% 2 1% 2 1. 4 %. Relative uncertainty in x Relative uncertainty in z Relative uncertainty in x · z Relative uncertainty in x 2
Page 1 and 2: Experimental Error 3 Experimental E
Page 3 and 4: 3-2 Significant Figures in Arithmet
Page 5 and 6: graph (such as Figure 3-3) to read
Page 7 and 8: eport several different readings. A
Page 9: Regardless of the initial and final
Page 13 and 14: The concentration of H is 6.17 (0.
Page 15 and 16: Problems 59 Reading (1.46 mm) Digit

Chapter 3 - Experimental Error - WH Freeman

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