Significant Figures CHM 235 - Dr. Skrabal Definition: Minimum # of digits needed to express a number in scientific notation without a loss of accuracy Example: Partial pressure of CO2 in atmosphere  0.000356 atm. This number has 3 sig. figs., but leading zeros are only place-keepers and can cause some confusion. So express in scientific notation: 3.56 x 10-4 atm This is much less ambiguous, as the 3 sig. figs. are clearly shown.

Avoiding ambiguity Consider the quantity 1000 g. A little ambiguous—how many sig. figs. are intended to be in this number? 1.000 x 103 g (4 sf) 1.00 x 103 g (3 sf) 1.0 x 103 g (2 sf) 1 x 103 g (1 sf) Using scientific notation takes away the ambiguity.

Rules for using sig. figs. in calculations Addition and Subtraction Answer goes to the same decimal place as the individual number containing the fewest number of sig. figs. to the right of decimal point. Example of addition: Formula weight of PbS: 207.2 + 32.066 = 239.266  round to 239.3 Example of subtraction: 4.5237 – 1.06 = 3.4637  round to 3.46

Adding and subtracting numbers in scientific notation First convert all numbers to same power, then apply rules for adding and subtracting. Example: 1.032 x 104  1.032 x 104 2.672 x 105  26.72 x 104 3.191 x 106  319.1 x 104 ---------------- 346.852 x 104  round to 346.9 x 104

About rounding When rounding, look at all digits to the right of the last digit you want to keep. If more than halfway to the next digit, round up. If more than halfway down to next digit, round down. Examples: (A) 4.9271 (round to 3 sf)  4.93 (B) 39.0324 (round to 4 sf)  39.03 (C) 5.43918 x 10-2 (round to 4 sf)  5.439 x 10-2

About rounding If exactly halfway, round to the nearest even digit. This avoids systematic round-off error. Examples: (A) 4.25 x 10-2 (round to 2 sf)  4.2 x 10-2 (B) 17.87500 (round to 4 sf)  17.88

Rules for using sig. figs. in calculations Multiplication and division The number of sig. figs. in the answer should be equal to the number of sig. figs. found in the individual number which contains the fewest number of sig. figs., regardless of whether or not the numbers are expressed in scientific notation or to what power they are raised. Examples: (A) (0.9987 g) (1.0032 mL g-1) = 1.0018958 mL  1.002 mL (B) (1.721) (1.8 x 10-4) = 3.09780 x 10-4  3.1 x 10-4 (C) 1.2215 x 10-3 / 0.831 = 1.4699158 x 10-3  1.47 x 10-3

Sig. figs. when using logs and antilogs Remember if n = 10a, log n = a; a is the logarithm (base 10) of n Example: 2 is the logarithm of 100 because 102 = 100 n is the antilogarithm of a Example: 100 is the antilog of 2

Sig. figs. when using logs and antilogs Also remember that the logarithm of any number consists of the: character –digits to the left of the decimal place and the mantissa—digits to the right of the decimal place. Example: log 339 = 2.530  the digit 2 is the character; the digits 530 form the mantissa

Sig. figs. when taking logarithms To have the correct number of sig. figs., the computed log of a number should have in its mantissa the same number of sig. figs. as appears in the number you are taking the logarithm of. Example: log 339 = 2.530  this has the correct # of sig. figs. because there are 3 sig. figs. in 339, so we keep 3 digits in the mantissa of the logarithm.

Sig. figs. when taking logarithms Practical example: pH Remember pH = - log [H+]. Given [H+] = 4.29 x 10-5 M, what is the pH (expressed with the correct # of sig. figs.)? - log(4.29 x 10-5) = 4.3675427 (not rounded yet) Since there are 3 sig. figs. in 4.29 x 10-5, the pH must be expressed with 3 sig. figs. in the mantissa of the log of that number. Correct answer: pH = 4.368

Sig. figs. when taking antilogarithms The antilog of a number should have the same number of sig. figs. as appears in the mantissa of the number you are taking the antilog of. Practical example: Given a pH of 8.72, what is [H+]? antilog (-8.72) = 10-8.72 = 1.9054607 x 10-9 (not rounded) Since there are 2 sig. figs. in the mantissa of -8.72, we must keep 2 sig. figs. in our answer. Correct answer: [H+] = 1.9 x 10-9 M