How Many In a Value Significant Figures In a Value Worksheet

Significant Figures Worksheet How many sig figs in a value? Measured and calculated values are always to be reported with the correct number of sig figs. All nonzero (e.g. 1-9) digits are automatically considered a sig fig. Trickier, meanwhile, are zeroes: Leading zeroes never count. Leading zeroes come before any nonzero number. For example: 1.5, 0.15, and 0.000015 all have two sig figs. Interior zeroes always count. Interior means they are between nonzero numbers. For example: 105, 10.5, 1.05, and 0.000105 all have three sig figs. Trailing zeroes only count if a decimal point is shown anywhere in the value. Trailing means that it comes at the very end of the value; it does not have to come after a decimal point. For example: 1050. , 1.050 , and 0.0001050 all have four sig figs. BUT, 1050 (with no decimal point shown!) only has three sig figs. Special case: exact numbers have an infinite number of sig figs. These are usually counting numbers. For example, a class might have 30 students. Since there is no uncertainty in the number, it is considered exact. So, you could just as well say there are 30.0000000…. students; this is why it has an infinite number of sig figs. How many atoms are in a molecule of methane (CH4)? There are exactly 5 atoms; this number also has infinite sig figs. Finally, when written in scientific notation, every digit shown before the “ x 10n “ is automatically a sig fig. So, it is easier to see how many sig figs are in a value when written is scientific notation. Note that converting a number from decimal to scientific notation (or vice versa) should never change the number of sig figs in the value. Practice: State how many sig figs in the value. 1. 37.5 8. 10000. 2. 1600 9. 8.20 x 10-3 3. 4080. 10. 4.05 x 107 4. 0.0037 11. The number of hydrogen atoms in 2 molecules of methane (CH4) 5. 0.0109 6. 4.280 7. 0.000001 12. The number of hydrogen atoms in 2 moles of methane Multiplication and Division The product of two (or more) values needs to be reported with the same number of sig figs as the input value with the fewest sig figs. Think of it as the weak link in the chain. The same rule is also used for division, or even a combination of multiplication and division! Examples: 1.07 x 0.039 = 0.04173 before rounding. The first value has 3 sig figs, the second has only 2. So: = 0.042 after rounding to 2 sig figs. 0.02080 x 200.0 = 4.16 before rounding. Both the first and second values have 4 sig figs, so: = 4.160 after rounding to 4 sig figs. Note that we had to tack on a trailing zero! (2.73 x 10-2) x (4.866 x 10-3) = 0.0001328418 before rounding. Only want 3 sig figs, so: = 0.000133 or, equivalently, 1.33 x 10-4 Practice: 1. 454 x 4.968 = 7. 48.0 x 27.0 = 2. 4.10 x 0.66 x 7600 = 8. (1.06 x 102) x (6.379 x 101) = 3. 295 x 63.578 x 600 = 9. Combustion of 1 molecule of CH4 will produce 2 molecules of H2O. How many molecules of water will be produced when 8 molecules of CH4 are combusted? 4. 140 ÷ 5.937 = 5. 101 ÷ 5.3 = 6. 50 ÷ 9.629 x 300. = 10. Combustion of 1 molecule of CH4 will produce 2 molecules of H2O. How many moles of water will be produced when 8.00 moles of CH4 are combusted? Addition and Subtraction The sum of two (or more) values is limited by the input whose least (i.e. last) significant figure (LSF) is in the highest decimal place. The answer must then be rounded to have its LSF occur in that same decimal place. At no point does it matter how many sig figs there are; the answer will often have more or fewer sig figs. All that matters is at which decimal place the least sig fig is located. The same is true for subtraction, or even a combination of both! Examples: 0.568 – 0.0439 = 0.5241 before rounding. The first input has its LSF in the “thousandths” position, while the second input has its LSF in the “ten-thousandth” position. So, the first input is less precise; it is the weak link! So: = 0.524 after rounding. 5.61 – 0.01 = 5.6 before rounding. Both inputs have their LSF in the “hundredths” position, so: = 5.60 after rounding. Note that we had to tack on a zero! (2.73 x 10-3) + (4.866 x 10-1) = 0.48933 before rounding. Let’s rewrite in decimal notation: (0.00273) + (0.4866) = 0.4893 after rounding. Here, it turns out that the weak link is the second input, even though it had more sig figs! Practice: 1. 61.1299 + 94.68 = 7. (1.48 – 1.36) x 100% = 1.48 2. 739.2 – 4.17 = 8. (0.507 – 0.43) x 100% = 0.507 3. 2.9982 + 1.2 + 37.9872 = 4. 150 + 12 = 9. How many atoms total are in 1 molecule of C3 H 8 ? 5. 48.0 – 0.01 = 6. 7500 – 500 = 10. How many moles of atoms total are in 4.0 moles of C3H8? Mass and Volume Measurements Sig figs using digital instruments (like a balance) are easy: write down every digit displayed! If you need a rough estimate, use the top-loading balance. If you need a precise measurement, use the analytical balances (the ones with the doors). Don’t round off any values, even trailing zeroes. Volume measurements, meanwhile, are often taken with glassware that is graduated, i.e. it has lines/markings etched into it. For these, you can often estimate beyond the marking, and potentially get another decimal place in your measurement. Always look for the meniscus to take your reading. Graduated Cylinder Buret Standard Deviations and Averages When you collect three or more measurements of the same quantity, the values of those measurements will likely be different from each other. If they are fairly similar, this indicates better precision, while very different values indicates poor precision. Consider the following measurements of mass: Trial 1 2 3 Mass (g) 7.3698 7.3246 7.3809 Average: Add the three masses together and divide by three. The general formula is: " 𝑥"# = % !#$ 𝑥! 𝑁 For the above data, the average is 7.3584. Standard Deviation: Take the difference of each mass from the average. Then, square each of those differences. Then, add up those squared-differences. Divide the sum by one-less than the number of trials. Finally, take the square-root of the result. The general formula is: " 𝑠𝑡𝑑. 𝑑𝑒𝑣. = -% ( 𝑥! − 𝑥 ")% 𝑁−1 !#$ The standard deviation is always rounded to one sig fig. For the above data, the standard deviation before rounding is 0.02982… But it should be rounded to just 0.03. Note: since the standard deviation represents a “plus-or-minus” to the average result, the average should be rounded to the same decimal place as the standard deviation. So, the above average (7.3584) is converted to the final result: Mass is 7.36 +/- 0.03 g. Try it with the following two tables of data. For each, report the average with the standard deviation. Trial 1 2 3 Volume (mL) 5.29 5.07 5.62 Trial 1 2 3 Mass (g) 4.37 x 10-4 3.68 x 10-4 5.05 x 10-4