Читать книгу U Can: Chemistry I For Dummies - Hren Chris - Страница 11

Doing chemistry means making a lot of measurements. The point of spending a pile of money on cutting-edge instruments is to make really good, really precise measurements. After you’ve got yourself some measurements, you roll up your sleeves, hike up your pants, and do some math.

Remember: When doing math in chemistry, you need to follow some rules to make sure that your sums, differences, products, and quotients honestly reflect the amount of precision present in the original measurements. You can be honest (and avoid the skeptical jeers of surly chemists) by taking things one calculation at a time, following a few simple rules. One rule applies to addition and subtraction, and another rule applies to multiplication and division.

Addition and subtraction

In addition and subtraction, round the sum or difference to the same number of decimal places as the measurement with the fewest decimal places. For example, suppose you’re adding the following amounts:

Your calculator will show 19.3645, but you round off to the hundredths place based on the 3.25, which has the fewest number of decimal places. You round the figure off to 19.36. (See the later section “Rounding off numbers” for the rounding rules.)

Multiplication and division

In multiplication and division, you report the answer to the same number of significant figures as the number that has the fewest significant figures. Remember that counted and exact numbers don’t count in the consideration of significant numbers. For example, suppose that you are calculating the density in grams per liter of an object that weighs 25.3573 (six sig figs) grams and has a volume of 10.50 milliliters (four sig figs). The setup looks like this:

Your calculator will read 2,414.981000. You have six significant figures in the first number and four in the second number (the 1,000 mL/L doesn’t count because it’s an exact conversion). You should have four significant figures in your final answer, so round the answer off to 2,415 g/L.

Notice the difference between the two rules. When you add or subtract, you assign significant figures in the answer based on the number of decimal places in each original measurement. When you multiply or divide, you assign significant figures in the answer based on the smallest number of significant figures from your original set of measurements.

Tip: Caught up in the breathless drama of arithmetic, you may sometimes perform multi-step calculations that include addition, subtraction, multiplication, and division, all in one go. No problem. Follow the normal order of operations, doing multiplication and division first, followed by addition and subtraction. At each step, follow the simple significant-figure rules, and then move on to the next step.

Rounding off numbers

Sometimes you have to round numbers at the end of a measurement to account for significant figures. Here are a couple of very simple rules to follow and remember:

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Rule 1: If the first number to be dropped is 5 or greater, drop it and all the numbers that follow it, and increase the last retained number by 1.

For example, suppose that you want to round off 237.768 to four significant figures. You drop the 6 and the 8. The 6, the first dropped number, is greater than 5, so you increase the retained 7 to 8. Your final answer is 237.8.

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Rule 2: If the first number to be dropped is less than 5, drop it and all the numbers that follow it, and leave the last retained number unchanged.

If you’re rounding 2.35427 to three significant figures, you drop the 4, the 2, and the 7. The first number to be dropped is 4, which is less than 5. The 5, the last retained number, stays the same. So you report your answer as 2.35.

Examples

Q. Express the following sum with the proper number of significant figures:

A. 671.1 miles. Adding the three values yields a raw sum of 671.05 miles. However, the 35.7 miles measurement extends only to the tenths place. Therefore, you round the answer to the tenths place, from 671.05 to 671.1 miles.

Q. Express the following product with the proper number of significant figures:

A. . Of the two measurements, one has two significant figures (27 feet) and the other has four significant figures (13.45 feet). The answer is therefore limited to two significant figures. You need to round the raw product, 363.15 feet². You could write 360 feet², but doing so may imply that the final 0 is significant and not just a placeholder. For clarity, express the product in scientific notation, as feet².

Practice Questions

1. Express the answer to this calculation using the appropriate number of significant figures:

2. Express the answer to this calculation using the appropriate number of significant figures:

3. Report the difference using the appropriate number of significant figures:

4. Express the answer to this multi-step calculation using the appropriate number of significant figures:

Practice Answers

1. 114.36 seconds. The trick here is remembering to convert all measurements to the same power of 10 before comparing decimal places for significant figures. Doing so reveals that seconds goes to the hundredths of a second, despite the fact that the measurement contains only two significant figures. The raw calculation yields 114.359 seconds, which rounds properly to the hundredths place (taking significant figures into account) as 114.36 seconds, or seconds in scientific notation.

2. inches. Here, you have to recall that defined quantities (1 foot is defined as 12 inches) have unlimited significant figures. So your calculation is limited only by the number of significant figures in the measurement 345.6 feet. When you multiply 345.6 feet by 12 inches per foot, the feet cancel, leaving units of inches:

The raw calculation yields 4,147.2 inches, which rounds properly to four significant figures as 4,147 inches, or inches in scientific notation.

3. –0.009 minutes. Here, it helps here to convert all measurements to the same power of 10 so you can more easily compare decimal places in order to assign the proper number of significant figures. Doing so reveals that minutes goes to the hundred-thousandths of a minute, and 0.009 minutes goes to the thousandths of a minute. The raw calculation yields –0.00863 minutes, which rounds properly to the thousandths place (taking significant figures into account) as –0.009 minutes, or minutes in scientific notation.

4. 2.80 feet. Following standard order of operations, you can do this problem in two main steps.

Following the rules of significant-figure math, the first step yields . Each product or quotient contains the same number of significant figures as the number in the calculation with the fewest number of significant figures.

After completing the first step, divide by 10.0 feet to finish the problem:

You write the answer with three sig figs because the measurement 10.0 feet contains three sig figs, which is the smallest available between the two numbers.