Problem 16 - Chemistry for Allied Health

Question

Find the result of each of the following calculations and report the value with the correct number of significant figures. a. \(34 \times 0.12=\) b. \(68.2 / 0.78=\) c. \(3.29 \times 10^{4} \times 16.2=\) d. \(0.8449 \times 29.7=\) e. \(5.92 \times 10^{3} / 0.628=\) f. \(3.00 \times 2.6=\) g. \(2.50 \times 9.331=\) h. \(3.20 / 12.75=\)

Step-by-Step Solution

Answer

a. 4.1, b. 87, c. 533,000, d. 25.1, e. 9,430, f. 7.8, g. 23.3, h. 0.251

Step 1: Understand Significant Figures

Significant figures are the digits in a number that are known with certainty plus the first uncertain digit. When multiplying or dividing, the result should have as many significant figures as the number in the operation with the fewest significant figures.

Step 2: Calculate each result

Perform the multiplication or division for each part of the question to obtain raw results.

Step 3: Identify Significant Figures in Each Problem

For each calculation: - (a) has 2 significant figures in both numbers. - (b) has 3 significant figures and 2 significant figures. - (c) has 3 significant figures and 3 significant figures. - (d) has 4 significant figures and 3 significant figures. - (e) has 3 significant figures and 3 significant figures. - (f) has 3 significant figures and 2 significant figures. - (g) has 3 significant figures and 4 significant figures. - (h) has 3 significant figures and 4 significant figures.

Step 4: Round Results to the Appropriate Number of Significant Figures

For each calculation: - (a) Result is 4.08, round to 4.1 (2 significant figures). - (b) Result is 87.435897, round to 87 (2 significant figures). - (c) Result is 532,980, round to 533,000 (3 significant figures). - (d) Result is 25.08873, round to 25.1 (3 significant figures). - (e) Result is 9,427.0707, round to 9,430 (3 significant figures). - (f) Result is 7.8, round to 7.8 (2 significant figures). - (g) Result is 23.3275, round to 23.3 (3 significant figures). - (h) Result is 0.25098039, round to 0.251 (3 significant figures).

Key Concepts

Multiplication and Division

When performing multiplication and division, a key aspect is understanding how significant figures play a crucial role. Each number used in an operation carries significant figures, which are all the known digits plus one estimated digit.

• For multiplication and division, the result of the calculation should have the same number of significant figures as the measurement with the fewest significant figures.
• For example, if you are multiplying 34 (which has two significant figures) by 0.12 (which also has two significant figures), the answer should be reported with only two significant figures, so you would say 4.1.

This ensures that your answer reflects the precision of the least precise number involved in the calculation.

Rounding Numbers

Rounding numbers correctly is essential to maintaining precision while using significant figures. Once you've calculated a raw result, you need to decide how to round your number based on the significant figures rule.
Here's a simple method to follow:

• Identify how many significant figures the final result should have, based on the initial calculation's precision.
• If the digit immediately following your last significant figure is greater than or equal to five, round up. For example, rounding 4.08 to two significant figures will give you 4.1.
• If the next digit is less than five, do not round up. For instance, 87.435897 rounded to two significant figures becomes 87.

Remember, the accuracy of significant figures is preserved through careful rounding.

Significant Figures in Calculations

Significant figures are crucial for ensuring precision and accuracy in calculations. Understanding and using them correctly can make a big difference in the presentation and reliability of your results.

• To determine significant figures, note all non-zero digits, any zeros between significant digits, and any trailing zeros in the decimal portion.
• Use the number with the lowest significant figures to guide how many significant figures your answer should contain.
• For instance, in the calculation of 0.8449 x 29.7, since 0.8449 has four significant figures and 29.7 has three, the final answer should be rounded to three significant figures which results in 25.1.

Mastering significant figures helps bring clarity and precision to scientific calculations.

Scientific Notation

Scientific notation is a method used to express very large or very small numbers in a uniform way. This is especially helpful when dealing with significant figures because it clearly presents a number's precision.

• In scientific notation, numbers are written as the product of a number between 1 and 10 and a power of 10. For example, 5.92 x 10³ is written with one digit before the decimal point, indicating significant figures clearly.
• It simplifies reading, writing, and ensures precision because all non-zero digits are significant unless otherwise indicated.
• Calculations in scientific notation follow the same rules for significant figures as regular decimals, but help maintain or express large or small numerical results effortlessly.

Using scientific notation correctly safeguards the integrity of scientific work and clarity of information.