Sig Fig Worksheet Chemistry Answers

Mastering Significant Figures: A Comprehensive Guide with Worked Examples and Practice Problems

Significant figures (sig figs) are a crucial concept in chemistry and other sciences. Understanding and correctly applying rules for significant figures ensures accuracy and precision in calculations and reporting experimental results. This comprehensive guide provides a detailed explanation of significant figures, including rules for determining them, performing calculations with them, and interpreting results. We'll cover everything you need to master sig figs, complete with worked examples and practice problems to solidify your understanding. This worksheet-style approach aims to help you confidently tackle any sig fig problem.

Understanding Significant Figures

Before diving into the rules, let's define what significant figures actually are. Significant figures represent the digits in a number that carry meaning contributing to its precision. They indicate the certainty of a measurement. A number with more significant figures implies a higher degree of precision than a number with fewer significant figures. For example, measuring a length as 10.5 cm is more precise than measuring it as 10 cm. The extra digit in "10.5" signifies greater accuracy in the measurement.

Rules for Determining Significant Figures

Here are the rules for determining the number of significant figures in a given number:

1. Non-zero digits are always significant. The digits 1, 2, 3, 4, 5, 6, 7, 8, and 9 are always significant. For example, in the number 245, all three digits are significant.

2. Zeros between non-zero digits are significant. In the number 205, the zero is significant. Similarly, in 1004, all four digits are significant.

3. Leading zeros (zeros to the left of the first non-zero digit) are not significant. These zeros simply indicate the position of the decimal point. For example, in 0.0025, only the 2 and 5 are significant.

4. Trailing zeros (zeros to the right of the last non-zero digit) are significant only if the number contains a decimal point. In the number 250, the zero is not significant. However, in 250., 250.0, and 250.00, all zeros are significant. The decimal point indicates that the measurement was precise enough to include those zeros.

5. Exact numbers have an infinite number of significant figures. These numbers are often obtained from definitions or counting, not measurements. For example, there are exactly 12 inches in a foot. This is not a measured quantity, so it doesn't limit the precision of calculations.

Significant Figures in Calculations

Applying significant figures correctly during calculations is just as important as determining them. Here's a breakdown of how sig figs affect different arithmetic operations:

Multiplication and Division

In multiplication and division, the result should have the same number of significant figures as the measurement with the fewest significant figures.

Example:

2.5 cm × 3.21 cm = 8.025 cm²

The number 2.5 has two significant figures, while 3.21 has three. Therefore, the result should be rounded to two significant figures: 8.0 cm².

Addition and Subtraction

In addition and subtraction, the result should have the same number of decimal places as the measurement with the fewest decimal places.

Example:

25.32 g + 1.2 g + 0.05 g = 26.57 g

25.32 has two decimal places, 1.2 has one decimal place, and 0.05 has two. The result must be rounded to one decimal place because 1.2 g has only one decimal place: 26.6 g.

Rounding

Rounding is essential when dealing with significant figures. The general rule is:

• If the digit to be dropped is less than 5, round down.
• If the digit to be dropped is 5 or greater, round up.

Example:

Rounding 8.025 to two significant figures gives 8.0.

Worked Examples: Sig Fig Worksheet Answers

Let's work through some examples to solidify your understanding. Consider this as your comprehensive sig fig worksheet with detailed answers.

Example 1:

Determine the number of significant figures in each of the following numbers:

a) 3500 Answer: 2 (Trailing zeros are not significant without a decimal point) b) 0.00450 Answer: 3 (Leading zeros are not significant; the trailing zero is) c) 2005 Answer: 4 (Zeros between non-zero digits are significant) d) 1.0050 Answer: 5 (All digits are significant) e) 6.02 x 10²³ Answer: 3 (Scientific notation; all digits in the coefficient are significant)

Example 2:

Perform the following calculations and express the answers with the correct number of significant figures:

a) 12.5 cm x 3.2 cm = ? Answer: 40 cm² (12.5 has 3 sig figs, 3.2 has 2 sig figs; result rounded to 2 sig figs) b) 25.6 g + 1.02 g = ? Answer: 26.6 g (25.6 has one decimal place, 1.02 has two; result rounded to one decimal place) c) 150.0 mL / 2.5 mL = ? Answer: 60 mL (150.0 has 4 sig figs, 2.5 has 2; result rounded to 2 sig figs) d) 1.87 m - 0.05 m = ? Answer: 1.82 m (Both have two decimal places, so the result retains two decimal places) e) (12.4 g / 3.2 mL) x 5.0 mL = ? Answer: 20 g/mL (12.4 has 3 sig figs, 3.2 has 2 sig figs, 5.0 has 2 sig figs. The result of the division (3.875) is limited to 2 sig figs (3.9) and then multiplied. The final result is rounded to 2 sig figs.)

Example 3:

A student measures the length and width of a rectangular block as 15.2 cm and 7.8 cm, respectively. Calculate the area of the block and report the answer with the correct number of significant figures.

Answer: Area = length × width = 15.2 cm × 7.8 cm = 118.56 cm². Since both measurements have two significant figures, the area should be rounded to two significant figures: 120 cm².

Example 4:

Three measurements of the mass of a substance are recorded as: 2.45 g, 2.48 g, and 2.47 g. Calculate the average mass and express the answer with the correct number of significant figures.

Answer: Average mass = (2.45 g + 2.48 g + 2.47 g) / 3 = 2.46666... g. The least precise measurement has two decimal places, so the average should be rounded to two decimal places: 2.47 g.

Frequently Asked Questions (FAQ)

Q: What happens if I have a calculation with multiple steps involving both multiplication/division and addition/subtraction?

A: Follow the order of operations (PEMDAS/BODMAS). Perform the operations within parentheses first, applying the appropriate significant figure rules. Then, proceed with the remaining operations, again applying the correct rules. Remember to round only at the very end of the entire calculation.

Q: How do I handle significant figures when using scientific notation?

A: All digits in the coefficient of a number expressed in scientific notation are significant. The exponent only affects the magnitude of the number, not its precision.

Q: Are significant figures important in real-world applications?

A: Absolutely! In scientific research, engineering, and other fields, accurately reporting measurements and calculations using significant figures is essential for reliable results and avoids misinterpretations. It reflects the precision of the instruments and methods used.

Q: Why are trailing zeros after a decimal point significant?

A: Trailing zeros after a decimal point show that the measurement was made with that level of precision. For instance, 10.00 g indicates that the measurement was precise to the hundredth of a gram, whereas 10 g only indicates precision to the nearest gram.

Conclusion

Mastering significant figures is a fundamental skill for anyone working with quantitative data, particularly in scientific fields. By understanding and applying the rules for determining and calculating with significant figures, you ensure the accuracy and reliability of your results. Consistent application of these rules is crucial for clear and precise communication of scientific findings. Remember to practice regularly using diverse problems to build your confidence and proficiency. This comprehensive guide and the worked examples serve as a valuable resource for your learning journey. With consistent practice, you'll become adept at handling significant figures confidently in any chemistry or scientific context.