Chemistry Significant Digits Answer Key

Mastering Significant Digits: A Comprehensive Guide with Worked Examples

Understanding significant figures (also known as significant digits) is crucial for anyone working with scientific measurements and calculations. Accurate reporting of measurements relies heavily on this concept, as it reflects the precision and uncertainty inherent in any experimental data. This comprehensive guide will delve into the rules governing significant figures, provide a wealth of examples, and clarify common misconceptions. By the end, you'll be confident in determining and using significant digits correctly in your calculations.

Introduction: Why Significant Digits Matter

Significant digits are the digits in a number that carry meaning contributing to its measurement precision. They indicate the reliability of a measurement. For instance, a measurement of 2.5 cm suggests a higher degree of precision than a measurement of 2 cm because the former specifies a value to the nearest tenth of a centimeter. Ignoring significant digits can lead to misleading conclusions and inaccurate results, especially when performing calculations involving multiple measurements. This is particularly critical in fields like chemistry, physics, and engineering where precise measurements are essential.

Rules for Determining Significant Digits

Several rules determine whether a digit is significant or not. Let's examine each rule with examples:

1. Non-zero digits are always significant: The digits 1, 2, 3, 4, 5, 6, 7, 8, and 9 are always significant regardless of their position in the number.

   Example: The number 234 has three significant figures. The number 1.2345 has five significant figures.

2. Zeros between non-zero digits are always significant: Zeros nestled between other significant digits contribute to the precision of the measurement.

   Example: The number 1002 has four significant figures. The number 20.05 has four significant figures.

3. Leading zeros are never significant: Leading zeros are those placed before the first non-zero digit in a number less than 1. They only serve to locate the decimal point.

   Example: The number 0.0025 has only two significant figures (2 and 5). The number 0.0102 has three significant figures (1, 0, and 2).

4. Trailing zeros in a number without a decimal point are ambiguous: Trailing zeros are those located at the end of the number. Without a decimal point, it is unclear whether they are significant or merely placeholders. Scientific notation should be used to clarify such numbers.

   Example: The number 100 could have one, two, or three significant figures depending on the precision of the measurement. To clarify, 1.00 x 10² indicates three significant figures, while 1 x 10² indicates only one.

5. Trailing zeros in a number with a decimal point are significant: Trailing zeros after a decimal point are always significant because they indicate the precision of the measurement.

   Example: The number 2.00 has three significant figures. The number 1.000 has four significant figures.

6. Exact numbers have infinite significant figures: Numbers obtained through counting (e.g., 12 apples) or defined values (e.g., 100 cm = 1 m) are considered exact and have infinite significant figures. They do not limit the significant figures in calculations.

Significant Figures in Calculations

When performing calculations with measurements, the result must reflect the precision of the least precise measurement involved. This is done using the following rules:

1. Addition and Subtraction: The result should have the same number of decimal places as the measurement with the fewest decimal places.

   Example: 25.2 cm + 1.23 cm + 0.005 cm = 26.435 cm which rounds to 26.4 cm (one decimal place).

2. Multiplication and Division: The result should have the same number of significant figures as the measurement with the fewest significant figures.

   Example: 25.2 cm x 1.2 cm = 30.24 cm², which rounds to 30 cm² (two significant figures).

Worked Examples: A Step-by-Step Approach

Let's work through several examples to solidify your understanding:

Example 1: Determine the number of significant figures in the following numbers:

• 345: Three significant figures
• 0.00345: Three significant figures
• 3040: Ambiguous (could be three or four). Scientific notation clarifies (3.04 x 10³ = three, 3.040 x 10³ = four)
• 3040.0: Five significant figures
• 1.00 x 10⁴: Three significant figures

Example 2: Perform the following calculations and express the answer with the correct number of significant figures:

• 12.5 cm + 3.2 cm + 0.12 cm = 15.82 cm (rounds to 15.8 cm)

• 25.2 g / 2.1 mL = 12 g/mL (two significant figures)

• 10.5 cm x 5.2 cm = 54.6 cm² (two significant figures)

Example 3: A student measures the mass of an object three times, obtaining the following results: 2.15 g, 2.16 g, 2.14 g. Calculate the average mass and express the answer with the correct number of significant figures.

• Sum of measurements: 6.45 g
• Average mass: 6.45 g / 3 = 2.15 g. Since '3' is an exact number, the number of significant figures is determined by the measurements, which have three significant figures. Therefore, the answer is 2.15 g.

Example 4: A rectangular block measures 5.2 cm in length, 3.1 cm in width, and 2.0 cm in height. Calculate the volume and express your answer with the correct number of significant figures.

• Volume = Length x Width x Height = 5.2 cm x 3.1 cm x 2.0 cm = 32.24 cm³
• The measurement with the fewest significant figures is 2.0 cm (two significant figures). Therefore, the answer rounds to 32 cm³.

Advanced Concepts and Exceptions

While the above rules cover the majority of cases, some situations warrant additional explanation:

• Rounding: When rounding numbers, consider the digit immediately following the last significant digit. If this digit is 5 or greater, round up. If it is less than 5, round down. If the digit is exactly 5, common practice is to round to the nearest even number.

• Scientific Notation: Using scientific notation (e.g., 1.23 x 10⁴) is crucial for unambiguously representing significant figures, particularly for very large or very small numbers. The number of significant figures is indicated by the digits in the coefficient (the number before the x 10 part).

• Logarithms and Antilogarithms: Specific rules apply when dealing with logarithms and antilogarithms, involving the number of significant figures in the mantissa (the fractional part of the logarithm).

Frequently Asked Questions (FAQ)

• Q: What is the difference between accuracy and precision in the context of significant figures?

  • A: Accuracy refers to how close a measurement is to the true value. Precision refers to how close repeated measurements are to each other. Significant figures reflect the precision of a measurement, not necessarily its accuracy.

• Q: Why are significant figures important in scientific reporting?

  • A: Significant figures convey the reliability and uncertainty associated with a measurement. Reporting an excessive number of digits implies a higher precision than is actually achieved, while reporting too few obscures the level of precision obtained.

• Q: Can I use a calculator to determine significant figures?

  • A: Calculators usually display many digits, often exceeding the appropriate number of significant figures. It is crucial to understand the rules of significant figures and apply them manually to obtain the correct answer. The calculator is a tool for performing the calculation, not for determining significant figures.

• Q: What happens if I make a mistake with significant figures in my calculations?

  • A: Incorrectly handling significant figures can lead to inaccurate conclusions and may affect the validity of your experimental results. It's essential to meticulously follow the rules to ensure accurate and reliable scientific reporting.

Conclusion: Mastering the Art of Significant Figures

Mastering significant figures is a crucial skill for anyone engaging with quantitative data in scientific fields. While the rules might seem initially complex, consistent practice and careful attention to detail will help you develop a strong understanding and apply them correctly. Remember, accurate reporting of measurements is paramount in scientific work, and using significant figures appropriately is a key component of achieving that accuracy. By diligently following the rules outlined above and practicing with various examples, you'll build the confidence and expertise necessary to handle significant figures confidently in your scientific endeavors. Understanding significant figures is not just about following rules; it’s about understanding the inherent uncertainty in measurements and communicating that uncertainty accurately and responsibly.