Problem 67 - General Chemistry: Principles and Modern Applications

Question

According to the rules on significant figures, the product of the measured quantities \(99.9 \mathrm{m}\) and \(1.008 \mathrm{m}\) should be expressed to three significant figures-\(101 \mathrm{m}^{2} .\) Yet, in this case, it would be more appropriate to express the result to four significant figures-\(100.7 \mathrm{m}^{2} .\) Explain why.

Step-by-Step Solution

Answer

In this case, it would be more appropriate to express the result to four significant figures because the rules of significant figures challenge the precision of our initial measurements when three significant figures are used. So, to represent the accuracy of the used values, it's better to keep an extra significant figure in the final answer.

Step 1: Understand the rules of significant figures in calculations

When multiplying or dividing, the result should have the same number of significant figures as the number in the calculation with the fewest significant figures. In our case, \(99.9 \mathrm{m}\) has three significant figures and \(1.008 \mathrm{m}\) has four. So, according to this rule, the result should be expressed with three significant figures.

Step 2: Calculate the Product

When we multiply these two numbers, we get \(99.9 \mathrm{m} * 1.008 \mathrm{m} = 100.792 \mathrm{m}^{2}\). The number 100.792 has six significant figures.

Step 3: Apply the rule of significant figures

According to the rules of significant figures in multiplication, our answer should be rounded to three significant figures, which gives us \(101 \mathrm{m}^{2}\).

Step 4: Analyze the discrepancy

In this case, however, providing the answer with three significant figures is misleading because it suggests a degree of inaccuracy in the initial measurements that was not there. The decimal places in the numbers 99.9m and 1.008m imply that these measurements were quite precise, and so a three-significant-figure answer implies a level of uncertainty that isn’t justified by the data.

Step 5: Conclude

Therefore, in situations where the significant figures rules might suggest reporting a measurement that falsely implies a high level of uncertainty, it might be more appropriate to keep an extra significant figure in the final answer.

Key Concepts

Multiplication and Division Rules

When performing multiplication or division with measured values, significant figures play a crucial role. The rule is quite straightforward: the resulting product or quotient should have the same number of significant figures as the measurement with the fewest significant figures. For instance, if we multiply 99.9 m (which has 3 significant figures) by 1.008 m (with 4 significant figures), our result must be limited to the precision of the least precise measurement, meaning it should have 3 significant figures. This rule helps maintain consistency in the precision across mathematical calculations involving measurements. It prevents us from assigning a false sense of accuracy to results derived from rougher measurements.

Precision in Measurement

Precision refers to how closely multiple measurements of the same thing are to each other. In scientific measurements, precision impacts how results should be reported. Ensuring that precision is upheld during calculations means acknowledging the inherent uncertainties in each measurement. For instance, the measurement 99.9 m implies a certain precision, highlighting that any value beyond the hundredth decimal is uncertain. This precision must be considered when performing calculations to prevent inaccuracies, which risk skewing the data analysis and interpretation. In our calculation of 99.9 m multiplied by 1.008 m, although the calculated result was 100.792 m², expressing the result as 101 m² would incorrectly suggest a limited precision not consistent with the original measurements.

Rounding Numbers

Rounding is the process of simplifying numbers while retaining their essence to fit the precision required by calculated measurements. During this process, rules help ensure that valuable information isn't lost due to unwanted rounding errors. Rounding numbers involves checking the digit immediately after your desired significant figure. If this digit is 5 or higher, round up; if it is lower than 5, round down. In the case of 100.792 m², rounding to three significant figures results in 101 m². However, judiciously deciding how much to round, especially when dealing with scientific data, helps convey the proper level of accuracy without misleading others about the data's precision.

Accuracy in Scientific Calculations

Accuracy in scientific calculations refers to how close a calculated or measured value is to its actual value. It reflects the correctness of a measurement as opposed to precision, which indicates consistency. Ensuring accuracy involves both proper measurement techniques and appropriate data processing through tools such as significant figure rules. However, rigidly applying these rules might sometimes mislead, implying an inaccurate degree of certainty in measurements. In this exercise's context, reporting the result as 100.7 m² suggests a more accurate portrayal of the initial data's precision. Selecting an appropriate level of accuracy conveys true measurement quality, aiding better scientific communication and interpretation.