Problem 109 - Chemistry Matter and Change

Question

When multiplying 602.4 \(\mathrm{m}\) by \(3.72 \mathrm{m},\) which value determines the number of significant figures in the answer? Explain.

Step-by-Step Solution

Answer

The value \(3.72\mathrm{m}\) determines the number of significant figures in the answer, as it has the least number of significant figures (3). When multiplying \(602.4\mathrm{m}\) and \(3.72\mathrm{m}\), the result is \(2240.928\mathrm{m^2}\), which is rounded to \(2240\mathrm{m^2}\) to maintain the correct number of significant figures.

Step 1: Identify the number of significant figures in each value

Examine both numbers and count the number of significant figures in each value. For the value \(602.4\mathrm{m}\), there are 4 significant figures. For the value \(3.72\mathrm{m}\), there are 3 significant figures.

Step 2: Identify the value with the least number of significant figures

Compare the number of significant figures for both values and determine which one has the least number. In this case, the value \(3.72\mathrm{m}\) has the least number of significant figures (3).

Step 3: Perform the multiplication

Multiply the two values, \(602.4\mathrm{m}\) and \(3.72\mathrm{m}\). \(602.4\mathrm{m} \times 3.72 \mathrm{m} = 2240.928 \mathrm{m^2}\)

Step 4: Round the answer to the correct number of significant figures

Round the result from Step 3 to the same number of significant figures as the value with the least number of significant figures (3). The correct number of significant figures in this case is determined by the value \(3.72\mathrm{m}\) which has 3 significant figures. Therefore, rounding the result from Step 3 to 3 significant figures gives: \(2240 \mathrm{m^2}\). In conclusion, the value determining the number of significant figures in the answer is \(3.72\mathrm{m}\), and the final result, rounded to the correct number of significant figures, is \(2240\mathrm{m^2}\).

Key Concepts

Multiplication in Chemistry

In chemistry, calculations often involve multiplying quantities, such as when determining area or volume from linear measurements. Multiplication of measurements requires careful attention to units and significant figures. When multiplying two or more measured values, the number of significant figures in the product is determined by the measurement with the fewest significant figures. This ensures that the resulting product does not imply greater precision than is justified by the least precise measurement.

• Review each number involved to count how many significant figures it has.
• Identify the number with the fewest significant figures, as this will dictate the precision of your final answer.
• Without this step, you risk indicating a misleading level of accuracy in scientific data.

Understanding these principles helps ensure that your calculations accurately reflect the reliability of the measurements you are working with.

Rounding Significant Figures

Rounding significant figures in calculations safeguards the proper presentation of data precision. In our example, when multiplying 602.4 m by 3.72 m, the intermediate unrounded product is 2240.928 m². However, since 3.72 m has only three significant figures, our final answer should also reflect this precision. Thus, we round 2240.928 m² to a concise and precise 2240 m².
This rounding process involves identifying the significant digits and then adjusting based on the digit that immediately follows the last significant figure.

• If this digit is 5 or greater, the last significant figure is increased by one.
• If it is less than 5, you leave the last significant figure as it is.

This method ensures consistency in reporting results and maintains the integrity of the scientific data being presented. It helps in making your results consistent with standard scientific reporting practices.

Precision in Measurements

In scientific measurements, precision refers to how closely repeated measurements agree with each other. It can also reflect the level of detail in the measurements taken. Precision is a key element in significant figures because it dictates how data is collected and interpreted.
Measurements like 602.4 m and 3.72 m provide different levels of precision. The measurement 602.4 m, with four significant figures, suggests a higher level of detail than 3.72 m, which has three significant figures. When communicating results, the challenge lies in balancing precision with the natural variability and limits of your measuring tools.

• Use consistent measuring instruments to capture detailed, repeatable data.
• Understand that increased significant figures offer more detailed data, leading to enhanced precision.

Precision does not always mean accuracy, but it helps ensure the work remains reliable when comparing the results of different measurements or experiments.