Question

Consider the calculation indicated below: \(\frac{2.21 \times 0.072333 \times 0.15}{4.995}\) Explain why the answer to this calculation should be reported to only two significant digits.

Short answer

The answer to the given calculation should be reported to only two significant digits because, in multiplication and division operations, the result must have the same number of significant digits as the number with the least significant digits involved in the operation. In this case, the number with the least significant digits was 0.15, which has two significant digits. Therefore, the final answer is 0.0048.

Step-by-step solution

Identify the number of significant digits in each value

First, we need to determine how many significant digits are in each value given in the calculation. The rule for counting significant digits is as follows: 1. Non-zero digits are always significant. 2. Any zeros between significant digits are significant. 3. Leading zeros are not significant. 4. Trailing zeros in a decimal fraction are significant. Applying these rules, we get: - 2.21 has three significant digits. - 0.072333 has five significant digits. - 0.15 has two significant digits. - 4.995 has four significant digits.

Determine the number of significant digits in the final answer

When dealing with multiplication and division, the result should be reported with the least number of significant digits present in any of the numbers involved in the operation. In this case, the number with the least significant digits is 0.15, which has 2 significant digits.

Perform the calculation and report the answer

Now, we will perform the given calculation and will round the result to the appropriate number of significant digits. Calculate the value: \(\frac{2.21 \times 0.072333 \times 0.15}{4.995} \approx \frac{0.023993485}{4.995} \approx 0.004802297\) Now, round the answer to two significant digits: The final answer is 0.0048.

Conclusion

The answer to the given calculation should be reported to only two significant digits because, in multiplication and division operations, the result must have the same number of significant digits as the number with the least significant digits involved in the operation. In this case, the number with the least significant digits was 0.15, which has two significant digits. Therefore, the final answer is 0.0048.

Key concepts

Rules for Counting Significant Digits

Understanding the rules for counting significant digits is essential in scientific calculations, ensuring precision and accuracy. Non-zero digits are always counted as significant, which means they represent a measure of precision in the reported number. For instance, in the number 123.45, all five digits are significant. Zeros situated between non-zero digits are also significant. Consider the number 1002; it contains four significant digits because the zeros are sandwiched between 1 and 2.

However, not all zeros are significant. Leading zeros, such as those in 0.0025, do not count towards the count of significant digits and are only placeholders, indicating the scale of the number. Contrastingly, trailing zeros in a decimal number are considered significant, as they indicate precision; for example, the number 2.300 has four significant digits. It's these rules that help determine the exactitude of measurements and calculations in scientific endeavors.

Multiplication and Division of Significant Figures

When it comes to multiplication and division of numbers with significant figures, the rule to remember is quite straightforward: the final answer should have the same number of significant figures as the number in the operation with the fewest significant figures.

Let's say we are multiplying 4.56 (three significant figures) with 1.2 (two significant figures). Although the actual product of these numbers is 5.472, we must round this to two significant figures—5.5—because 1.2 has only two significant digits. This rule keeps our results consistent with the precision of the least precise number used in our calculations.

Rounding to Significant Digits

Rounding to significant digits is a crucial step in ensuring that the reported value reflects the precision of the measurement. When rounding, we follow regular rounding rules, but specifically to the significant digit level. If the digit immediately after the last significant one is 5 or greater, we round up. If it's less than 5, we round down.

For example, if we start with 0.004802297 and we need to round to two significant digits, we look at the third significant figure, which is 2 (the third non-zero number). Since the digit after it (the '9') is greater than 5, we round the '2' up to '3'. However, since we're only allowed two significant digits, our number becomes just 0.0048.

Significant Figures in Chemistry Calculations

In chemistry, dealing with significant figures is pivotal when calculating measurements, concentrations, and other quantitative data. This strict adherence to significant digits ensures the accuracy and reliability of experimental results and subsequent interpretations. For example, when calculating molarity, which involves measurements of mass and volume, chemists must be keen on maintaining the correct number of significant figures to report a precise concentration. They use the same rules and principles of significant figures as in the general math to prevent any misrepresentation of a chemical's concentration in a solution and to stay true to the precision of their instruments.