Problem 71 - Chemistry The Science in Context

Question

Perform each of the following calculations, and express the answer with the correct number of significant figures (only the highlighted values are exact): a. \(0.6274 \times 1.00 \times 10^{3} /\left[2.205 \times(2.54)^{3}\right]=\) b. \(6 \times 10^{-18} \times\left(1.00 \times 10^{3}\right) \times 12=\) c. \((4.00 \times 58.69) /\left(6.02 \times 10^{23} \times 6.84\right)=\) d. \([(26.0 \times 60.0) / 43.53] /\left(1.000 \times 10^{4}\right)=\)

Step-by-Step Solution

Answer

a. \(0.6274 \times 1.00 \times 10^{3} /\left[2.205 \times(2.54)^{3}\right]\) b. \(6 \times 10^{-18} \times\left(1.00 \times 10^{3}\right) \times 12\) c. \((4.00 \times 58.69) /\left(6.02 \times 10^{23} \times 6.84\right)\) d. \([(26.0 \times 60.0) / 43.53] /\left(1.000 \times 10^{4}\right)\) Answer: a. \(17.8\) b. \(7 \times 10^{-15}\) c. \(5.70 \times 10^{-22}\) d. \(3.58 \times 10^{-3}\)

Step 1: Analyze significant figures

Determine the number of significant figures in each value: 4 in \(0.6274\), 3 in \(1.00\), 4 in \(2.205\), and 3 in \(2.54\).

Step 2: Perform the calculations

Follow the order of operations and perform the calculations: \(0.6274 \times 1.00 \times 10^{3} = 627.4\) and \((2.54)^{3} = 16.387064\). Now divide: \(627.4/[2.205 \times 16.387064] = 17.809963\).

Step 3: Round the answer to the correct number of significant figures

The original values have 4, 3, 4, and 3 significant figures. For multiplication and division, we use the smallest number of significant figures, which is 3. Therefore, we round our answer to 3 significant figures: \(17.809963 \approx 17.8\). a. \(= 17.8\) b. \(6 \times 10^{-18} \times\left(1.00 \times 10^{3}\right) \times 12=\)

Step 1: Analyze significant figures

Determine the number of significant figures in each value: 1 in \(6\), 3 in \(1.00\), and 2 in \(12\).

Step 2: Perform the calculations

Multiply the values: \(6 \times 10^{-18} \times(1.00 \times 10^{3}) \times 12 = 7.2 \times 10^{-15}\).

Step 3: Round the answer to the correct number of significant figures

The original values have 1, 3, and 2 significant figures. Use the smallest number of significant figures, which is 1. Therefore, we round our answer to 1 significant figure: \(7.2 \times 10^{-15} \approx 7 \times 10^{-15}\). b. \(= 7 \times 10^{-15}\) c. \((4.00 \times 58.69) /\left(6.02 \times 10^{23} \times 6.84\right)=\)

Step 1: Analyze significant figures

Determine the number of significant figures in each value: 3 in \(4.00\), 4 in \(58.69\), 3 in \(6.02\), and 3 in \(6.84\).

Step 2: Perform the calculations

Perform the multiplication and division: \((4.00 \times 58.69) = 234.76\) and \((6.02 \times 10^{23} \times 6.84) = 4.117368 \times 10^{23}\). Now divide: \(234.76 / (4.117368 \times 10^{23}) = 5.702406 \times 10^{-22}\).

Step 3: Round the answer to the correct number of significant figures

The original values have 3, 4, 3, and 3 significant figures. Use the smallest number of significant figures, which is 3. Therefore, we round our answer to 3 significant figures: \(5.702406 \times 10^{-22} \approx 5.70 \times 10^{-22}\). c. \(= 5.70 \times 10^{-22}\) d. \([(26.0 \times 60.0) / 43.53] /\left(1.000 \times 10^{4}\right)=\)

Step 1: Analyze significant figures

Determine the number of significant figures in each value: 3 in \(26.0\), 3 in \(60.0\), 4 in \(43.53\), and 4 in \(1.000\).

Step 2: Perform the calculations

Perform the multiplication and division: \((26.0 \times 60.0) = 1560\) and \(1560/43.53 = 35.839495\). Now divide: \(35.839495 / (1.000 \times 10^{4}) = 3.5839495 \times 10^{-3}\).

Step 3: Round the answer to the correct number of significant figures

The original values have 3, 3, 4, and 4 significant figures. Use the smallest number of significant figures, which is 3. Therefore, we round our answer to 3 significant figures: \(3.5839495 \times 10^{-3} \approx 3.58 \times 10^{-3}\). d. \(= 3.58 \times 10^{-3}\)

Key Concepts

Scientific Notation

Scientific notation is a way of expressing numbers that are too large or too small to be conveniently written in decimal form. It is often used by scientists, engineers, and mathematicians to handle such numbers easily. For example, the speed of light in a vacuum is approximately 299,792,458 meters per second, which can be written in scientific notation as \(2.99792458 \times 10^8\) m/s.

In scientific notation, a number is written as a product of two numbers: a coefficient and \(10\) raised to a power, known as the exponent. The coefficient must be a number greater than or equal to 1 and less than 10, and the exponent indicates how many places the decimal point has moved. This exponent is positive for numbers greater than one and negative for numbers less than one.

For instance, the number 0.00045 written in scientific notation becomes \(4.5 \times 10^{-4}\). Notice how the decimal point has been moved four places to the right, which is reflected in the negative exponent. This notation is not only compact but also simplifies mathematical calculations, as working with exponents becomes easier when multiplying or dividing numbers in scientific notation.

Significant Figures Calculation

Significant figures, or 'sig figs,' are essential in chemistry and other sciences for accurately communicating the precision of measurements and calculations. Each digit in a significant figure represents a certain amount of precision in the measurement. To calculate significant figures, start by identifying all non-zero digits as significant. Zeros in between non-zero digits are also significant. Leading zeros are not significant since they merely indicate the position of the decimal point. However, trailing zeros in a decimal number are significant as they indicate measured or estimated precision.

For example, in the number \(0.6274\), all of the digits are significant, giving it four significant figures. When performing calculations, the significant figures in the final result should reflect the least precise measurement used in the calculation. This approach ensures that the reported result is not overly precise, considering the initial data.

Rounding to Significant Figures

Rounding to significant figures is a crucial step after performing a calculation in order to ensure that the answer reflects the precision of the inputs. When rounding, the number of significant figures you round to depends on the operation performed. For multiplication and division, the result should have the same number of significant figures as the measurement with the least number of significant figures. In addition, when adding or subtracting, it is the decimal place with the least precision that dictates the number of decimal places in the result.

To properly round a number, look at the digit immediately to the right of the last significant figure you want to keep. If this digit is five or higher, you round up the last significant figure. If it's lower than five, you leave it as is. For example, to round \(17.809963\) to three significant figures, you would consider the fourth digit, '0.' Since '0' is less than five, the rounded result is simply \(17.8\).

Understanding how to round correctly is paramount in providing reliable and credible results in scientific and mathematical work.