Question

Evaluate each of the following and write the answer to the appropriate number of significant figures. a. \(212.2+26.7+402.09\) b. \(1.0028+0.221+0.10337\) c. \(52.331+26.01-0.9981\) d. \(2.01 \times 10^{2}+3.014 \times 10^{3}\)

Short answer

The short answers for each part of the question are: a. \(641.0\) b. \(1.327\) c. \(77.34\) d. \(3.215 \times 10^3\)

Step-by-step solution

a. Calculate the sum

Add the numbers together: \(212.2 + 26.7 + 402.09 = 640.99\) When adding or subtracting, the answer must have the same number of decimal places as the measurement with the least number of decimal places. In this case, the number with the least number of decimal places is \(26.7\), so our result should have one decimal place.

a. Apply significant figures rule and write the answer

Our result should have one decimal place, so the final answer for a is \(641.0\).

b. Calculate the sum

Add the numbers together: \(1.0028 + 0.221 + 0.10337 = 1.32717\) When adding, the answer must have the same number of decimal places as the measurement with the least number of decimal places. In this case, the number with the least number of decimal places is \(0.221\), so our result should have three decimal places.

b. Apply significant figures rule and write the answer

Our result should have three decimal places, so the final answer for b is \(1.327\).

c. Calculate the sum and difference

Perform the operations: \(52.331 + 26.01 - 0.9981 = 77.3429\) When adding or subtracting, the answer must have the same number of decimal places as the measurement with the least number of decimal places. In this case, the number with the least number of decimal places is \(26.01\), so our result should have two decimal places.

c. Apply significant figures rule and write the answer

Our result should have two decimal places, so the final answer for c is \(77.34\).

d. Calculate the sum

Add the numbers given in scientific notation: \((2.01 \times 10^{2}) + (3.014 \times 10^{3})\) Before adding them, we need to express the numbers with the same exponent: \((2.01 \times 10^{2}) = (0.201 \times 10^{3})\) Now we can add them: \((0.201 \times 10^3) + (3.014 \times 10^3) = (0.201 + 3.014) \times 10^3 = 3.215 \times 10^3\)

d. Write the answer with appropriate significant figures

Our result has the same significant figures as the original numbers, so the final answer for d is \(3.215 \times 10^3\).

Key concepts

Calculating Significant Figures

Understanding significant figures in chemistry is vital since they reflect the precision of a measurement or calculation. Calculating significant figures involves identifying all the digits in a number that contribute to its accuracy. Here's what you need to know:

The first non-zero digit in a number is the first significant figure, and all digits to the right up to and including the last non-zero digit are also significant. This includes any zeroes between non-zero digits. However, leading zeroes (zeroes before the first non-zero digit) are not counted as significant. For example, in the number 0.00452, the significant figures are the digits 4, 5, and 2.

When combining measurements with different degrees of precision, the precision of the resultant number must match the least precise measurement. This ensures that the calculated result does not suggest a greater degree of precision than the measurements allow.

Scientific Notation Operations

Chemists often use scientific notation to handle very large or very small numbers. When performing operations in scientific notation, the rules of significant figures still apply. Adding or subtracting numbers requires the exponents to be the same. Align the decimal places and then, preserving the significant figures, perform the addition or subtraction.

For multiplication and division, manage the exponents separately and then multiply or divide the significant digits, rounding the result to the correct number of significant figures. Remember, the result can only be as precise as the least precise number involved in the operation.

Significant Figures Rules

There are specific rules for determining the number of significant figures in the results of arithmetic operations. When adding or subtracting, the result should have the same number of decimal places as the least precise measurement. With multiplication or division, the answer should have the same number of significant figures as the measurement with the fewest significant figures. It’s crucial to consider these rules during calculations to communicate the precision of your results accurately.

Arithmetic with Significant Figures

When doing arithmetic with significant figures, it's essential to apply the rules systematically to each calculation. If you're adding or subtracting, first align the decimal points, then round the final result to the least number of decimal places from the numbers involved. During multiplication or division, perform the operation as usual, then round the answer to the least number of significant figures found in the original numbers. Understanding and applying these principles helps maintain the integrity of data precision during arithmetic operations in chemistry.