Problem 78 - Introductory Chemistry Atoms First

Question

To the correct number of significant figures, what is the product of each mathematical operation? Use scientific notation when necessary. (No units shown means a number is exact.) (a) \(2.30 \mathrm{~cm} \times 2\) (b) \(2 \mathrm{~m} \times 2.000 \mathrm{~m}\) (c) \(1001 \mathrm{~J} \times 10\) (d) \(124 \mathrm{~mm} \div 0.1 \mathrm{~mm}\)

Step-by-Step Solution

Answer

(a) 4.60 cm (b) 4 m (c) 10010 J (d) 1240

Step 1: Identify the operation and significant figures

The operation is multiplication: 2.30 cm × 2. Since 2.30 has three significant figures, our final answer should have three significant figures as well.

Step 2: Perform the operation

Multiply 2.30 by 2: \(2.30 \times 2 = 4.6\)

Step 3: Write the answer with the correct number of significant figures

The final answer is 4.6 cm (with three significant figures). Part (b)

Step 1: Identify the operation and significant figures

The operation is multiplication: 2 m × 2.000 m. Since the 2 m has one significant figure, our final answer should have only one significant figure.

Step 2: Perform the operation

Multiply 2 by 2.000: \(2 \times 2.000 = 4\)

Step 3: Write the answer with the correct number of significant figures

The final answer is 4 m (with one significant figure). Part (c)

Step 1: Identify the operation and significant figures

The operation is multiplication: 1001 J × 10. Since 1001 has four significant figures, our final answer should have four significant figures as well.

Step 2: Perform the operation

Multiply 1001 by 10: \(1001 \times 10 = 10010\)

Step 3: Write the answer with the correct number of significant figures

The final answer is 10010 J (with four significant figures). Part (d)

Step 1: Identify the operation and significant figures

The operation is division: 124 mm ÷ 0.1 mm. Since 124 has three significant figures, our final answer should have three significant figures as well.

Step 2: Perform the operation

Divide 124 by 0.1: \(124 \div 0.1 = 1240\)

Step 3: Write the answer with the correct number of significant figures

The final answer is 1240 (with three significant figures). Since no units are given for the second number (0.1), we will not include any units in the final answer.

Key Concepts

Scientific Notation

Scientific notation is a method of expressing numbers that are too large or too small to be conveniently written in decimal form. It is often used in science and engineering to simplify calculations and to signify the precision of measurements. In scientific notation, a number is written as the product of a number between 1 and 10 and a power of 10. For example, the number 650,000 is written in scientific notation as \(6.5 \times 10^5\).

This form is particularly useful when dealing with significant figures, as it makes it easy to identify which digits are significant. For instance, \(6.50 \times 10^5\) signifies that the number has three significant figures. Understanding scientific notation allows students to confidently handle data in various scientific calculations by clearly conveying the precision of the numbers involved.

Multiplication and Division

When performing multiplication and division in scientific calculations, it is crucial to consider significant figures to maintain the accuracy of your results. The rule of thumb is that the number of significant figures in the product or quotient should match the number in the least precise factor or divisor used in your calculation.

For instance, when you multiply \(2.30\) (which has three significant figures) by an exact number like 2, the product, \(4.60\), reflects the precision of the less precise term, \(2.30\). Similarly, dividing two numbers will result in a value with a number of significant figures that corresponds to the least precise number. This awareness ensures that multiplications and divisions in scientific settings do not artificially inflate or deflate the implied accuracy of the result.

Accuracy in Chemistry Calculations

Accuracy in chemistry calculations depends heavily on correctly accounting for significant figures. Every digit in a chemist's measurement is assumed to be precise, except for the last digit, which is an estimate. For example, in the volume measurement \(12.5\) mL, the digits \(1\) and \(2\) are certain, whereas \(5\) is estimated.

In chemistry, when mixing reagents or scaling a reaction, even a tiny inaccuracy can affect the outcome. Thus, chemists must be meticulous in recording significant figures during measurements to communicate the reliability of their data. When reporting the results of a calculation, using the correct number of significant figures imparts the degree of certainty (or uncertainty) that accompanied the initial measurements, preserving the integrity of the scientific processes.