Problem 49 - Chemistry The Central Science In Si Units

Question

Carry out the following operations and express the answers with the appropriate number of significant numbers. (a) \(43.029+0.02348\) (b) \(952.72-73.4201\) (c) \(\left(2.93 \times 10^{3}\right)(0.732)\) (d) \(0.06324 / 0.624\)

Step-by-Step Solution

Answer

(a) 43.052; (b) 879.30; (c) 2.14 × 10³; (d) 0.101.

Step 1: Addition with Significant Figures

First, add the numbers in (a):\[ 43.029 + 0.02348 = 43.05248 \]Next, determine the significant figures. The number with the least decimal places is 43.029, which has three decimal places. Therefore, round the result to three decimal places: \[ 43.05248 \approx 43.052 \] Thus, the answer is 43.052.

Step 2: Subtraction with Significant Figures

Next, perform the subtraction in (b):\[ 952.72 - 73.4201 = 879.2999 \]Identify which number limits the significant figures: 952.72 has two decimal places. Therefore, round the result to two decimal places:\[ 879.2999 \approx 879.30 \]Thus, the answer is 879.30.

Step 3: Multiplication with Significant Figures

Perform the multiplication in (c):\[ (2.93 \times 10^{3}) \times 0.732 = 2144.76 \]Count the number of significant figures in the factors: 2.93 has three significant figures, and 0.732 also has three significant figures. Therefore, the product should have three significant figures:\[ 2144.76 \approx 2.14 \times 10^{3} \]Thus, the answer is \(2.14 \times 10^{3}\).

Step 4: Division with Significant Figures

Finally, perform the division in (d):\[ 0.06324 / 0.624 = 0.1013461538 \]Check the number of significant figures: 0.06324 has four significant figures, while 0.624 has three significant figures. Therefore, the result should have three significant figures:\[ 0.1013461538 \approx 0.101 \]Thus, the answer is 0.101.

Key Concepts

Addition and Subtraction

When performing addition or subtraction, it's important to focus on decimal places. The rule for significant figures in these operations is to align your answer with the number having the least decimal places. This ensures that the uncertainty in the measurement is reflected accurately.
Let's break this down:

• Addition Example: With 43.029 and 0.02348, when added, the sum is 43.05248. Since 43.029 has three decimal places, our result should be rounded to three decimal places, giving 43.052.
• Subtraction Example: Consider 952.72 minus 73.4201. The result is 879.2999. Here, 952.72 has the fewest decimal places at two, so we round to two decimal places, producing 879.30.

Remember, this rule emphasizes measuring precision and ensures your answer isn't falsely more precise than your inputs.

Multiplication and Division

For multiplication and division, the rule for significant figures changes. Instead of looking at decimal places, count the total number of significant figures in each number involved in the calculation. The final result should match the number with the fewest significant figures used in the operation.
Here's how it works:

• Multiplication Example: Multiplying 2.93 by 0.732 yields 2144.76. Both numbers have three significant figures, so our answer is also expressed with three significant figures: in this case, 2.14 × 10^3.
• Division Example: Dividing 0.06324 by 0.624 gives 0.1013461538. Since 0.624 has three significant figures, the quotient is rounded to three significant figures, resulting in 0.101.

This method ensures that the result's precision accurately reflects the measurements used, staying consistent with scientific accuracy and reliability.

Rounding Rules

Rounding is crucial when dealing with significant figures. It helps to maintain the integrity of the data by reflecting its precision. The basic rounding rule is: if a digit is less than 5, round down, and if it's 5 or above, round up.
Consider these examples:

• For the number 43.05248 rounded to three decimal places, look at the fourth decimal digit (4). It's less than 5, so 43.052 stays.
• In 879.2999 rounded to two decimals, the third digit (9) is more than 5, so it rounds up, giving 879.30.
• With 0.1013461538 rounded to three digits, the fourth digit also forces rounding up, resulting in 0.101.

Rounding brings simplicity to our results and aligns numerical data presentation with the precision of original measurements.