Problem 27 - Chemistry: Principles and Reactions

Question

Calculate the following to the correct number of significant figures. Assume that all these numbers are measurements. (a) \(x=17.2+65.18-2.4\) (b) \(x=\frac{13.0217}{17.10}\) (c) \(x=(0.0061020)(2.0092)(1200.00)\) (d) \(x=0.0034+\frac{\sqrt{(0.0034)^{2}+4(1.000)\left(6.3 \times 10^{-4}\right)}}{(2)(1.000)}\)

Step-by-Step Solution

Answer

Question: Determine the results of the following operations and make sure the final answer contains the correct number of significant figures: a) 17.2 + 65.18 - 2.4 b) 13.0217 ÷ 17.10 c) 0.0061020 × 2.0092 × 1200.00 d) (0.0034 ± √[(0.0034)² + 4 × 1.000 × (6.3 × 10⁻⁴)]) / 2(1.000) Answer: a) 80.0 b) 0.7623 c) 14.682 d) 0.0031

Step 1: Perform the addition and subtraction

Initially, perform the mathematical operations directly: 17.2 + 65.18 - 2.4 = 79.98

Step 2: Correct for significant figures in addition and subtraction

The number 17.2 has the least decimal places (1 decimal place), so the final answer also needs to have 1 decimal place. Therefore, the result is 80.0. ##Part (b): Division##

Step 1: Perform the division

Initially, perform the division directly: 13.0217 ÷ 17.10 = 0.762307603686636

Step 2: Correct for significant figures in division

The number 17.10 has the least significant figures (4 significant figures), so the final answer also needs to have 4 significant figures. Therefore, the result is 0.7623. ##Part (c): Multiplication##

Step 1: Perform the multiplication

Initially, perform the multiplication directly: 0.0061020 × 2.0092 × 1200.00 = 14.68158784

Step 2: Correct for significant figures in multiplication

The number 2.0092 has the least significant figures (5 significant figures), so the final answer also needs to have 5 significant figures. Therefore, the result is 14.682. ##Part (d): A Complex Operation Involving Both Addition, Division, and Square Root##

Step 1: Compute the square root term

First compute the square root part of the equation: sqrt[ (0.0034)² + 4 × 1.000 × (6.3 × 10⁻⁴) ] = 0.002818913

Step 2: Perform the addition, division

Then add 0.0034 and divide by 2(1.000) => [0.0034 + 0.002818913] ÷ 2 = 0.003109457

Step 3: Correct for significant figures

Here, the tricky part is defining the least significant figure. In this case, we follow the term with the least accurate figure, namely 0.0034, which has 2 significant figures. Therefore, the result should have 2 significant figures, and it comes out to 0.0031.

Key Concepts

Addition and Subtraction with Significant Figures

When dealing with addition and subtraction, it's important to focus on decimal places rather than total significant figures. The rule is that your result should have as few decimal places as the number in your calculations with the least decimal precision. For example, if you add 17.2 + 65.18 - 2.4, you first perform the operations to get 79.98. But here, 17.2 has only one decimal place, so the answer must also feature only one decimal place, rounding 79.98 to 80.0.

Simple steps to follow include:

• Identify the number with the least number of decimal places.
• Perform the addition/subtraction as usual.
• Round off the result to match the least number of decimal places.

This ensures your calculation stays true to the precision of the least precise measurement involved.

Division with Significant Figures

In division, the rule for determining significant figures in your result is different from that in addition and subtraction. Here, the key is the total number of significant figures in each number. For example, dividing 13.0217 by 17.10 results in 0.762307603686636. Even though the number appears daunting, your answer needs to match the number with the fewest significant figures among those being divided.

In this case, 17.10 has four significant figures. Thus, round your answer to four significant figures: 0.7623. To apply this method:

• Perform the division using a calculator for precise results.
• Count the significant figures in each of the numbers involved.
• Match the final result's significant figures to the number with the fewest significant figures.

This way ensures the division reflects the accuracy of your least accurate number.

Multiplication and Significant Figures

Multiplication follows similar rules to division when it comes to significant figures. The critical point lies in identifying which number in your calculation has the fewest significant figures, and ensuring your final result doesn't exceed that. For example, when multiplying 0.0061020 × 2.0092 × 1200.00, you calculate 14.68158784 as the direct result.

Since 2.0092 represents the number with five significant figures, so should your result: 14.682. Here are the steps to handle multiplication with significant figures:

• Execute the multiplication as you normally would.
• Count significant figures in each operand.
• Round the calculated result to match the least number of significant figures.

By doing this, you maintain an accurate reflection of the numbers' precision involved in the multiplication.

Square Root Operations and Significant Figures

Combining operations like addition, division, and square roots necessitates special attention to significant figures. In complex equations, such as calculating \(x=0.0034+\frac{\sqrt{(0.0034)^{2}+4(1.000)(6.3 \times 10^{-4})}}{(2)(1.000)}\), each operation needs to properly consider significant figure rules.

First, compute the square root, considering precision from each contributing term. When you add to this result and divide, your final result should reflect the least precise term in the entire expression. For example, the initial calculation yields 0.003109457, but the limiting factor is 0.0034, which only includes two significant figures. Therefore, the final rounded result should also reflect those two figures and become 0.0031.

• Calculate square root operations within the expression.
• Proceed through addition or division following above-discussed rules.
• Ensure the result reflects the least precise measurement throughout the computation.

Through this approach, even complex operations can retain appropriate precision based on given measurements.