Problem 110 - General Chemistry

Question

Assuming all of the numbers are measured quantities, perform the following arithmetic setups and report the answers to the correct number of significant figures. a \(\frac{9.345-9.005}{9.811}\) b \(\frac{9.345+9.005}{9.811}\) c \((7.50+7.53) \times 3.71\) d) \(0.71 \times 0.36+17.36\)

Step-by-Step Solution

Answer

a) 0.0346, b) 1.871, c) 55.7, d) 17.62

Step 1: Solve a)

Start with the subtraction in the numerator: \(9.345 - 9.005 = 0.340\). Both numbers have three decimal places, so the result should be reported with three decimal places, leading to 0.340. Now divide by the denominator: \(\frac{0.340}{9.811} = 0.034645\). The numerator has three significant figures and the denominator has four, so the final answer is rounded to three significant figures: 0.0346.

Step 2: Solve b)

Begin with the addition in the numerator: \(9.345 + 9.005 = 18.350\). Both numbers have three decimal places, so the result should have three decimal places, maintaining 18.350. Next, divide by the denominator: \(\frac{18.350}{9.811} = 1.87094\). The numerator has five significant figures and the denominator four, so round the result to four significant figures: 1.871.

Step 3: Solve c)

First, perform the addition inside the parentheses: \(7.50 + 7.53 = 15.03\). Each term inside the sum has two decimal places, so the result must also have two decimal places. Then multiply by 3.71: \(15.03 \times 3.71 = 55.7313\). The multiplicands have three significant figures, so round the result to three significant figures: 55.7.

Step 4: Solve d)

Calculate the multiplication: \(0.71 \times 0.36 = 0.2556\). The multiplicands have two significant figures, so round to two significant figures: 0.26. Add the result to 17.36: \(0.26 + 17.36 = 17.62\). The sum must be reported with two decimal places since it's the smallest decimal place count from the addends, resulting in 17.62.

Key Concepts

Arithmetic Operations

When performing arithmetic operations, accuracy is key. It’s important to manage significant figures correctly to ensure our calculations reflect the precision of measurements. In mathematical operations:

• Addition and Subtraction: Focus on the decimal places. The result should have the same number of decimal places as the term with the fewest decimal places. For example, in the calculation of 9.345 - 9.005, both numbers have three decimal places. Thus, our answer has three decimal places too.
• Multiplication and Division: Count the significant figures. The number of significant figures in the result is determined by the input number with the fewest significant figures. In our division from Step 1, 0.340 and 9.811, 0.340 with three significant figures dictates the precision here, leading us to round the answer to three significant figures.

Paying close attention to these rules helps produce results that best represent the uncertainty inherent to measurements.

Rounding Rules

Rounding plays a crucial role in expressing answers with appropriate precision. Always adhere to specific rounding rules to maintain accuracy:

• If the digit to be dropped is less than 5, keep the last remaining figure the same. For instance, from a calculated value of 0.340 to 0.034645, the outcome rounds to 0.0346, since the next digit is less than 5.
• If the digit is 5 or greater, round up the last remaining figure. For example, the calculated value 55.7313 becomes 55.7 after rounding because the next digit is 7, larger than 5.
• Always ensure consistency in rounding to stay accurate and reliable in approximating values.

These rules ensure that the precision of your answer properly matches the limitations of the given data.

Significant Figures in Chemistry

In chemistry, correctly determining significant figures is imperative for precision and accuracy as you deal with measurements and calculations. The concept of significant figures allows chemists to communicate the precision of their measures effectively:

• Importance: They indicate how precise a measurement is, which is critical in conveying accurate experimental data.
• Application: When calculating amounts, concentrations, or any chemical property, knowing how to properly apply significant figures ensures the results reflect the true precision of the measurements used.
• Examples in Practice: Processes like determining molar concentrations or reaction yields often rely heavily on the precise use of significant figures, as seen in solving problems like dividing sums as shown in the original exercise.

A keen understanding and application of these figures help ensure effectiveness in chemical experiments and calculations, reducing errors and improving reliability.