Question

Perform the following arithmetic and round off the answers to the correct number of significant figures. Include the correct units with the answers. (a) \(3.58 \mathrm{~g} / 1.739 \mathrm{~mL}\) (b) \(4.02 \mathrm{~mL}+0.001 \mathrm{~mL}\) (c) \((22.4 \mathrm{~g}-8.3 \mathrm{~g}) /(1.142 \mathrm{~mL}-0.002 \mathrm{~mL})\) (d) \((1.345 \mathrm{~g}+0.022 \mathrm{~g}) /(13.36 \mathrm{~mL}-8.4115 \mathrm{~mL})\) (e) \((74.335 \mathrm{~m}-74.332 \mathrm{~m}) /(4.75 \mathrm{~s} \times 1.114 \mathrm{~s})\)

Short answer

The rounded answers are (a) 2.06 g/mL, (b) 4.021 mL, (c) 12.4 g/mL, (d) 0.14 g/mL, (e) 5.67e-4 m/s^2.

Step-by-step solution

- Solve for Density with Significant Figures

Calculate the density by dividing mass by volume: Density = Mass / Volume = 3.58 g / 1.739 mL.The result to the least number of decimal places in the given numbers is 2.058 g/mL when not rounded.Round to the appropriate number of significant figures: 3.58 has three significant figures, and 1.739 has four, so the final answer should be to three significant figures, yielding 2.06 g/mL.

- Add Volumes with Significant Figures

Sum the volumes: 4.02 mL + 0.001 mL = 4.021 mL.Since the value with the least number of decimal places used has three decimal places, round the result to three decimal places: 4.021 mL.

- Calculate Difference in Mass and Volume, then Divide

Subtract first the masses, then the volumes and divide: (22.4 g - 8.3 g) / (1.142 mL - 0.002 mL) = 14.1 g / 1.14 mL.The final result rounded to three significant figures (least from the data provided) is 12.4 g/mL.

- Add Masses, Subtract Volumes, then Divide

Compute the sum of the masses and the difference of the volumes: (1.345 g + 0.022 g) / (13.36 mL - 8.4115 mL).The result before rounding is 0.13645 g/mL.Considering the least number of decimal places (1.345 g has four and 13.36 mL has two), round to two decimal places if we're dealing with addition of masses and subtraction of volumes. Final answer: 0.14 g/mL.

Key concepts

Density Calculation

Understanding density is fundamental in chemistry, as it relates the mass of a substance to its volume. To calculate density, you divide the mass by the volume (\( \text{Density} = \frac{\text{Mass}}{\text{Volume}} \)). It's essential to consider the significant figures in both the mass and volume to ensure the calculation's accuracy. For instance, if we have a mass of 3.58 g and a volume of 1.739 mL, the density calculation yields 2.058 g/mL when not rounded. However, we need to match the least number of significant figures in the given measurements, which is three in this case, leading to a final density of 2.06 g/mL.

When facing issues with understanding density, remember, it's not just about the formula; think of it as how much matter is packed within a certain space. More matter in the same space means higher density.

Volume Addition

When you add volumes in chemistry, maintaining the accuracy of significant figures is key. Let's take 4.02 mL plus 0.001 mL as an example. The result is 4.021 mL, but it's essential to consider the precision of our initial measurements. The lowest number of decimal places amongst the values to be added guides the final rounding off. In this case, since 4.02 mL has three decimal places, the sum should be rounded to the same level of precision, resulting in 4.021 mL.

Always remember that when adding or subtracting, it's the decimal places that count for significant figures and not just the number of digits!

Mass-Volume Ratio

In chemistry, the mass-volume ratio gives us the density of a substance, but it can also be used to compare changes in mass and volume separately before performing a final density calculation. Consider a two-step problem: First, find the difference in mass and volume, then divide. For instance, when solving \( (22.4 \text{g} - 8.3 \text{g}) / (1.142 \text{mL} - 0.002 \text{mL}) \), the subtraction yields 14.1 g and 1.14 mL. Now, divide these to get the initial density without rounding which, in this case, should be rounded to three significant figures, yielding a final result of 12.4 g/mL.

Keep in mind that the mass-volume ratio is a pivotal concept to understand as it acts as the foundation for understanding various chemical properties.

Rounding Off Significant Figures

Correctly rounding off significant figures ensures precision and accuracy in chemical calculations. It's a systematic way to express a number to the level of precision that the measurement allows. For example, when given a result of 0.13645 g/mL, and the least precise number you are working with in the calculation has two decimal places (as with 13.36 mL in subtraction), then you would round the final result to two decimal places, resulting in 0.14 g/mL.

It's crucial to learn proper rounding rules because it reflects the precision of your tools and the certainty of your measurements. Not rounding or rounding incorrectly can significantly affect the reliability of your results in chemistry.

Units Conversion

Converting units is an everyday task in the chemistry lab. To convert units, you'll need a conversion factor—a ratio that represents how many of one unit equals another. For instance, converting from milliliters to liters requires the conversion factor of 1000 mL per liter. Chemical equations and formulas must have matching units across all terms to ensure the validity of the calculations. Understanding and correctly applying unit conversions can prevent errors and are vital when working with formulas that have different units for different components.

Mastery of unit conversions will make interpreting and solving chemical problems much simpler and is indispensable for a successful chemistry student.