Question

Divide the following measurements and round off the answer: (a) \(66.3 \mathrm{~g} / 7.5 \mathrm{~mL}\) (b) \(12.5 \mathrm{~g} / 4.1 \mathrm{~mL}\) (c) \(42.620 \mathrm{~g} / 10.0 \mathrm{~mL}\) (d) \(91.235 \mathrm{~g} / 10.00 \mathrm{~mL}\)

Short answer

(a) 8.8 g/mL, (b) 3.0 g/mL, (c) 4.26 g/mL, (d) 9.124 g/mL.

Step-by-step solution

Understand the Problem

For each division problem, we will divide the mass by the volume to find the density, giving a result in grams per milliliter \(\mathrm{g/mL}\). We also need to round the result to the correct number of significant figures.

Divide and Determine Significant Figures for (a)

Calculate \(66.3 \mathrm{~g} / 7.5 \mathrm{~mL}\). First, divide 66.3 by 7.5 to get approximately 8.84 \[\mathrm{g/mL}\]. The number with the fewest significant figures is 7.5 with two digits, so round the result to two significant figures, giving 8.8 \[\mathrm{g/mL}\].

Divide and Determine Significant Figures for (b)

Calculate \(12.5 \mathrm{~g} / 4.1 \mathrm{~mL}\). First, divide 12.5 by 4.1 to get approximately 3.04878 \[\mathrm{g/mL}\]. The number with the fewest significant figures is 4.1 with two digits, so round the result to two significant figures, giving 3.0 \[\mathrm{g/mL}\].

Divide and Determine Significant Figures for (c)

Calculate \(42.620 \mathrm{~g} / 10.0 \mathrm{~mL}\). First, divide 42.620 by 10.0 to get approximately 4.262 \[\mathrm{g/mL}\]. The number with the fewest significant figures is 10.0 with three digits, so round the result to three significant figures, giving 4.26 \[\mathrm{g/mL}\].

Divide and Determine Significant Figures for (d)

Calculate \(91.235 \mathrm{~g} / 10.00 \mathrm{~mL}\). First, divide 91.235 by 10.00 to get approximately 9.1235 \[\mathrm{g/mL}\]. The number with the fewest significant figures is 10.00 with four digits, so round the result to four significant figures, giving 9.124 \[\mathrm{g/mL}\].

Key concepts

Density Calculations

When we talk about density, we're looking at how much mass is packed into a certain volume.
In simpler words, it's a way of saying how "heavy" or "light" something is for a given space it occupies. To calculate density, you divide the mass by the volume. In formula terms, it's given by: \[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \] This operation gives the density in units like grams per milliliter (\(\text{g/mL}\)) or kilograms per liter.
For example, if you have 66.3 grams of a substance and it takes up 7.5 mL of space, its density would be: \[ \frac{66.3 \text{g}}{7.5 \text{mL}} = 8.84 \text{g/mL} \] In real-world scenarios, density helps us to identify materials. If you know the density of a substance, you can often figure out what kind of material it is just by seeing how it compares to known densities.

Division of Measurements

Dividing measurements like mass and volume is essential in determining densities.
When doing these calculations, it’s crucial to pay attention to the significant figures. Significant figures are the digits that carry meaningful information about the precision of a measurement.Why is this important? Let's explore:- When you divide or multiply measurements, the resulting answer should have no more significant figures than the measurement with the least number of significant figures.
- If you divide 12.5 grams by 4.1 mL, you're working with numbers that have three and two significant figures, respectively. The least in this division is two, so your answer should also have two significant figures: 3.0 \(\text{g/mL}\). This ensures that your final result is as reliable as the least precise measurement you started with. So, pay close attention to the numbers you are working with and round appropriately after division.

Rounding Numbers

Rounding is a critical skill in scientific calculations, especially when significant figures are involved.
It's all about making your numbers clearer and more readable while still showing their precision limits. Here's a quick guide on how to round numbers when dealing with significant figures:

• If the digit right after your rounding place is less than 5, just drop it. For instance, rounding 4.262 to three significant figures gives 4.26.
• If it's 5 or more, increase the last retained digit by one. Consider 9.1235 rounded to four significant figures, yielding 9.124.

Each calculation will dictate how many figures you keep based on the number with the fewest significant figures in your measurements.
Learning to round correctly not only ensures precise calculations but is also a valuable skill across various scientific disciplines.