Problem 100 - Chemistry Principles and Practice

Question

A Consider two liquids: liquid \(A\), with a density of \(0.98 \mathrm{~g} / \mathrm{mL}\), and liquid \(\mathrm{B}\), with a density of \(1.03 \mathrm{~g} / \mathrm{mL}\). Notice that one density is known to have two significant figures and the other to have three. Calculate the volume of liquid \(A\) in a sample that weighs \(9.9132 \mathrm{~g}\); be sure to express your result to the proper number of significant digits. Calculate the volume of the same mass of liquid \(\mathrm{B}\), again making sure that you have the appropriate number of significant figures. Recording the number of significant figures is only one way to estimate the uncertainty. Repeat the calculations of volume by using the minimum and maximum values of density to calculate maximum and minimum volumes. The range between the two is also a measure of uncertainty. Compare the estimated uncertainties in the two liquids as measured by the two techniques. Do all estimates give the same answer? Should they? Explain any disagreements.

Step-by-Step Solution

Answer

The volume of liquid A is 10 mL, and for liquid B it is 9.62 mL, both accounting for significant figures. The non-significant figures method shows lower uncertainty for volumes and reflects realistic approximations compared to the more conservative significant figures estimate.

Step 1: Calculate the Volume of Liquid A

To calculate the volume of liquid A in a sample that weighs 9.9132 g, use the formula for density \( \text{Density} = \frac{\text{Mass}}{\text{Volume}} \), rearranged as \( \text{Volume} = \frac{\text{Mass}}{\text{Density}} \).Given:- Mass = 9.9132 g- Density of A = 0.98 g/mLSubstitute the given values into the formula:\[ \text{Volume of A} = \frac{9.9132 \text{ g}}{0.98 \text{ g/mL}} = 10.12 \text{ mL} \]Since the density of liquid A is given with two significant figures, the volume should be reported with two significant figures, thus the volume is 10 mL.

Step 2: Calculate the Volume of Liquid B

Using the same method as in Step 1 for liquid B, but with different values:Given:- Mass = 9.9132 g- Density of B = 1.03 g/mLCalculate the volume:\[ \text{Volume of B} = \frac{9.9132 \text{ g}}{1.03 \text{ g/mL}} \approx 9.624 \text{ mL} \]Since the density of liquid B is given with three significant figures, the volume should be reported with three significant figures, so the volume is 9.62 mL.

Step 3: Estimate Uncertainty Using Significant Figures

The uncertainty in volume due to significant figures arises because the calculation can only be as precise as the least precise measurement used. - For liquid A, the uncertainty is indicated by the two significant figures, implying a range of 10 mL ± 0.5 mL. - For liquid B, it's 9.62 mL with uncertainty indicated by three significant figures, implying a more precise measurement than liquid A.

Step 4: Estimate Maximum and Minimum Volumes for Liquid A

For liquid A, calculate the volumes using a minimum density of 0.975 g/mL (0.98 - 0.005) and a maximum density of 0.985 g/mL (0.98 + 0.005):- Maximum volume for minimum density: \[ \text{Volume}_{\text{max}} = \frac{9.9132 \text{ g}}{0.975 \text{ g/mL}} = 10.17 \text{ mL} \]- Minimum volume for maximum density: \[ \text{Volume}_{\text{min}} = \frac{9.9132 \text{ g}}{0.985 \text{ g/mL}} = 10.06 \text{ mL} \]The range of values is 10.17 mL - 10.06 mL = 0.11 mL.

Step 5: Estimate Maximum and Minimum Volumes for Liquid B

For liquid B, calculate similar ranges using a minimum density of 1.025 g/mL and a maximum density of 1.035 g/mL:- Maximum volume for minimum density: \[ \text{Volume}_{\text{max}} = \frac{9.9132 \text{ g}}{1.025 \text{ g/mL}} \approx 9.669 \text{ mL} \]- Minimum volume for maximum density: \[ \text{Volume}_{\text{min}} = \frac{9.9132 \text{ g}}{1.035 \text{ g/mL}} \approx 9.578 \text{ mL} \]The range of values is 9.67 mL - 9.58 mL = 0.09 mL.

Step 6: Compare Uncertainty Estimates

Compare the two estimates of uncertainty: the significant figures method and the minimum/maximum range method. - For liquid A: - Significant figures imply uncertainty of 0.5 mL. - Range method suggests uncertainty of 0.11 mL. - For liquid B: - Significant figures suggest minimal implied uncertainty. - Range method suggests 0.09 mL. The range method provides a smaller uncertainty for liquid A than the significant figures and matches expectation for liquid B. This discrepancy arises as significant figures often overstate uncertainty when density values have approximations.

Key Concepts

Density Measurement

When talking about density, we refer to a fascinating physical property of a substance that describes how much mass is contained in a given volume. Mathematically, density is expressed as \[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \] In our exercise, we have two liquids, A and B, with densities of 0.98 g/mL and 1.03 g/mL, respectively. These densities help us understand the mass-to-volume relationship in each liquid, a crucial piece of information for comparing and calculating volumes.
**Significant Figures in Density:** - Liquid A: Two significant figures (0.98).
- Liquid B: Three significant figures (1.03).
These significant figures reveal the precision of our density measurements.Understanding and correctly applying such figures ensures that our calculations, like those involving volume or mass, are accurate to the level intended by the measurement.

Volume Calculation

Calculating volume is largely about rearranging the formula for density to find the unknown: \[ \text{Volume} = \frac{\text{Mass}}{\text{Density}} \]
In this problem, we find the volume of each liquid given a mass and the known density:
**For Liquid A:** - Mass: 9.9132 g
- Density: 0.98 g/mL
- Volume = 10.12 mL, adjusted to 10 mL considering two significant figures.
**For Liquid B:** - Mass: 9.9132 g
- Density: 1.03 g/mL
- Volume = 9.624 mL or 9.62 mL when rounded to three significant figures.
Remember, the accuracy of your answer depends on the precision of your measurements, highlighted by the significant figures in the density. This simple rearrangement allows us to easily calculate how much space a given mass of liquid occupies when its density is known.

Uncertainty Estimation

Estimating uncertainty involves understanding the limitations inherent in our measurements. When using density with significant figures to find volume, the number of figures gives an approximate uncertainty range.
1. **Using Significant Figures:** - For liquid A, two significant figures suggest an uncertainty of approximately ±0.5 mL.
- For liquid B, three significant figures imply a smaller inherent variability in measurement.
2. **Using Range Method:** - Liquid A's density can vary between 0.975 and 0.985 g/mL, translating to volume uncertainties from 10.06 to 10.17 mL.
- For liquid B, densities range from 1.025 to 1.035 g/mL, leading to volume measures between 9.578 and 9.669 mL.
The range method often offers a more nuanced understanding of uncertainty than significant figures alone, allowing for more precise consideration of measurement variability.

Liquid Properties

Liquids are unique in how their properties, like density, influence their behavior and usage in practical applications. Understanding a liquid's density helps us anticipate how it will react under different conditions, its buoyancy, and how it's used in design and engineering.
Here are a few key properties of liquids: - **Consistency and Flow:** Depending on density, liquids like our two examples here, can vary in how easily they flow or pour.
- **Buoyancy:** Denser liquids may allow for greater buoyancy of objects compared to less dense liquids.
- **Volume Manipulation:** Modifying volume and density interdependently can give insights into chemical structure or concentration.
These essential liquid properties play a crucial role in a multitude of scientific and industrial applications, from beverage production to chemical manufacturing.