Problem 148 - General Chemistry

Question

A 33.0 -g sample of an unknown liquid at \(20.0^{\circ} \mathrm{C}\) is heated to \(120^{\circ} \mathrm{C}\). During this heating, the density of the liquid changes from \(0.854 \mathrm{~g} / \mathrm{cm}^{3}\) to \(0.797 \mathrm{~g} / \mathrm{cm}^{3}\). What volume would this sample occupy at \(120^{\circ} \mathrm{C} ?\)

Step-by-Step Solution

Answer

The liquid occupies approximately 41.40 cm³ at 120°C.

Step 1: Find Initial Volume

To calculate the initial volume of the liquid at 20°C, we use the formula for density, which is \( \text{density} = \frac{\text{mass}}{\text{volume}} \). Rearranging for volume, we have \( \text{volume} = \frac{\text{mass}}{\text{density}} \). Substitute the given values: mass \( = 33.0 \) g and density \( = 0.854 \text{ g/cm}^3 \). Thus, the initial volume \( V_1 = \frac{33.0 \text{ g}}{0.854 \text{ g/cm}^3} \approx 38.63 \text{ cm}^3 \).

Step 2: Calculate Volume at Final Temperature

The change in density due to the temperature increase affects the volume of the liquid. At the final temperature (120°C), the density is given as \( 0.797 \text{ g/cm}^3 \). Using the formula \( \text{volume} = \frac{\text{mass}}{\text{density}} \), we calculate the final volume: \( V_2 = \frac{33.0 \text{ g}}{0.797 \text{ g/cm}^3} \approx 41.40 \text{ cm}^3 \).

Key Concepts

Thermal Expansion

When a material is heated, its particles move more vigorously and tend to take up more space. This leads to thermal expansion, which refers to the increase in volume of substances as temperature rises. For liquids, this behavior can be somewhat predictable.
The underlying principle here is that, when temperature changes, it influences the kinetic energy of the particles. As a result, the volume increases while mass remains constant. This results in a change in density. This phenomenon explains why the volume of the unknown liquid in the exercise increased when heated from \(20.0^{\circ}C\) to \(120.0^{\circ}C\). Thermal expansion is crucial in real-world scenarios like engine cooling systems and thermometers.

Density Change

Density is defined as mass per unit volume. When thermal expansion occurs, this density changes. As the volume of a substance increases due to heating, the density decreases if the mass remains unchanged.
In the given exercise, the density of the liquid decreased from \(0.854 \text{ g/cm}^3\) to \(0.797 \text{ g/cm}^3\) as the temperature increased. This happens because the same mass (33.0 g) is now spread out over a larger volume due to expansion. Understanding density changes help in various applications like metallurgy and the pharmaceutical industry, where precise density measurements are crucial for processing materials.

Mass and Volume Relationship

Mass and volume are interlinked by the property of density. Density gives us a direct way to correlate mass and volume: \( \text{density} = \frac{\text{mass}}{\text{volume}} \). This relationship is pivotal in calculating the volume of a substance when mass and density are known.
In the exercise, knowing the mass and initial density allowed us to calculate the initial volume of the unknown liquid. Later, by substituting the new density value, we could find the volume at a higher temperature. The constancy of mass despite changes in density and volume is due to the mass conservation principle. This is a fundamental concept in physics and chemistry, ensuring that calculations adhere to natural laws.