Problem 81 - Chemistry The Central Science In Si Units

Question

Water has a density of \(0.997 \mathrm{~g} / \mathrm{cm}^{3}\) at \(25^{\circ} \mathrm{C} ;\) ice has a density of \(0.917 \mathrm{~g} / \mathrm{cm}^{3}\) at \(-10^{\circ} \mathrm{C} .\) (a) If a soft-drink bottle whose volume is \(1.50 \mathrm{~L}\) is completely filled with water and then frozen to \(-10^{\circ} \mathrm{C},\) what volume does the ice occupy? (b) Can the ice be contained within the bottle?

Step-by-Step Solution

Answer

(a) Ice occupies approximately 1630.64 cm³. (b) No, the ice cannot be contained within the bottle.

Step 1: Convert Volume to Consistent Units

The given volume of the bottle is in liters, but the densities of water and ice are given in grams per cubic centimeter. We need to convert the volume of the bottle from liters to cubic centimeters. Since 1 liter is equal to 1000 cubic centimeters, we convert 1.50 liters to cubic centimeters: \[ 1.50 \text{ L} = 1.50 \times 1000 \text{ cm}^3 = 1500 \text{ cm}^3. \]

Step 2: Calculate the Mass of Water

The mass of water in the bottle can be calculated using the density of water and the volume of the bottle. The formula for mass is given by: \[ \text{mass} = \text{density} \times \text{volume}. \] Substitute the given density of water (0.997 g/cm³) and the volume we converted: \[ \text{mass} = 0.997 \text{ g/cm}^3 \times 1500 \text{ cm}^3 = 1495.5 \text{ g}. \]

Step 3: Determine the Volume of Ice

The mass of the ice will be the same as the mass of the water because mass is conserved in this process. To find the volume that this mass of ice occupies, we use the ice density: \[ \text{volume of ice} = \frac{\text{mass}}{\text{density}}. \] Using the mass of 1495.5 g and the density of ice (0.917 g/cm³): \[ \text{volume of ice} = \frac{1495.5 \text{ g}}{0.917 \text{ g/cm}^3} \approx 1630.64 \text{ cm}^3. \]

Step 4: Analyze Whether Ice Fits in the Bottle

The volume of the bottle is 1500 cm³, but the volume of ice is approximately 1630.64 cm³. As the volume of ice is greater than the volume of the bottle, the ice cannot be fully contained within 1.50 L bottle.

Key Concepts

Volume Conversion

When working with different units, it's essential to ensure consistency, particularly in science. In this scenario, we have a volume measured in liters, while densities are provided in grams per cubic centimeter. To resolve this disparity, we convert all measurements to a single unit.

A liter is equivalent to 1000 cubic centimeters (cm³). This conversion factor is crucial. By applying it to the bottle's volume of 1.50 liters, we find the volume in cubic centimeters:

• 1 liter = 1000 cm³
• 1.50 liters = 1.50 x 1000 = 1500 cm³

Volume conversion ensures the same unit is used throughout calculations, which simplifies problem-solving processes.

Mass Calculation

After converting the volume, calculating the mass of a substance requires understanding the relationship between mass, density, and volume. The formula is \[ \text{mass} = \text{density} \times \text{volume}. \] In this case, using the density of water (\(0.997 \text{ g/cm}^3\)) and the volume (\(1500 \text{ cm}^3\)) allows us to find the mass of the water:

• Mass = 0.997 g/cm³ x 1500 cm³
• Mass = 1495.5 g

Remember that understanding this equation is crucial because it helps calculate how much matter is present in a given space. It's practical for determining quantities of substances in varied scientific problems.

Phase Change

Phase change refers to the transition between different states of matter, such as from liquid to solid. During this process, the mass is conserved, meaning it doesn't change, but volume can vary significantly due to different densities in each state.

Consider water freezing into ice. To find out how much space ice occupies, you apply the formula: \[ \text{volume of ice} = \frac{\text{mass}}{\text{density}}. \] With a mass of 1495.5 g and the density of ice (\(0.917 \text{ g/cm}^3\)), we find:

• Volume of ice = 1495.5 g / 0.917 g/cm³
• Volume of ice ≈ 1630.64 cm³

This increase in volume during the phase change explains why frozen water can expand beyond its original container's capacity, a common property observed with water compared to most substances.