Problem 75 - Chemistry: The Central Science

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

The ice occupies a volume of approximately \(1.6306 L\), which is greater than the original soft-drink bottle's volume of \(1.50 L\). Therefore, the ice cannot be contained within the bottle.

Step 1: Convert the volume of water to mass

To determine the mass of the water, we will use the density formula. \(Density = \frac{Mass}{Volume}\) But we need the mass, so we rearrange the formula to solve for mass: \(Mass = Density * Volume\) Now, substitute the given density of water at 25°C (0.997 g/cm³) and the volume of water (1.50 L) into the formula. Note that 1 L = 1000 cm³. \(Mass = 0.997 \frac{g}{cm^3} * 1.50*1000 cm^3\) Now, calculate the mass of the water. \(Mass ≈ 1495.5 g\)

Step 2: Calculate the volume of the ice formed

To calculate the volume of ice formed, we will use the same density formula as earlier. \(Density = \frac{Mass}{Volume}\) This time, we need to solve for the volume: \(Volume = \frac{Mass}{Density}\) Now, substitute the given density of ice at -10°C (0.917 g/cm³) and the mass calculated in step 1 (1495.5 g) into the formula. \(Volume = \frac{1495.5 g}{0.917 \frac{g}{cm^3}}\) Now, calculate the volume of the ice. \(Volume ≈ 1630.6 cm^3\)

Step 3: Convert the volume of ice to liters

To compare the volume of the ice to the bottle's original volume, convert the volume of ice from cm³ to liters. Note that 1 L = 1000 cm³. \(Volume = \frac{1630.6 cm^3}{1000}\) \(Volume ≈ 1.6306 L\)

Step 4: Determine if the ice can be contained within the bottle

Now, let's determine if the volume of ice can be contained within the bottle. Compare the volume of ice formed (1.6306 L) to the original volume of the bottle (1.50 L). Since the volume of ice (1.6306 L) is greater than the original volume of the bottle (1.50 L), the ice cannot be fully contained within the bottle. The bottle is not large enough to hold the expanded volume of the frozen water.

Key Concepts

Understanding Density and Its Calculation

The concept of density is crucial when studying the physical properties of substances, including the water and ice in our exercise. Density is defined as a substance's mass per unit volume and is often expressed in units like grams per cubic centimeter (g/cm³).

The density formula, written as \(Density = \frac{Mass}{Volume}\), allows us to compare how much mass is packed into a given volume of different substances. In the context of our exercise, water at \(25^{\text{\textdegree}}C\) has a density of \(0.997 \mathrm{~g} / \mathrm{cm}^{3}\) whereas ice at \(-10^{\text{\textdegree}}C\) has a density of \(0.917 \mathrm{~g} / \mathrm{cm}^{3}\). The differing densities are why ice floats on water; it's less dense and therefore has a larger volume for the same mass.

To solve for the mass or volume using the density formula, one must rearrange the equation accordingly. To find mass, multiply density by volume, and to find volume, divide mass by density. This manipulation of the formula is key to solving our exercise and understanding how density is applied in real-world scenarios.

Volume-Mass Calculations for Different States of Matter

Volume-mass calculation is an essential skill in science, especially when dealing with changes in a substance's state. The mass of a substance remains constant during state change, but its volume can change significantly because of the different densities in each state.

To calculate mass from volume, we multiply the volume by the substance's density, as seen in Step 1 of our exercise solution. Conversely, to find the volume given the mass and density, we divide the mass by the density, as demonstrated in Step 2.

It is important to pay attention to unit conversion during these calculations. In our exercise, the volume initially provided in liters (L) was converted to cubic centimeters (cm³) to match the density unit. Similarly, the calculated volume of ice in cm³ was converted back to liters to compare with the bottle's capacity. These conversions ensure accurate calculations and avoid potential misunderstandings when comparing different measurement units.

State Change of Water and Its Impact on Volume

The state change of water, from liquid to solid (freezing) or solid to liquid (melting), comes with a notable change in volume due to the differences in density. In most substances, freezing leads to a decrease in volume as molecules pack together more tightly, but water uniquely expands when it freezes.

When water turns into ice, its density decreases from \(0.997 \mathrm{~g} / \mathrm{cm}^{3}\) to \(0.917 \mathrm{~g} / \mathrm{cm}^{3}\). This decrease in density explains why the volume of ice is greater than the same mass of water, an outcome we observed in our example when the frozen water could not be contained within the same bottle.

The state change of water is a critical concept, not only in chemistry and physics but also in understanding natural phenomena such as the formation of icebergs and the expansion of water pipes during winter. Recognizing how temperature affects the state and properties of water is essential for a comprehensive understanding of the material's behavior in different conditions.