Problem 69 - 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

(a) The volume of ice that forms when the water is frozen is approximately \(1631.6 \mathrm{cm}^3\). (b) No, the ice cannot be contained within the bottle.

Step 1: Find the mass of water in the bottle

First, let's find the mass of water in the soft-drink bottle before it freezes. We can use the formula: Mass = Density × Volume The density of water at 25°C is given as 0.997 g/cm³, and the volume of the bottle is 1.50 L. First, we will convert the volume from liters to cubic centimeters (1 L = 1000 cm³): Volume = 1.50 L × (1000 cm³ / 1 L) = 1500 cm³ Now, let's find the mass of water: Mass of water = (0.997 g/cm³) × (1500 cm³) = 1495.5 g

Step 2: Find the mass of ice after freezing

The mass of ice after freezing will remain the same as the mass of water before freezing (mass conservation). Mass of ice = Mass of water = 1495.5 g

Step 3: Find the volume of ice after freezing

Now that we have the mass of ice, we can find its volume using the density of ice at -10°C, which is given as 0.917 g/cm³. We can use the formula: Volume = Mass / Density Volume of ice = (1495.5 g) / (0.917 g/cm³) ≈ 1631.6 cm³

Step 4: Compare the volume of ice with the volume of the bottle

We need to determine if the volume of ice can be contained within the bottle. To do this, we compare the volume of ice with the volume of the soft-drink bottle: Volume of ice = 1631.6 cm³ Volume of bottle = 1500 cm³ Since the volume of ice (1631.6 cm³) is greater than the volume of the bottle (1500 cm³), the ice cannot be contained within the bottle.

Step 5: Answer

(a) The volume of ice that forms when the water is frozen is approximately 1631.6 cm³. (b) No, the ice cannot be contained within the bottle.

Key Concepts

Mass Conservation Principle

One of the fundamental ideas in physics and chemistry is the mass conservation principle, which states that mass in an isolated system is neither created nor destroyed. This principle applies to the freezing of water as well.

When water turns into ice, even though it undergoes a state change from liquid to solid, no mass is lost in the process. The mass of the frozen water (ice) is equal to the mass of the liquid water before it was frozen. In the given problem, the mass remains constant at 1495.5 grams. By understanding this principle, you know that for any calculations involving mass before and after a state change, you can safely assume that the mass will be the same.

State Changes of Water

Water is a unique substance with three common states: solid (ice), liquid (water), and gas (vapor). When water changes from one state to another, like from liquid to solid, this process is known as a state change or phase transition.

At a molecular level, the state change from water to ice involves the molecules slowing down and becoming more rigidly structured, forming a crystalline solid. Notably, when water freezes, it expands in volume, which is unusual as most substances contract upon solidification. This expansion explains why ice has a lower density than water. In the example provided, the state change results in ice occupying a greater volume than the liquid water did, even though the mass remains unchanged.

Temperature Effects on Density

Density is defined as mass per unit volume. It's a crucial concept in understanding matter's characteristics and is temperature-dependent. As temperature increases or decreases, the density of substances can change; generally, substances expand when heated and contract when cooled.

In the case of water, its maximum density occurs at approximately 4°C. Above and below this temperature, the density of water decreases. This is why ice, which forms at 0°C or lower, is less dense than liquid water. Therefore, ice floats on water. In calculations, it's essential to use the correct density for the given temperature, like in our exercise where the density of water at 25°C and the density of ice at -10°C are used to determine the mass and volume accordingly.

Volume Conversion

Converting between different units of volume is a common task in many scientific calculations. It is crucial to represent volumes in consistent units to perform calculations correctly. For instance, in the provided example, before we could calculate the mass of water, we needed to convert the volume of the soft-drink bottle from liters to cubic centimeters, using the conversion factor 1 L = 1000 cm³.

When working with volume conversions, always keep track of the units and use conversion factors where necessary. It's also essential to understand that converting units doesn't change the quantity of the substance, just how that quantity is expressed. In other words, the amount of space the substance occupies remains the same, irrespective of the units used to measure that volume.