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?

Short answer

The volume of ice formed after freezing the water in the soft-drink bottle is $1630.86{\mathrm{cm}}^3$. Since the volume of the bottle is only $1500{\mathrm{cm}}^3$, the ice cannot be contained within the bottle.

Step-by-step solution

Calculate the mass of water in the bottle

Firstly, we need to calculate the mass of water in the soft-drink bottle. We know the volume of water in the bottle and its density, so we can use the following formula to calculate the mass: $$mass=density\times volume$$ Since the volume of water in the bottle is given in liters, let's convert it to cubic centimeters (cm³) before proceeding: $$1.50\mathrm{L}=1500{\mathrm{cm}}^3$$ Now, we can calculate the mass of water using the provided density: $$mass=0.997\frac{\mathrm{g}}{{\mathrm{cm}}^3}\times 1500{\mathrm{cm}}^3$$

Compute the mass of water in the bottle

After calculating the mass of water, we can now compute it: $$mass=0.997\frac{\mathrm{g}}{{\mathrm{cm}}^3}\times 1500{\mathrm{cm}}^3=1495.5\mathrm{g}$$

Calculate the volume of ice

Now that we have the mass of water, we can use the density of ice to find the volume of ice formed. We can use the same formula and rearrange it to find the volume: $$volume=\frac{mass}{density}$$ So now, we can find the volume of ice formed: $$volume=\frac{1495.5\mathrm{g}}{0.917\frac{\mathrm{g}}{{\mathrm{cm}}^3}}$$

Compute the volume of ice

Now, let's compute the volume of the ice using the mass of water and the density of ice: $$volume=\frac{1495.5\mathrm{g}}{0.917\frac{\mathrm{g}}{{\mathrm{cm}}^3}}=1630.86{\mathrm{cm}}^3$$

Can the ice be contained within the bottle?

In the last step, we need to compare the volume of the ice formed to the volume of the soft-drink bottle: The volume of ice is: $1630.86{\mathrm{cm}}^3$ The volume of the bottle is: $1500{\mathrm{cm}}^3$ Since the volume of ice ($1630.86{\mathrm{cm}}^3$) formed is higher than the volume of the bottle ($1500{\mathrm{cm}}^3$), the ice cannot be contained within the bottle.

Key concepts

Volume Expansion

When water turns into ice, it experiences a fascinating phenomenon called volume expansion. This means that the volume of water increases when it freezes. The reason behind this expansion is the arrangement of water molecules in ice.
This structure causes ice to occupy more space than liquid water. In our given exercise, we observe that when 1.50 liters (or 1500 cm³) of water turns into ice, it occupies a volume of approximately 1630.86 cm³.
This increase in volume is due to the lower density of ice compared to water. Understanding volume expansion is crucial because it explains why water-filled containers can break when frozen.

• Water molecules form a hexagonal lattice in ice, making it less dense.
• This results in the expansion of volume, around 9% more compared to liquid water.
• This physical change happens because ice occupies more space than the same mass of liquid water.

Knowing about volume expansion can help in predicting how water behaves in different temperature scenarios, crucial for practical applications like storing liquids in cold climates.

Mass Calculation

Before we can understand how much the volume expands upon freezing, it's essential to compute the mass of the water we started with. This step is straightforward since we know the density and the volume of the water.
Using the density of water, 0.997 g/cm³, and the volume of 1500 cm³ (1.50 L), the mass can be calculated using the formula: $$mass=density\times volume$$ Substituting in the values, we find: $$mass=0.997\frac{\mathrm{g}}{{\mathrm{cm}}^3}\times 1500{\mathrm{cm}}^3=1495.5\mathrm{g}$$This calculation is vital because it is the conservative property during phase transition.
The mass of a substance remains unchanged whether in liquid or solid form. Understanding how to calculate mass helps maintain consistency in measurements and is essential for further calculations like determining the subsequent volume of ice.

Density Comparison

Density is a key concept when discussing water and ice because it explains why ice floats and why water expands when it freezes. At 25°C, water has a density of approximately 0.997 g/cm³.
Ice, on the other hand, has a density of 0.917 g/cm³ at -10°C. The density of a substance is its mass per unit volume, so substances with lower densities float above those with higher densities.
This is why ice floats on water; ice is less dense compared to liquid water. Let's compare:

• Ice density = 0.917 g/cm³
• Water density = 0.997 g/cm³

This difference explains the volume expansion when water freezes. Since ice occupies more space for the same mass, its density is lower.
Density comparison is crucial in explaining natural phenomena like iceberg flotation and is a key point in understanding how and why the soft-drink bottle cannot contain ice when water freezes inside it.