Question

The densities of ice and water at \(0^{\circ} \mathrm{C}\) and 1 bar are \(0.96\) and \(0.99 \mathrm{~g} / \mathrm{cm}^{3}\), respectively. If the percentage of occupied space in ice is \(x\), then the percentage of empty space in water is (a) \(\frac{32}{33} x\) (b) \(\frac{33}{32} x\) (c) \(100-\frac{33}{32} x\) (d) \(100-\frac{32}{33} x\)

Short answer

The correct answer is (c) \(100-\frac{33}{32} x\).

Step-by-step solution

Understand the Relationship Between Density and Volume

The density of a substance is the mass per unit volume. Given that ice has a lower density than water, this implies it has a larger volume for the same mass. Since densities are given, we can use these to determine the relative occupied and empty space in both ice and water.

Calculate the Ratio of Volumes of Ice and Water

Let the mass of both ice and water be the same (1 g for simplicity). For ice, V_ice = mass / density_ice = 1 g / 0.96 g/cm^3. For water, V_water = mass / density_water = 1 g / 0.99 g/cm^3. Calculate the ratio V_ice / V_water to compare the volumes.

Determine the Percentage of Occupied Space

If the percentage of occupied space in ice is x, then the volume of ice that is actually occupied by water (when it was liquid) is x% of V_ice. Since we've assumed the same mass for water and ice, the occupied volume in ice would be equivalent to the entire volume of water.

Express the Empty Space in Water in Terms of x

Since the occupied space in water is its total volume and in ice the same mass occupies a larger volume (due to its lower density), the empty space will be the additional volume ice has compared to water. This is the ratio of volumes subtracted from 1 times 100 to convert it to a percentage: 1 - (V_water / V_ice) * 100 = empty space in water.

Find the Ratio of Empty Space in Water to Occupied Space in Ice

After finding the ratio V_ice / V_water, you can determine the ratio of the empty space in water to the occupied space in ice which is akin to finding the ratio of empty space to occupied space for the same mass in different states (solid vs liquid).

Solve for the Correct Option

Now, use this ratio and symbolic representation x to express the empty space in water in terms of the occupied space in ice. Match the derived expression with the given options to find the correct answer.

Key concepts

Density Calculations

Density is a fundamental concept in physical chemistry, representing the mass of a substance per unit volume. It is commonly expressed in grams per cubic centimeter (\text{g/cm}^3) for solid and liquid substances. To calculate the density of a substance, you use the formula \( \text{Density} = \frac{\text{Mass}}{\text{Volume}} \). When you are given the densities of two substances, like ice and water, you can compare their volumes while keeping the mass constant.

For instance, with ice having a density of \(0.96 \text{ g/cm}^3\) and water having a density of \(0.99 \text{ g/cm}^3\), we can deduce that for the same mass, ice will occupy a greater volume, demonstrating its lower density compared to water. Understanding this concept is pivotal in solving problems related to the physical properties of substances, as changes in density can indicate phase changes or differences in the structural arrangement of molecules.

Volume Ratio Comparison

Comparing the volume ratios of substances is essential when determining the relationship between their densities. As noted in the solution steps, if you consider the same mass for ice and water, you can calculate their respective volumes. With volume (\text{V}) inversely proportional to density, a lower density means a larger volume.

To compare volumes ratio of ice to water, you use the formula \( \frac{\text{V}_{\text{ice}}}{\text{V}_{\text{water}}} \). If we assume a mass of 1 gram, then for ice and water, the volumes are calculated using their respective densities. The resulting volume ratio serves as a stepping stone to understanding how the same mass of substance occupies different volumes in different states, in this case, as ice and water. Having the volume ratio allows you to transition into calculating the percentage of empty space in water based on the occupied space in ice, which is an application of this comparison.

Physical Properties of Ice and Water

Understanding the physical properties of ice and water is key to solving many physical chemistry problems, especially those related to phase changes and density. Ice is less dense than water, which is why it floats - an anomaly for solids that are typically denser than their liquid form. This is due to the crystalline structure of ice, where water molecules are arranged in a specific hexagonal pattern, creating more space between molecules compared to liquid water.

The slightly expanded structure of ice compared to water leads to the question tackled in our exercise: how does the 'empty' space in water compare to the 'occupied' space in ice? Empty space refers to the volume that is not occupied by water molecules. Since ice has a unique property of expanding upon freezing, this 'empty' space is what makes it less dense and often leads to a crucial understanding in various scientific disciplines such as climatology, geology, and environmental science. Analyses of these properties not only help us solve numerical problems but also deepen our understanding of water's role in nature and various processes.