Question

Assuming that the density of water is \(.9971\left(\mathrm{~g} / \mathrm{cm}^{3}\right)\) at \(25^{\circ} \mathrm{C}\) and that of ice at \(0^{\circ}\) is \(917\left(\mathrm{~g} / \mathrm{cm}^{3}\right)\), what percent of a water jug at \(25^{\circ} \mathrm{C}\) should be left empty so that, if the water freezes, it will just fill the jug?

Short answer

To solve this problem, calculate the density ratio between ice and water, which is approximately \(0.9196\). Subtract this ratio from 1 and multiply by 100 to get the percent change, which is approximately \(8.04\%\). Therefore, to ensure that the water just fills the jug when it freezes, you should leave about \(8.04\%\) of the jug empty.

Step-by-step solution

Find the density ratio (ice to water)

First, we need to find the ratio between the density of ice and the density of water. We can do it by dividing the density of ice by the density of water: Density ratio (ice to water) \(= \frac{\text{Density of ice}}{\text{Density of water}}\)

Calculate the density ratio

Now, plug the given values for the density of ice and the density of water into the equation: Density ratio (ice to water) \(= \frac{917\frac{\mathrm{g}}{\mathrm{cm}^3}}{.9971\frac{\mathrm{g}}{\mathrm{cm}^3}} = 0.9196\)

Determine the percent change

Now, subtract the density ratio from 1, and multiply by 100 to get the percent change: Percent change \(= (1 - \text{Density ratio}) \times 100\) Percent change \(= (1 - 0.9196) \times 100 = 8.04\%\)

Interpret the result

This means that the water will expand by approximately 8.04% when it freezes. Therefore, to ensure that the water just fills the jug when it freezes, you should leave about 8.04% of the jug empty.

Key concepts

Density of Water

Water is a vital substance in chemistry and our daily lives, largely characterized by its density. Density, a measure of the mass per unit volume of a substance, can vary with temperature. For water at 25 degrees Celsius, its density is approximately .9971 grams per cubic centimeter (.9971 g/cm³ ). Understanding the density of water is crucial for a myriad of applications, ranging from buoyancy to the calculation of flow rates in hydrodynamics.

It's interesting to note that water has a unique property; its maximum density occurs at around 4 degrees Celsius, above its freezing point. Below this temperature, the density starts to decrease, a fascinating anomaly that explains why ice floats on water. This concept is essential when considering the expansion of water upon freezing, as it pertains to the original exercise.

Density of Ice

Ice, the solid state of water, exhibits a lower density compared to liquid water. This is counterintuitive, as solids are often denser than their liquid counterparts. For our purposes, the density of ice at 0 degrees Celsius is given as 917 grams per cubic centimeter (917 g/cm³). This lower density is the result of water molecules forming a crystal structure when frozen, which takes up more space than the random arrangement in its liquid state.

The real-life implications of this phenomenon are significant. It explains why icebergs float and how aquatic life can survive beneath the ice-covered surfaces of water bodies during the winter. This concept ties directly into the step-by-step solution of the original exercise, where the difference in densities of ice and water determines how much volume increases when water turns into ice.

Percent Change Calculation

Percent change is a mathematical concept used to describe the degree of change over time in the context of statistics, economics, and, in our case, physical changes of substances. The formula for percent change is given by ((new value − original value) / original value) * 100 %, which provides the change in terms of a percentage.

In the context of the solved exercise, this calculation helps us understand by what percentage the volume of water will increase as it freezes. By establishing the percent change, students can practically apply this concept to predict the expansion of water in a container, thereby preventing it from breaking as the water turns to ice. The exercise uses the density ratio between water and ice to find that the volume change, hence the percent change, is approximately 8.04%. This tells us that the water jug should be left 8.04% empty to accommodate for the expansion when the water freezes.