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?

The concept of density is crucial in understanding how different substances interact in relation to their mass and volume. Density, defined as the mass per unit volume of a substance, is expressed as \( \text{density} = \frac{\text{mass}}{\text{volume}} \). It's an intrinsic property that doesn’t depend on the amount of the material, which means that one cubic centimeter of water will have the same density as a whole ocean of water, assuming temperature and pressure remain constant.

For instance, water at \(25^\circ C\) has a density of \(0.997 \frac{\text{g}}{\text{cm}^3}\). In the comparison of substances like water and ice, ice has a lower density (\(0.917 \frac{\text{g}}{\text{cm}^3}\) at \(-10^\circ C\)), which is why ice floats on water. Understanding density is essential as it helps predict whether a substance will float or sink when placed in a fluid and is crucial to many applications in science and engineering, such as shipbuilding and material selection.

The mass-volume relationship is a way to quantify the amount of substance contained in a given space. It correlates directly to the concept of density. To find the mass of an object when its volume and density are known, the formula \( \text{mass} = \text{density} \times \text{volume} \) is used, as seen in our bottle example. Conversely, to find the volume from the known mass and density, the formula is rearranged to \( \text{volume} = \frac{\text{mass}}{\text{density}} \).

This relationship is paramount in many scientific calculations, including those in medicine for drug dosages, cooking recipes for ingredient measurements, and industry for mixing and manufacturing products. It also forms the basis for calculations involving gas laws in chemistry, where volume and mass are related under different conditions of pressure and temperature.

Water is an extraordinary substance because it behaves differently from most materials when it freezes: it expands. This phenomenon is known as thermal expansion. When water is cooled down to its freezing point, its molecules arrange in a crystalline structure that occupies more space than when the water is in its liquid form. This expansion is why ice has a lower density and why it floats on water, which is a unique property significant for aquatic life, as it insulates the water below and provides a habitat for creatures in cold environments.

The understanding of thermal expansion is not only important in nature but also in everyday applications like plumbing, where pipes could burst from freezing water, and the design of structures that must withstand temperature variations. In our exercise, thermal expansion causes the water when frozen to occupy a greater volume than in its liquid state at \(25^\circ C\), leading to a volume that exceeds the bottle’s capacity.

In science, various units are utilized for measuring the same quantities. Being able to convert between these units is essential for proper calculation and understanding. Conversions allow for standardized communication across different regions and fields of study. The conversion factor between liters and cubic centimeters is a prime example, where \(1 \text{L} = 1000 \text{cm}^3\). This conversion is used in our problem to ascertain the bottle's volume in cubic centimeters to use the given densities effectively.

It's important to ensure that units in an equation match so as to not distort the computation. Unit conversion is involved in multiple aspects of daily life and technical worlds alike, from cooking measurements to fuel economy in cars (miles per gallon vs. liters per 100 km). Clarity in unit conversions can prevent misunderstandings and errors in many practical applications.