Problem 158 - General Chemistry

Question

An ice cube measures \(3.50 \mathrm{~cm}\) on each edge and weighs \(39.45 \mathrm{~g}\). a Calculate the density of ice. b Calculate the mass of \(400.4 \mathrm{~mL}\) of water in an ice cube.

Step-by-Step Solution

Answer

a. 0.9206 g/cm³; b. 400.4 g

Step 1: Understand the Density Formula

Density is calculated by the formula \( \text{Density} = \frac{\text{Mass}}{\text{Volume}} \). We need to apply this formula to find the density of the ice cube given its mass and volume.

Step 2: Calculate Volume of the Ice Cube

Since the ice cube is a cube, its volume is given by \( \text{Volume} = \text{side}\,\text{length}^3 = (3.50\,\mathrm{cm})^3 = 42.875\,\mathrm{cm}^3 \).

Step 3: Calculate the Density of Ice

Using the formula for density, substitute the mass of the ice cube (39.45 g) and its volume (42.875 cm³): \( \text{Density} = \frac{39.45\,\mathrm{g}}{42.875\,\mathrm{cm}^3} \approx 0.9206\,\mathrm{g/cm}^3 \).

Step 4: Convert Volume of Ice Pieces to Volume of Water

When ice melts, it retains the same volume in terms of water, thus 400.4 mL of ice is equivalent to 400.4 mL of water because 1 mL = 1 cm³.

Step 5: Calculate the Mass of Water in Ice Pieces

The density of water is approximately \( 1\,\mathrm{g/cm}^3 \). Therefore, the mass of 400.4 mL of water is calculated by \( \text{Mass} = \text{Density} \times \text{Volume} = 1\,\mathrm{g/cm}^3 \times 400.4\,\mathrm{cm}^3 = 400.4\,\mathrm{g} \).

Key Concepts

Density Formula

The concept of density is fundamental in understanding how different substances relate in terms of mass and volume. Density is a measurement of how much mass is contained in a given volume. The formula to calculate density is:

• \( \text{Density} = \frac{\text{Mass}}{\text{Volume}} \)

This simple equation tells us how compact or concentrated a substance is. If an object is heavier for its size, it has a higher density, and if it's lighter for its size, it has a lower density. Calculating density helps in predicting whether an object will float or sink in a fluid. Essentially, knowing the density allows you to compare the heaviness of various materials irrespective of their size.

Volume Calculation

Volume measures the space that an object occupies. In this exercise, we have an ice cube which is a perfect cube. For a cube, the volume is calculated by cubing the length of one of its sides:

• \( \text{Volume} = \text{side length}^3 \)

In our example, the side length is given as 3.50 cm. Therefore, the volume of the cube is \( (3.50 \mathrm{~cm})^3 = 42.875 \mathrm{~cm}^3 \). Knowing how to calculate the volume of different shapes is essential because it plays a critical role in determining various physical properties, such as density.

Mass Calculation

Mass is essentially the amount of matter in an object, usually measured in grams or kilograms. In this exercise, the mass of the ice cube is given as 39.45 g. To find the mass of water represented in the ice, we apply the same principles for calculating the ice's mass but consider the properties of water. When applying the density formula to calculate the mass, it is expressed as:

• \( \text{Mass} = \text{Density} \times \text{Volume} \)

This calculation is crucial when converting between different states of matter, such as ice to water, because it helps predict how much of the substance is present in various forms. Understanding how to manipulate these calculations allows you to uncover a wealth of information about the object or substance in question.

Properties of Water

Water is one of the most studied substances due to its unique properties. One vital property is its density, which is typically about \(1 \,\mathrm{g/cm}^3\) at standard temperature and pressure. This density value means that 1 milliliter (mL) of water weighs exactly 1 gram (g), making it an easy substance for conversion calculations.
Water's higher density compared to ice allows ice to float on water. This difference also plays a role in the natural environment and ecosystems. When calculating, we frequently use the properties of water to simplify and check calculations, as was done in the exercise, to convert the volume and density of ice to that of water. Understanding water's properties aids in a wide range of scientific analyses and real-world applications.