Question

The following densities are given at \(20^{\circ} \mathrm{C}\) : water, \(0.998 \mathrm{g} / \mathrm{cm}^{3} ;\) iron, \(7.86 \mathrm{g} / \mathrm{cm}^{3} ;\) aluminum, \(2.70 \mathrm{g} / \mathrm{cm}^{3}\). Arrange the following items in terms of increasing mass. (a) a rectangular bar of iron,$$81.5 \mathrm{cm} \times 2.1 \mathrm{cm} \times 1.6 \mathrm{cm}$$ (b) a sheet of aluminum foil,$$12.12 \mathrm{m} \times 3.62 \mathrm{m} \times 0.003 \mathrm{cm}$$ (c) 4.051 L of water

Short answer

The items arranged in order of increasing mass are: the aluminum sheet, the iron bar, and then the water.

Step-by-step solution

Calculate the volume and mass of the iron bar

First, we need to calculate the volume of the iron bar. This can be done by multiplying its dimensions: \(81.5 \, \mathrm{cm} \times 2.1 \, \mathrm{cm} \times 1.6 \, \mathrm{cm} = 273.42\, \mathrm{cm}^{3}\). Next, we multiply the volume we just calculated by the density of iron (7.86 \, \mathrm{g} / \, \mathrm{cm}^{3}) to give the mass: \(273.42 \, \mathrm{cm}^{3} \times 7.86 \, \mathrm{g} / \mathrm{cm}^{3} = 2149.16 \, \mathrm{g}\).

Calculate the volume and mass of the aluminum sheet

Similarly, we find the volume of the aluminum sheet by multiplying its dimensions: \(12.12 \, \mathrm{m} \times 3.62 \, \mathrm{m} \times 0.003 \, \mathrm{cm} = 132.10 \, \mathrm{cm}^{3}\). We then multiply that by the density of aluminum (2.70 \, \mathrm{g} / \, \mathrm{cm}^{3}) to give the mass: \(132.10 \, \mathrm{cm}^{3} \times 2.70 \, \mathrm{g} / \mathrm{cm}^{3} = 356.67 \, \mathrm{g}\). Note that we converted the dimensions from meters to centimeters so that the units match the density's units.

Calculate the mass of the water

Here, you can use the density directly because the volume is provided in liters and 1 liter is equivalent to 1000 cm\(^3\). So, to find the mass, multiply the volume by the density: \(4.051 \, \mathrm{L} \times 0.998 \, \mathrm{g}/\mathrm{cm}^{3} = 4042.948 \, \mathrm{g}\).

Arrange the items in order of increasing mass

Finally, arrange the items in order of increasing mass: Aluminum Sheet (356.67 g) < Iron Bar (2149.16 g) < Water (4042.948 g).

Key concepts

Volume and Mass Relationships

The relationship between volume and mass is essential in understanding how different substances interact. Volume is the measure of how much space an object occupies, while mass is the measure of how much matter it contains.

To determine the mass of an object, one can use the known relationship of mass, volume, and density, given by the formula:

• Mass = Density × Volume

This formula shows how mass directly depends on the volume of the object and its density, which implies that if you know any two properties, you can solve for the third.

In the context of our exercise, calculating the volume of materials like iron and aluminum involved multiplying their physical dimensions to derive a numerical value representing the space they occupy. Once obtained, these values are manipulated through multiplication with their respective densities to find mass. This straightforward relationship allows us to understand and compare materials based on their physical dimensions and inherent properties.

Material Properties

Different materials have unique properties that affect how they interact with the world around them. These properties include density, conductivity, malleability, and more, defining their utility in various applications.

Density is a critical material property in this context, providing insight into how 'heavy' a material feels for a given volume. Materials like iron, which have a high density of 7.86 g/cm³, are often heavier and compact, meaning more mass fits into a smaller space. In contrast, aluminum, with a lower density of 2.70 g/cm³, is lighter and thus often used in industries where weight is a concern, such as in aerospace.

Knowing these properties helps us predict how materials behave under different conditions and determine the best materials for specific functions, such as when selecting materials for building structures or designing lightweight transportation devices.

Density of Materials

Density is a fundamental concept that describes how much mass an object contains relative to its volume, mathematically expressed as:

• Density = Mass/Volume

It provides an easy-to-understand framework to compare and contrast different substances.

In our exercise, we see that density plays a crucial role in determining which item among the iron bar, aluminum foil, and water has the greatest mass. By calculating the mass using the formula, considering the densities of each material, we can place them in order of increasing mass accurately.

Understanding density allows us to make informed decisions regarding material selection and use, helping us to choose the most efficient materials needed for specific tasks. It also aids in explaining why some objects float while others sink and is valuable in various scientific and engineering disciplines.