Question

The thin outer layer of Earth, called the crust, contains only 0.50 percent of Earth's total mass and yet is the source of almost all the elements (the atmosphere provides elements such as oxygen, nitrogen, and a few other gases). Silicon (Si) is the second most abundant element in Earth's crust (27.2 percent by mass). Calculate the mass of silicon in kilograms in Earth's crust. (The mass of Earth is \(5.9 \times 10^{21}\) tons. 1 ton \(=2000 \mathrm{lb}\); \(1 \mathrm{lb}=453.6 \mathrm{~g} .)\)

Short answer

The mass of silicon in Earth's crust is approximately \(3.65 \times 10^{21}\) kg.

Step-by-step solution

Convert the mass of the Earth to kilograms

To begin, convert the mass of Earth from tons to kilograms. There are 2,000 lbs in a ton, and 453.6 g in a lb. Therefore, 1 ton = \(2000 \times 453.6\ g\), and, as 1 kg = 1000 g, 1 ton = \( \frac{2000 \times 453.6}{1000}\ kg\). Then, the mass of Earth is \(5.9 \times 10^{21}\ \times \frac{2000 \times 453.6}{1000} = 2.68 \times 10^{24}\) kg.

Calculate the mass of the crust

Next, determine the mass of the crust, which is said to be 0.50 percent of Earth's total mass. Therefore, the mass of the crust is \(2.68 \times 10^{24} \times 0.005 = 1.34 \times 10^{22}\) kg.

Calculate the mass of silicon

Finally, the mass of silicon in the crust can be calculated. The problem states that silicon comprises 27.2 percent of the crust's mass by mass. As a percentage, this is equivalent to 0.272. Therefore, the mass of silicon is \(1.34 \times 10^{22} \times 0.272 = 3.65 \times 10^{21}\) kg.

Key concepts

Understanding Mass Conversion

The concept of mass conversion is essential for solving many science problems, especially when dealing with Earth's mass or celestial bodies. A common task is converting mass from one unit to another, like tons to kilograms. This concept is crucial as various regions use different units, and scientific calculations often require consistent units for accuracy.

To convert mass, use conversion factors. For instance, one ton equals 2000 pounds, and one pound equals 453.6 grams. Therefore, to convert tons to kilograms, which is more scientific, the formula is:

• First, convert tons to pounds by multiplying the mass in tons by 2000.
• Then, convert pounds to grams by multiplying by 453.6.
• Finally, convert grams to kilograms. Since 1 kg = 1000 grams, divide by 1000.

Understanding these steps makes it easier to work with massive quantities, like Earth's mass.

Identifying Abundant Elements in Earth's Crust

Earth's crust is the outermost layer, which, although thin, contains a variety of elements. Identifying the most abundant elements helps us understand more about Earth's resources and composition.

The abundance of elements in the crust is influenced by geological and cosmochemical factors. Silicon, oxygen, aluminum, and iron are among the most common.

Here's a look at some common elements:

• Oxygen: Makes about 46.6% of Earth's crust by weight.
• Silicon: Approximately 27.2% of the crust.
• Aluminum: Around 8.1%.
• Iron: Constitutes about 5%.

Understanding these abundances aids in resource management and various fields of study such as mineralogy and geology.

The Role of Silicon in Earth's Crust

Silicon is not only one of the most abundant elements in Earth's crust, coming in second after oxygen, but it plays a crucial role in the formation of rocks and minerals. This element is pivotal in creating silicate minerals, which make up a large portion of the Earth's crust.

Silicate minerals like quartz, feldspar, and mica are vital for understanding the structure and components of the crust.

Key roles of silicon include:

• Formation of Silicates: Silicates form the building blocks for many rocks, thanks to silicon's ability to form stable bonds with oxygen.
• Impact on Earth's Resources: Silicon is used in industries ranging from electronics to glass-making, illustrating its economic significance.
• Geological Importance: Helps geologists understand tectonic processes and mineral composition.

Silicon's prevalence and functionality make it indispensable for both natural processes and human applications.