Question

Silicon is widely used in the semi-conductor industry. Its isotopes and their atomic masses are: Si-28 ( \(27.9769 \mathrm{amu}\) ), Si-29 ( \(28.9765 \mathrm{amu}\) ), and Si-30 ( \(29.9794\) amu). Which is the most abundant?

Short answer

Answer: Si-28

Step-by-step solution

Determine the atomic mass of Silicon

We will first find the atomic mass of Silicon. According to periodic table, the atomic mass of Silicon is approximately \(28.0855\) amu. This value is a weighted average of the masses of its isotopes.

Set up the equation to represent the weighted average

To represent the weighted average, let the fraction of each isotope (Si-28, Si-29 and Si-30) be represented by x, y, and z, respectively. We can set up the equation: $$28.0855 = 27.9769x + 28.9765y + 29.9794z$$ Since x, y, and z are fractions of the isotopes, their sum will be equal to 1: $$x+y+z=1$$

Solve for one variable and express it in terms of others

We will solve for one of the variables in terms of the others. Let's solve for z: $$z = 1-x-y$$ Now, substitute z into the weighted average equation: $$28.0855 = 27.9769x + 28.9765y + 29.9794(1-x-y)$$

Rearrange the equation to make it easier to analyze

Rearrange the equation in order to group terms with x and y: $$1.0026x - 0.9779y = 0.1061$$

Analyze the equation to identify which isotope is most abundant

From the rearranged equation, we can see that if the value of x (Si-28 fraction) is more significant than the value of y (Si-29 fraction), then the value of 1.0026x would be greater than 0.9779y. Similarly, if y is more significant than x, then 0.9779y would be greater than 1.0026x. Now, if Si-30 was the most abundant, increasing its fraction (increasing z) value would decrease both x and y. However, this would decrease the right side of the equation, which is a fixed value (0.1061). This means that an increased fraction of Si-30 would not conform to the equation. Hence, Si-30 cannot be the most abundant isotope. Now, let's compare Si-28 and Si-29. To satisfy the equation, increasing x (increasing Si-28 fraction) and decreasing y (decreasing Si-29 fraction) would make the left side of the equation greater, while increasing y (increasing Si-29 fraction) and decreasing x (decreasing Si-28 fraction) would make the left side of the equation smaller. Therefore, Si-28 (having a higher fraction) would have a higher contribution to the atomic mass and thus be the most abundant isotope. #Conclusion# The most abundant isotope of Silicon is Si-28.

Key concepts

Atomic Mass

The atomic mass of an element is a key concept in understanding chemistry.
For any given element, the atomic mass is not just a simple average of its isotopic masses; rather, it is a weighted average.
This means that it takes into account the relative abundance of each isotope of that element.
In our case, the atomic mass of silicon is approximately 28.0855 amu as provided by the periodic table.
Why is this important? Because knowing the atomic mass allows us to calculate how much of each isotope contributes to the element's overall mass. Each isotope of silicon has a specific atomic mass: Si-28 is 27.9769 amu, Si-29 is 28.9765 amu, and Si-30 is 29.9794 amu.
Understanding these values and their contributions allows for more precise applications in scientific fields, such as semiconductor design.

Weighted Average

The concept of weighted average is crucial when calculating the atomic mass from isotopes.
A weighted average alters the typical arithmetic average by assigning different "weights" or significance to the numbers involved, based on their relative abundance.
This ensures a more accurate representation of the actual mass found in nature.
In the context of silicon, if we let the fractions of the isotopes Si-28, Si-29, and Si-30 be represented as variables x, y, and z respectively, we then set up an equation for silicon's weighted atomic mass:

• Overall weighted average = Sum of (isotopic mass * fraction abundance)
• For silicon: \(28.0855 = 27.9769x + 28.9765y + 29.9794z\)

This equation, along with the constraint \(x + y + z = 1\), allows us to determine not only the total atomic mass but the contributions of individual isotopes to that mass.

Silicon Isotopes Analysis

Analyzing silicon's isotopes helps us understand the element more deeply, particularly in industrial applications.
Silicon has three main isotopes: Si-28, Si-29, and Si-30, each with its distinct mass.
In silicon's isotopic analysis, understanding the weighted average allows us to figure out which isotope has a higher natural abundance based on their masses.Using the weighted average equation:

• 26.0855 = 27.9769x + 28.9765y + 29.9794z
• x+y+z=1
• Rearranging gives: \(1.0026x - 0.9779y = 0.1061\)

Through analysis of these isotopic contributions, we can establish that Si-28 is the most abundant isotope, crucial for silicon's wide application in electronic and semiconductor industries.

Abundance of Isotopes

The abundance of isotopes is essential for understanding the contribution of each isotope to the overall mass of the element.
Abundance refers to how frequently or in what proportion each isotope is found in nature. In our silicon example, Si-28 being more abundant means it has a greater impact on silicon's atomic mass compared to Si-29 and Si-30.
The relationship between the isotopic abundance and mass is evident in the equation, which shows that for silicon, a higher contribution of Si-28 leads to the atomic mass being closer to its own mass, making it the dominant isotope.
This characteristic is vital in applications where precise material properties are necessary, such as semiconductor manufacturing, as it directly affects silicon's performance. Hence, knowing which isotopes are more abundant allows for optimized use of silicon based on its natural composition.