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Chapter 5: Problem 160

Gallium arsenide, GaAs, has gained widespread use in semiconductor devices that convert light and electrical signals in fiber-optic communications systems. Gallium consists of \(60 . \%^{69} \mathrm{Ga}\) and \(40 . \%^{71} \mathrm{Ga}\). Arsenic has only one naturally occurring isotope, \(^{75}\)As. Gallium arsenide is a polymeric material, but its mass spectrum shows fragments with the formulas GaAs and \(\mathrm{Ga}_{2} \mathrm{As}_{2}\). What would the distribution of peaks look like for these two fragments?

Short Answer

Expert verified
For GaAs, the mass spectrum would show two peaks: a peak at a mass of 144 with a relative abundance of 60.0% (corresponding to the \(^{69}\)Ga-\(^{75}\)As combination), and a peak at a mass of 146 with a relative abundance of 40.0% (corresponding to the \(^{71}\)Ga-\(^{75}\)As combination). For Ga\(_2\)As\(_2\), the mass spectrum would show three peaks: a peak at a mass of 288 with a relative abundance of 36.0% (corresponding to the \(^{69}\)Ga-\(^{69}\)Ga-\(^{75}\)As-\(^{75}\)As combination), a peak at a mass of 290 with a relative abundance of 24.0% (corresponding to the \(^{69}\)Ga-\(^{71}\)Ga-\(^{75}\)As-\(^{75}\)As combination), and a peak at a mass of 292 with a relative abundance of 16.0% (corresponding to the \(^{71}\)Ga-\(^{71}\)Ga-\(^{75}\)As-\(^{75}\)As combination).

Step by step solution

01

Identify possible combinations and calculate their masses

First, let's identify all possible combinations of isotopes for both GaAs and Ga\(_2\)As\(_2\). For GaAs, we have the following combinations: 1. \(^{69}\)Ga-\(^{75}\)As 2. \(^{71}\)Ga-\(^{75}\)As Now let's calculate the mass of each combination: 1. Mass of \(^{69}\)Ga-\(^{75}\)As: \( 69 + 75 \) = 144 2. Mass of \(^{71}\)Ga-\(^{75}\)As: \( 71 + 75 \) = 146 Next, we'll do the same for Ga\(_2\)As\(_2\). Here, we have the following combinations: 1. \(^{69}\)Ga-\(^{69}\)Ga-\(^{75}\)As-\(^{75}\)As 2. \(^{69}\)Ga-\(^{71}\)Ga-\(^{75}\)As-\(^{75}\)As 3. \(^{71}\)Ga-\(^{71}\)Ga-\(^{75}\)As-\(^{75}\)As Again, let's calculate the mass of each combination: 1. Mass of \(^{69}\)Ga-\(^{69}\)Ga-\(^{75}\)As-\(^{75}\)As: \( 2(69) + 2(75) \) = 288 2. Mass of \(^{69}\)Ga-\(^{71}\)Ga-\(^{75}\)As-\(^{75}\)As: \( 69 + 71 + 2(75) \) = 290 3. Mass of \(^{71}\)Ga-\(^{71}\)Ga-\(^{75}\)As-\(^{75}\)As: \( 2(71) + 2(75) \) = 292
02

Determine the relative abundances of each combination

We know that \(^{69}\)Ga has a relative abundance of 60.0% and \(^{71}\)Ga has a relative abundance of 40.0%. Now, we can find the relative abundances of each isotope combination for both fragments. For GaAs: 1. Relative abundance of \(^{69}\)Ga-\(^{75}\)As: \( 60.0\% \) 2. Relative abundance of \(^{71}\)Ga-\(^{75}\)As: \( 40.0\% \) For Ga\(_2\)As\(_2\): 1. Relative abundance of \(^{69}\)Ga-\(^{69}\)Ga-\(^{75}\)As-\(^{75}\)As: \( (60.0\%)^2 = 36.0\% \) 2. Relative abundance of \(^{69}\)Ga-\(^{71}\)Ga-\(^{75}\)As-\(^{75}\)As: \( 60.0\% \times 40.0\% = 24.0\% \) 3. Relative abundance of \(^{71}\)Ga-\(^{71}\)Ga-\(^{75}\)As-\(^{75}\)As: \( (40.0\%)^2 = 16.0\% \)
03

Describe the distribution of peaks

Using the information obtained in the previous steps, we can describe the distribution of peaks in the mass spectrum for the different fragments. For GaAs, the mass spectrum would show two peaks: 1. A peak at a mass of 144 with a relative abundance of 60.0% (corresponding to the \(^{69}\)Ga-\(^{75}\)As combination). 2. A peak at a mass of 146 with a relative abundance of 40.0% (corresponding to the \(^{71}\)Ga-\(^{75}\)As combination). For Ga\(_2\)As\(_2\), the mass spectrum would show three peaks: 1. A peak at a mass of 288 with a relative abundance of 36.0% (corresponding to the \(^{69}\)Ga-\(^{69}\)Ga-\(^{75}\)As-\(^{75}\)As combination). 2. A peak at a mass of 290 with a relative abundance of 24.0% (corresponding to the \(^{69}\)Ga-\(^{71}\)Ga-\(^{75}\)As-\(^{75}\)As combination). 3. A peak at a mass of 292 with a relative abundance of 16.0% (corresponding to the \(^{71}\)Ga-\(^{71}\)Ga-\(^{75}\)As-\(^{75}\)As combination).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Gallium Arsenide
Gallium arsenide (GaAs) is a semiconductor material prized for its superb electronic properties. Unlike silicon, GaAs has a direct band gap, which means that it can efficiently emit light. That's why it's widely used in applications like LEDs, laser diodes, and high-speed electronic devices.

The beauty of GaAs lies in its capacity to operate at higher frequencies and offer greater bandwidth, which is critical in fiber-optic communications and 5G networks. This makes it an important material in the fields of telecommunications and advanced electronics.

Moreover, GaAs can also tolerate higher operating temperatures than silicon, further boosting its appeal in high-power and high-frequency devices.
Semiconductor Devices
Semiconductor devices are the foundation of modern electronics. They control the flow of electricity in a variety of sophisticated gadgets, from simple diodes to complex microprocessors. These components are integral to countless applications like computing, communications, and automation.

In these devices, materials like gallium arsenide are used to manufacture components that have superior speed and efficiency compared to traditional silicon. Because of these properties, semiconductor devices made from GaAs are employed in critical areas where rapid signal processing is key, including satellite communications, radar systems, and certain cellular networks.
Isotope Abundance
Isotope abundance refers to the relative proportion of an element's isotopes found naturally. Each element can have several isotopes, differing in the number of neutrons in their nuclei. Understanding isotope abundance is critical in mass spectrum analysis because it helps predict the pattern of peaks representing various isotopes.

For example, gallium has two stable isotopes: Ga-69 and Ga-71, with natural abundances of about 60% and 40%, respectively. In mass spectrum analysis, these percentages help us predict the abundance of various molecular fragments. With isotopes, we witness the marvel of nature's fingerprint in the elemental compositions, which is especially crucial in advanced fields like semiconductor physics and materials science.

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