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Chapter 3: Problem 40

Gallium arsenide, GaAs, has gained widespread use in semiconductor devices that convert light and electrical signals in fiberoptic 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 fragment, the distribution of peaks will have a 60% abundance for \(^{69}\)Ga - \(^{75}\)As and a 40% abundance for \(^{71}\)Ga - \(^{75}\)As. For Ga₂As₂ fragment, the distribution of peaks will have a 36% abundance for \(^{69}\)Ga - \(^{69}\)Ga - \(^{75}\)As - \(^{75}\)As, a 24% abundance for \(^{69}\)Ga - \(^{71}\)Ga - \(^{75}\)As - \(^{75}\)As, a 24% abundance for \(^{71}\)Ga - \(^{69}\)Ga - \(^{75}\)As - \(^{75}\)As, and a 16% abundance for \(^{71}\)Ga - \(^{71}\)Ga - \(^{75}\)As - \(^{75}\)As.

Step by step solution

01

Determine the possible combinations of isotopes in GaAs fragment

Let's list down all possible combinations of Gallium and Arsenic isotopes in GaAs fragment: 1. \(^{69}\)Ga - \(^{75}\)As 2. \(^{71}\)Ga - \(^{75}\)As
02

Calculate the relative abundances for GaAs fragment

We multiply the abundances of the individual isotopes in each combination: 1. \(^{69}\)Ga - \(^{75}\)As: 0.60 * 1 = 0.60 or 60% 2. \(^{71}\)Ga - \(^{75}\)As: 0.40 * 1 = 0.40 or 40%
03

Determine the possible combinations of isotopes in Ga₂As₂ fragment

Let's list down all possible combinations of Gallium and Arsenic isotopes in Ga₂As₂ fragment: 1. \(^{69}\)Ga - \(^{69}\)Ga - \(^{75}\)As - \(^{75}\)As 2. \(^{69}\)Ga - \(^{71}\)Ga - \(^{75}\)As - \(^{75}\)As 3. \(^{71}\)Ga - \(^{69}\)Ga - \(^{75}\)As - \(^{75}\)As 4. \(^{71}\)Ga - \(^{71}\)Ga - \(^{75}\)As - \(^{75}\)As
04

Calculate the relative abundances for Ga₂As₂ fragment

We multiply the abundances of the individual isotopes in each combination: 1. \(^{69}\)Ga - \(^{69}\)Ga - \(^{75}\)As - \(^{75}\)As: (0.60 * 0.60) * (1 * 1) = 0.36 or 36% 2. \(^{69}\)Ga - \(^{71}\)Ga - \(^{75}\)As - \(^{75}\)As: (0.60 * 0.40) * (1 * 1) = 0.24 or 24% 3. \(^{71}\)Ga - \(^{69}\)Ga - \(^{75}\)As - \(^{75}\)As: (0.40 * 0.60) * (1 * 1) = 0.24 or 24% 4. \(^{71}\)Ga - \(^{71}\)Ga - \(^{75}\)As - \(^{75}\)As: (0.40 * 0.40) * (1 * 1) = 0.16 or 16% Now we have the distribution of peaks for both fragments: For GaAs fragment: 1. \(^{69}\)Ga - \(^{75}\)As: 60% 2. \(^{71}\)Ga - \(^{75}\)As: 40% For Ga₂As₂ fragment: 1. \(^{69}\)Ga - \(^{69}\)Ga - \(^{75}\)As - \(^{75}\)As: 36% 2. \(^{69}\)Ga - \(^{71}\)Ga - \(^{75}\)As - \(^{75}\)As: 24% 3. \(^{71}\)Ga - \(^{69}\)Ga - \(^{75}\)As - \(^{75}\)As: 24% 4. \(^{71}\)Ga - \(^{71}\)Ga - \(^{75}\)As - \(^{75}\)As: 16%

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