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Sagot :
### Part A: Predicting the Average Atomic Mass
Given the three naturally occurring isotopes of silicon and their corresponding atomic masses and percent abundances, let’s analyze where the average atomic mass of silicon would lie:
[tex]\[ \begin{array}{|l|r|r|} \hline \text{Isotope} & \text{Atomic Mass (amu)} & \text{Percent Abundance} \\ \hline \text{Silicon-28} & 27.98 & 92.21\% \\ \hline \text{Silicon-29} & 28.98 & 4.70\% \\ \hline \text{Silicon-30} & 29.97 & 3.09\% \\ \hline \end{array} \][/tex]
First, let's note the percent abundances. Silicon-28 has a significantly higher abundance (92.21\%) compared to Silicon-29 (4.70\%) and Silicon-30 (3.09\%).
### Prediction Reasoning
1. Dominant Isotope: Silicon-28 is the most abundant isotope by far. When one isotope is present in a much greater percentage than the others, the weighted average atomic mass will be heavily influenced by that isotope’s mass.
2. Proximity of Other Isotopes: Although Silicon-29 and Silicon-30 have higher masses, their combined abundances still only account for [tex]\(4.70\% + 3.09\% = 7.79\%\)[/tex] of the total. This relatively low percentage means they have a smaller impact on the average atomic mass.
### Conclusion
Given the high abundance of Silicon-28 (27.98 amu and 92.21\%), we can predict that the average atomic mass for silicon would be closer to the atomic mass of Silicon-28 than to Silicon-29 or Silicon-30.
### Calculation of Average Atomic Mass and Closest Isotope
We calculate the weighted average atomic mass through the following steps:
1. Convert percent abundances to fractions:
[tex]\[ \text{Silicon-28: } \frac{92.21}{100} = 0.9221 \][/tex]
[tex]\[ \text{Silicon-29: } \frac{4.70}{100} = 0.0470 \][/tex]
[tex]\[ \text{Silicon-30: } \frac{3.09}{100} = 0.0309 \][/tex]
2. Weighted average formula:
[tex]\[ \text{Average Atomic Mass} = (27.98 \times 0.9221) + (28.98 \times 0.0470) + (29.97 \times 0.0309) \][/tex]
Using these weights,
[tex]\[ \text{Average Atomic Mass} = (27.98 \times 0.9221) + (28.98 \times 0.0470) + (29.97 \times 0.0309) = 28.08849 \, \text{amu} \][/tex]
3. Determining the Closest Isotope:
By comparing the average atomic mass (28.08849 amu) with the atomic masses of the isotopes (27.98, 28.98, 29.97):
[tex]\[ |27.98 - 28.08849| = 0.10849 \][/tex]
[tex]\[ |28.98 - 28.08849| = 0.89151 \][/tex]
[tex]\[ |29.97 - 28.08849| = 1.88151 \][/tex]
The smallest absolute difference is with Silicon-28, meaning it is the closest isotope.
### Final Answer
- The average atomic mass of silicon is approximately 28.08849 amu.
- The closest isotope to this average atomic mass is Silicon-28 with an atomic mass of 27.98 amu.
Given the three naturally occurring isotopes of silicon and their corresponding atomic masses and percent abundances, let’s analyze where the average atomic mass of silicon would lie:
[tex]\[ \begin{array}{|l|r|r|} \hline \text{Isotope} & \text{Atomic Mass (amu)} & \text{Percent Abundance} \\ \hline \text{Silicon-28} & 27.98 & 92.21\% \\ \hline \text{Silicon-29} & 28.98 & 4.70\% \\ \hline \text{Silicon-30} & 29.97 & 3.09\% \\ \hline \end{array} \][/tex]
First, let's note the percent abundances. Silicon-28 has a significantly higher abundance (92.21\%) compared to Silicon-29 (4.70\%) and Silicon-30 (3.09\%).
### Prediction Reasoning
1. Dominant Isotope: Silicon-28 is the most abundant isotope by far. When one isotope is present in a much greater percentage than the others, the weighted average atomic mass will be heavily influenced by that isotope’s mass.
2. Proximity of Other Isotopes: Although Silicon-29 and Silicon-30 have higher masses, their combined abundances still only account for [tex]\(4.70\% + 3.09\% = 7.79\%\)[/tex] of the total. This relatively low percentage means they have a smaller impact on the average atomic mass.
### Conclusion
Given the high abundance of Silicon-28 (27.98 amu and 92.21\%), we can predict that the average atomic mass for silicon would be closer to the atomic mass of Silicon-28 than to Silicon-29 or Silicon-30.
### Calculation of Average Atomic Mass and Closest Isotope
We calculate the weighted average atomic mass through the following steps:
1. Convert percent abundances to fractions:
[tex]\[ \text{Silicon-28: } \frac{92.21}{100} = 0.9221 \][/tex]
[tex]\[ \text{Silicon-29: } \frac{4.70}{100} = 0.0470 \][/tex]
[tex]\[ \text{Silicon-30: } \frac{3.09}{100} = 0.0309 \][/tex]
2. Weighted average formula:
[tex]\[ \text{Average Atomic Mass} = (27.98 \times 0.9221) + (28.98 \times 0.0470) + (29.97 \times 0.0309) \][/tex]
Using these weights,
[tex]\[ \text{Average Atomic Mass} = (27.98 \times 0.9221) + (28.98 \times 0.0470) + (29.97 \times 0.0309) = 28.08849 \, \text{amu} \][/tex]
3. Determining the Closest Isotope:
By comparing the average atomic mass (28.08849 amu) with the atomic masses of the isotopes (27.98, 28.98, 29.97):
[tex]\[ |27.98 - 28.08849| = 0.10849 \][/tex]
[tex]\[ |28.98 - 28.08849| = 0.89151 \][/tex]
[tex]\[ |29.97 - 28.08849| = 1.88151 \][/tex]
The smallest absolute difference is with Silicon-28, meaning it is the closest isotope.
### Final Answer
- The average atomic mass of silicon is approximately 28.08849 amu.
- The closest isotope to this average atomic mass is Silicon-28 with an atomic mass of 27.98 amu.
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