Naturally occurring silicon exists as 3 isotopes: 28Si, 29Si, and 30Si, whose atomic masses are 27.9769, 28.9865, and 29.9838, respectively. The most abundant isotope is 28Si, which accounts for 92.23% of naturally occurring silicon. Given that the average atomic mass of silicon is 28.0855 amu, calculate the percentages of 29Si and 30Si in nature.
Please explain step by step
Well, let's start by calculating the atomic mass of each isotope using its relative abundance and atomic mass.
The atomic mass of an isotope is calculated by multiplying its abundance (as a decimal) by its atomic mass, and then summing up these values for all the isotopes.
We know that the most abundant isotope, 28Si, accounts for 92.23% of naturally occurring silicon. Therefore, the abundance of 28Si is 0.9223, and its atomic mass is 27.9769 amu.
Next, we have 29Si, whose abundance we don't know yet, but we can represent it as x. Its atomic mass is 28.9865 amu.
Finally, we have 30Si, with an abundance of 1 - (0.9223 + x), as it is the remaining percentage. Its atomic mass is 29.9838 amu.
Now, we can set up an equation to represent the average atomic mass of silicon:
(0.9223 * 27.9769) + (x * 28.9865) + (1 - (0.9223 + x)) * 29.9838 = 28.0855
Simplifying this equation, we have:
25.81452437 + 28.9865x + 29.9838 - 29.9838x = 28.0855
Combining the like terms, we get:
54.79832437 - x = 28.0855
Subtracting 54.79832437 from both sides:
-x = -26.71282437
Dividing both sides by -1, we find that x = 26.71282437.
Therefore, the abundance of 29Si is approximately 26.71%.
To calculate the abundance of 30Si, we subtract the abundances of 28Si and 29Si from 100%:
Abundance of 30Si = 100% - (92.23% + 26.71%)
= 100% - 118.94%
= -18.94%
Oops! Looks like we made a mistake in our calculations. The calculation for the abundance of 30Si is coming out to be negative, which is not possible as it represents a percentage.
So, it seems like there might be an error in the given data or calculation. I apologize for the inconvenience caused.
To calculate the percentages of 29Si and 30Si in nature, we need to use the information provided about the abundance of the three isotopes and their atomic masses.
Step 1: Determine the mass of each isotope.
- The atomic masses are given as follows:
- Mass of 28Si = 27.9769 amu
- Mass of 29Si = 28.9865 amu
- Mass of 30Si = 29.9838 amu
Step 2: Calculate the abundance of 28Si.
- The question states that 28Si accounts for 92.23% of naturally occurring silicon.
- To convert this percentage to a decimal, divide by 100: 92.23 / 100 = 0.9223
Step 3: Calculate the abundance of the remaining isotopes (29Si and 30Si).
- Since the percentages of all isotopes must add up to 100%, we can calculate the remaining percentage as follows:
Remaining percentage = 100% - abundance of 28Si
Remaining percentage = 100% - 92.23% = 7.77%
Step 4: Calculate the abundance of 29Si and 30Si.
- To find the percentage of each isotope, divide the remaining percentage equally among them.
- Since there are two remaining isotopes (29Si and 30Si), divide the remaining percentage by 2:
Abundance of each isotope = Remaining percentage divided by 2
Abundance of each isotope = 7.77% / 2 = 3.885%
Step 5: Convert the abundance of 29Si and 30Si back to decimal form.
- To convert the percentages back to decimals, divide by 100:
Abundance of 29Si = 3.885% / 100 = 0.03885
Abundance of 30Si = 3.885% / 100 = 0.03885
Step 6: Calculate the atomic mass of silicon.
- The atomic mass of silicon is given as 28.0855 amu.
- Since the atomic mass is calculated as the weighted average of the isotopes, we can use the following formula:
(Mass of isotope 1 × abundance of isotope 1) + (Mass of isotope 2 × abundance of isotope 2) + ... = Atomic mass
(27.9769 amu × 0.9223) + (28.9865 amu × 0.03885) + (29.9838 amu × 0.03885) = 28.0855 amu
(25.78422587) + (1.125116525) + (1.16362273) = 28.0855 amu
28.073965125 ≈ 28.0855 amu
Therefore, the percentages of 29Si and 30Si in nature are approximately 3.885%.
Answer?
28Si = 92.23% and 27.9769 amu
29Si = X% and 28.9865 amu
30Si = 100%-92.23%-X% = 7.77%-X and 29.9838 amu
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(0.9223*27.9769)+(X)(28.9865)+(0.0777)(29.9838 = 28.0955
Solve for X and 7.77-X