To find the atomic mass of hafnium based on the given distribution of isotopes, we can calculate the weighted average of the isotopic masses using their relative abundances.
Here’s how you can do it:
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List the isotopes and their relative abundances:
- \( \text{Mass} = 170 \) with \( 5 \) atoms
- \( \text{Mass} = 177 \) with \( 19 \) atoms
- \( \text{Mass} = 178 \) with \( 27 \) atoms
- \( \text{Mass} = 179 \) with \( 14 \) atoms
- \( \text{Mass} = 180 \) with \( 35 \) atoms
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Calculate total number of atoms: \[ 5 + 19 + 27 + 14 + 35 = 100 \text{ atoms} \]
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Calculate the weighted average: \[ \text{Average atomic mass} = \frac{(170 \times 5) + (177 \times 19) + (178 \times 27) + (179 \times 14) + (180 \times 35)}{100} \]
Now, let's compute each term:
- For mass \( 170 \): \( 170 \times 5 = 850 \)
- For mass \( 177 \): \( 177 \times 19 = 3363 \)
- For mass \( 178 \): \( 178 \times 27 = 4794 \)
- For mass \( 179 \): \( 179 \times 14 = 2506 \)
- For mass \( 180 \): \( 180 \times 35 = 6300 \)
Now sum these values: \[ 850 + 3363 + 4794 + 2506 + 6300 = 17913 \]
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Divide by total number of atoms (100): \[ \text{Average atomic mass} = \frac{17913}{100} = 179.13 \]
Thus, the atomic mass of hafnium based on the given isotopic distribution is approximately 179.13 amu.