Calculate the atomic mass for iron given the data for its natural isotopes. Fe-54: 53.940 amu 5.82% Fe-56: 55.935 amu 91.66% Fe-57: 56.935 amu 2.19% Fe-58: 57.933 amu 0.33%
The Correct Answer and Explanation is:
To calculate the atomic mass of iron (Fe), we need to use the weighted average formula for isotopes. The atomic mass is calculated by multiplying the atomic mass of each isotope by its relative abundance (as a fraction), and then summing these products.
Here is the data for the isotopes of iron:
- Fe-54: 53.940 amu with an abundance of 5.82%
- Fe-56: 55.935 amu with an abundance of 91.66%
- Fe-57: 56.935 amu with an abundance of 2.19%
- Fe-58: 57.933 amu with an abundance of 0.33%
First, convert the percentage abundances to decimal form:
- Fe-54: 5.82% = 0.0582
- Fe-56: 91.66% = 0.9166
- Fe-57: 2.19% = 0.0219
- Fe-58: 0.33% = 0.0033
Now, calculate the weighted sum:Atomic mass=(53.940 amu×0.0582)+(55.935 amu×0.9166)+(56.935 amu×0.0219)+(57.933 amu×0.0033)\text{Atomic mass} = (53.940 \, \text{amu} \times 0.0582) + (55.935 \, \text{amu} \times 0.9166) + (56.935 \, \text{amu} \times 0.0219) + (57.933 \, \text{amu} \times 0.0033)Atomic mass=(53.940amu×0.0582)+(55.935amu×0.9166)+(56.935amu×0.0219)+(57.933amu×0.0033)
Perform the multiplication for each term:53.940×0.0582=3.13753.940 \times 0.0582 = 3.13753.940×0.0582=3.13755.935×0.9166=51.31955.935 \times 0.9166 = 51.31955.935×0.9166=51.31956.935×0.0219=1.24656.935 \times 0.0219 = 1.24656.935×0.0219=1.24657.933×0.0033=0.19157.933 \times 0.0033 = 0.19157.933×0.0033=0.191
Add these values together:3.137+51.319+1.246+0.191=55.893 amu3.137 + 51.319 + 1.246 + 0.191 = 55.893 \, \text{amu}3.137+51.319+1.246+0.191=55.893amu
Thus, the atomic mass of iron is approximately 55.89 amu.
Explanation:
This weighted average accounts for both the mass and the relative abundance of each isotope of iron. Since the most abundant isotope (Fe-56) contributes the most to the atomic mass, it has the greatest influence on the result. The atomic mass calculated here is very close to the standard atomic mass of iron, which is 55.845 amu.
