Alright, let's determine the atomic mass of the metal and identify the element based on its isotopic abundances and isotopic masses.
### Step-by-Step Solution
1. Isotopic Abundances and Masses:
We are provided with the relative abundances and isotopic masses:
- Relative Abundances: [tex]\(0.06917, 0.07854, 0.11223, 0.24113, 0.39861, 0.10032\)[/tex]
- Isotopic Masses (in atomic mass units - amu): [tex]\(62.930, 64.927, 66.927, 67.928, 68.926, 70.925\)[/tex]
2. Calculating the Weighted Average:
To calculate the atomic mass of the element, we take the weighted average of its isotopes. The formula is:
[tex]\[
\text{Atomic Mass} = \sum (\text{Relative Abundance of Isotope} \times \text{Isotopic Mass})
\][/tex]
Let's compute this step by step:
- For the first isotope:
[tex]\[
0.06917 \times 62.930 = 4.3513931
\][/tex]
- For the second isotope:
[tex]\[
0.07854 \times 64.927 = 5.10473238
\][/tex]
- For the third isotope:
[tex]\[
0.11223 \times 66.927 = 7.51620921
\][/tex]
- For the fourth isotope:
[tex]\[
0.24113 \times 67.928 = 16.37940464
\][/tex]
- For the fifth isotope:
[tex]\[
0.39861 \times 68.926 = 27.49237286
\][/tex]
- For the sixth isotope:
[tex]\[
0.10032 \times 70.925 = 7.08860624
\][/tex]
3. Summing the Contributions:
Now, add all the individual contributions together to get the total atomic mass:
[tex]\[
4.3513931 + 5.10473238 + 7.51620921 + 16.37940464 + 27.49237286 + 7.08860624 = 67.93271939
\][/tex]
Thus, the calculated atomic mass of the element is approximately [tex]\(67.93 \ \text{amu}\)[/tex].
4. Identifying the Element:
Now we need to match this calculated atomic mass with known atomic masses from the periodic table. Checking the periodic table, an atomic mass of approximately [tex]\(67.93 \ \text{amu}\)[/tex] closely matches the element Gallium (Ga), which has an average atomic mass around [tex]\(69.723 \ \text{amu}\)[/tex].
Hence, based on the provided isotopic data and the calculated average atomic mass, the element is identified as Gallium (Ga).
