To calculate the average atomic mass of a metal using data from a mass spectrum, follow these detailed steps:
1. Identify the isotopes and their respective masses: From the problem, we have two isotopes for the metal:
- Isotope 1: [tex]\( ^{63} \text{Cu} \)[/tex]
- Isotope 2: [tex]\( ^{65} \text{Cu} \)[/tex]
2. Obtain the atomic mass of each isotope: From the given data:
- The mass of [tex]\( ^{63} \text{Cu} \)[/tex] is 62.9296 amu.
- The mass of [tex]\( ^{65} \text{Cu} \)[/tex] is 64.9278 amu.
3. Find the relative abundance of each isotope: The relative abundances given are:
- Relative abundance of [tex]\( ^{63} \text{Cu} \)[/tex]: 69.15%
- Convert percent to decimal: 69.15% = 0.6915
- Relative abundance of [tex]\( ^{65} \text{Cu} \)[/tex]: 30.85%
- Convert percent to decimal: 30.85% = 0.3085
4. Calculate the weighted average atomic mass: This is done using the formula for the average atomic mass, which is the sum of the products of each isotope's mass and its relative abundance.
[tex]\[
\text{Average atomic mass} = (\text{mass of } ^{63} \text{Cu} \times \text{relative abundance of } ^{63} \text{Cu}) + (\text{mass of } ^{65} \text{Cu} \times \text{relative abundance of } ^{65} \text{Cu})
\][/tex]
Substitute the given values:
[tex]\[
\text{Average atomic mass} = (62.9296 \times 0.6915) + (64.9278 \times 0.3085)
\][/tex]
5. Calculate each component separately:
- For [tex]\( ^{63} \text{Cu} \)[/tex]:
[tex]\[
62.9296 \times 0.6915 = 43.5420724
\][/tex]
- For [tex]\( ^{65} \text{Cu} \)[/tex]:
[tex]\[
64.9278 \times 0.3085 = 20.0039723
\][/tex]
6. Sum the components to get the average atomic mass:
[tex]\[
\text{Average atomic mass} = 43.5420724 + 20.0039723 = 63.5460447 \text{ amu}
\][/tex]
7. Identify the element: Now compare the calculated average atomic mass with values in the periodic table. The closest element to the calculated average atomic mass of 63.5460447 amu is Copper (Cu).
Therefore, the average atomic mass of the metal is 63.5460447 amu, and the element is identified as Copper (Cu).
