QuestionCalculate the average mass of silver isotopes: ${ }^{107} \mathrm{Ag}$ (106.90509 amu, 51.84\%) and ${ }^{109} \mathrm{Ag}$ (108.905 amu, 48.16\%).

Studdy Solution

STEP 1

Assumptions1. The mass of ${ }^{107} \mathrm{Ag}$ isotope is106.90509 amu. The abundance of ${ }^{107} \mathrm{Ag}$ isotope is51.84%
3. The mass of ${ }^{109} \mathrm{Ag}$ isotope is108.905 amu4. The abundance of ${ }^{109} \mathrm{Ag}$ isotope is48.16%
5. The average mass of a sample of isotopes can be calculated by multiplying the mass of each isotope by its relative abundance and then summing these products.

STEP 2

First, we need to convert the percentage abundance of each isotope to a decimal. We can do this by dividing the percentage by100.
$Decimal\, abundance = Percentage\, abundance /100$

STEP 3

Now, plug in the given values for the abundance of ${ }^{107} \mathrm{Ag}$ and ${ }^{109} \mathrm{Ag}$ to calculate the decimal abundance.
$Decimal\, abundance\, of\, { }^{107} \mathrm{Ag} =51.84 /100$ $Decimal\, abundance\, of\, { }^{109} \mathrm{Ag} =48.16 /100$

STEP 4

Calculate the decimal abundance of each isotope.
$Decimal\, abundance\, of\, { }^{107} \mathrm{Ag} =51.84 /100 =0.5184$ $Decimal\, abundance\, of\, { }^{109} \mathrm{Ag} =48.16 /100 =0.4816$

STEP 5

Next, we need to calculate the contribution of each isotope to the average mass. We can do this by multiplying the mass of each isotope by its decimal abundance.
$Contribution\, to\, average\, mass = Mass\, of\, isotope \\times Decimal\, abundance$

STEP 6

Now, plug in the given values for the mass and decimal abundance of ${ }^{107} \mathrm{Ag}$ and ${ }^{109} \mathrm{Ag}$ to calculate the contribution of each isotope to the average mass.
$Contribution\, of\, { }^{107} \mathrm{Ag} =106.90509 \, amu \\times0.5184$ $Contribution\, of\, { }^{109} \mathrm{Ag} =108.905 \, amu \\times0.4816$

STEP 7

Calculate the contribution of each isotope to the average mass.
$Contribution\, of\, { }^{107} \mathrm{Ag} =106.90509 \, amu \\times0.5184 =55.419 \, amu$ $Contribution\, of\, { }^{109} \mathrm{Ag} =108.905 \, amu \\times0.4816 =52.459 \, amu$

STEP 8

Finally, we can calculate the average mass of the sample by adding the contributions of each isotope.
$Average\, mass = Contribution\, of\, { }^{107} \mathrm{Ag} + Contribution\, of\, { }^{109} \mathrm{Ag}$

STEP 9

Plug in the values for the contributions of ${ }^{107} \mathrm{Ag}$ and ${ }^{109} \mathrm{Ag}$ to calculate the average mass.
$Average\, mass =55.419 \, amu +52.459 \, amu$

STEP 10

Calculate the average mass of the sample.
$Average\, mass =55.419 \, amu +52.459 \, amu =107.878 \, amu$ The average mass of the natural sample of silver is107.878 amu.

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