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semiconductors are materials electrical conductivity between conductor and insulators and are the foundation of modern electronics. silicon is one element that exhibit semiconducting properties. it has three stable isotopes which are Si-28,Si-29,I-30 with isotopic masses 27.9769 amu, 28.9765 amu, 29.9738 respectively. the most abundant isotope has a percent abundance of 92.23%. if the average atomic mass is 28.0854 amu, what are the percent abundances of each of the isotopes? ​

Sagot :

Given:

[tex]\text{isotopic mass}_{Si-28} = \text{27.9769 amu}[/tex]

[tex]\text{isotopic mass}_{Si-29} = \text{28.9765 amu}[/tex]

[tex]\text{isotopic mass}_{Si-30} = \text{29.9738 amu}[/tex]

[tex]\text{percent abundance}_{Si-28} = 92.23\%[/tex]

[tex]\text{average atomic mass = 28.0854 amu}[/tex]

Unknown:

[tex]\text{percent abundance}_{Si-29}[/tex]

[tex]\text{percent abundance}_{Si-30}[/tex]

Solution:

Step 1: Represent a variable for the unknown percent abundances.

[tex]\text{percent abundance}_{Si-29} = x[/tex]

[tex]\text{percent abundance}_{Si-30} = 100 - (92.23 + x) = 7.77 - x[/tex]

Step 2: Solve for the value of x.

Note: The average atomic mass is multiplied by 100 because the abundances are expressed in percent.

[tex]\text{average atomic mass} = (\text{isotopic mass}_{Si-28})(\text{percent abundance}_{Si-28}) + (\text{isotopic mass}_{Si-29})(\text{percent abundance}_{Si-29}) + (\text{isotopic mass}_{Si-30})(\text{percent abundance}_{Si-30})[/tex]

[tex]28.0854 × 100 = (27.9769)(92.23) + (28.9765)(x) + (29.9738)(7.77 - x)[/tex]

[tex]2808.54 = 2580.3095 + 28.9765x + 232.8964 - 29.9738x[/tex]

[tex]29.9738x - 28.9765x = 2580.3095 + 232.8964 - 2808.54[/tex]

[tex]0.9973x = 4.6659[/tex]

[tex]x = \frac{4.6659}{0.9973}[/tex]

[tex]x = 4.68\%[/tex]

Step 3: Determine the values of unknown percent abundances.

For Si-29

[tex]\text{percent abundance}_{Si-29} = x[/tex]

[tex]\boxed{\text{percent abundance}_{Si-29} = 4.68\%}[/tex]

For Si-30

[tex]\text{percent abundance}_{Si-30} = 7.77 - x[/tex]

[tex]\text{percent abundance}_{Si-30} = 7.77 - 4.68[/tex]

[tex]\boxed{\text{percent abundance}_{Si-30} = 3.09\%}[/tex]

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