- Convert the percentage abundances of each isotope into decimal form by dividing each by 100.
- Multiply each isotope's mass by its decimal abundance to find the weighted contribution of each.
- Calculate the sum of these weighted contributions to obtain the average atomic mass.
- The average atomic mass of the element is \(\boxed{28.1 \text{ amu}}\).
Explanation
- Analysis and Data Conversion
The problem requires calculating the average atomic mass of an element based on its isotopic composition. The average atomic mass is calculated as the weighted average of the masses of its naturally occurring isotopes, where the weight for each isotope is its fractional abundance. We are given the following data:
- Isotope 1: Mass \(M_1 = 28.0 \text{ amu}\), Abundance \(A_1 = 92.230\%\)
- Isotope 2: Mass \(M_2 = 29.0 \text{ amu}\), Abundance \(A_2 = 4.683\%\)
- Isotope 3: Mass \(M_3 = 30.0 \text{ amu}\), Abundance \(A_3 = 3.087\%\)
To perform the calculation, we first convert the percentage abundances into decimal form by dividing each by \(100\).
- \(A_1 = 0.92230\)
- \(A_2 = 0.04683\)
- \(A_3 = 0.03087\)
The sum of these abundances is \(0.92230 + 0.04683 + 0.03087 = 1.00000\), confirming the data represents the full isotopic distribution.""",number:1,title:
- Calculation of Average Atomic Mass
The average atomic mass is given by the formula:
\[\text{Average Atomic Mass} = \sum (M_i \times A_i) = (M_1 \times A_1) + (M_2 \times A_2) + (M_3 \times A_3)\]
Substituting the values:
\[\text{Average Atomic Mass} = (28.0 \times 0.92230) + (29.0 \times 0.04683) + (30.0 \times 0.03087)\]
Calculating each product:
- \(28.0 \times 0.92230 = 25.8244\)
- \(29.0 \times 0.04683 = 1.35807\)
- \(30.0 \times 0.03087 = 0.9261\)
Summing these values:
\[25.8244 + 1.35807 + 0.9261 = 28.10857 \text{ amu}\]
Rounding to a reasonable number of decimal places matching the provided options, we obtain \(28.1 \text{ amu}\). This corresponds to the first option.""",number:2,title:
- Final Answer Selection
Comparing our calculated result of \(28.10857 \text{ amu}\) with the given options, we find that the average atomic mass of the element is \(28.1 \text{ amu}\). Thus, the correct choice is the first one. \(\boxed{28.1 \text{ amu}}\).""",number:3,title:
Examples
The concept of calculating average atomic mass is essential in chemistry for determining the macroscopic properties of elements. For instance, if you are analyzing a sample of silicon (which consists of isotopes Si-28, Si-29, and Si-30), knowing the average atomic mass allows you to calculate the molar mass, which is a fundamental value for stoichiometry in chemical reactions. By applying weighted averages, scientists can accurately predict how substances will behave in various chemical environments.