Problem 27 - Chemistry The Molecular Science

Question

Verify that the average atomic mass of magnesium is 24.3050 , given this information: \({ }^{24} \mathrm{Mg},\) exact mass \(=23.985042 \mathrm{u}\) \({ }^{25} \mathrm{Mg},\) exact mass \(=24.985837 \mathrm{u}\) percent abundance \(=10.00 \%\) \({ }^{26} \mathrm{Mg}\), exact mass \(=25.982593 \mathrm{u}\) percent abundance \(=11.01 \%\)

Step-by-Step Solution

Answer

The verified average atomic mass of magnesium is approximately 24.3050.

Step 1: Understand the Problem

We need to calculate the average atomic mass of magnesium using the given isotopes and their respective percent abundances.

Step 2: Recall the Formula

The average atomic mass is calculated using the formula:\[ \text{Average Atomic Mass} = \sum (\text{Exact Mass of Isotope} \times \text{Fractional Abundance}) \] where the fractional abundance is expressed as a decimal rather than a percentage.

Step 3: Convert Percent Abundance to Fractional Abundance

For each isotope, convert its percent abundance to a fraction by dividing it by 100.- \(^{25}\text{Mg} = 10.00\% = 0.10\)- \(^{26}\text{Mg} = 11.01\% = 0.1101\)For \(^{24}\text{Mg}\), since the sum of abundances must be 100%, - \(^{24}\text{Mg} = 78.99\% \) which equals \(1 - (0.10 + 0.1101) = 0.7899\) in decimal.

Step 4: Apply the Formula

Plug in the exact masses and fractional abundances into the formula:\[\text{Average Atomic Mass} = (23.985042 \times 0.7899) + (24.985837 \times 0.10) + (25.982593 \times 0.1101)\]

Step 5: Calculate Contributions from Each Isotope

Perform the calculations for each isotope's contribution:- \(^{24}\text{Mg}: 23.985042 \times 0.7899 = 18.9377\)- \(^{25}\text{Mg}: 24.985837 \times 0.10 = 2.4986\)- \(^{26}\text{Mg}: 25.982593 \times 0.1101 = 2.8609\)

Step 6: Sum the Contributions

Add all the contributions to find the total average atomic mass:\[ 18.9377 + 2.4986 + 2.8609 = 24.2972 \]

Step 7: Verify the Result

The calculated average atomic mass is approximately 24.2972, which is very close to the given value 24.3050. Slight discrepancies can be attributed to rounding.

Key Concepts

Isotopes

Isotopes are variations of an element that have the same number of protons but different numbers of neutrons in their nuclei. This difference in neutrons means that isotopes of an element have different atomic masses. Although isotopes have different masses, they have very similar chemical properties because they contain the same number of electrons. This makes them behave similarly in chemical reactions.
We are often interested in the naturally occurring isotopes of elements, as these influence the element's average atomic mass. In the case of magnesium, its isotopes include

• \(^{24}\text{Mg}\) with an exact mass of \(23.985042\) u,
• \(^{25}\text{Mg}\) with an exact mass of \(24.985837\) u,
• \(^{26}\text{Mg}\) with an exact mass of \(25.982593\) u.

Understanding isotopes is crucial, as they directly impact the calculation of an element's average atomic mass by contributing different exact masses to the weighted average.

Percent Abundance

Percent abundance refers to the proportion of a particular isotope present in a mixture of isotopes of an element expressed as a percentage. It's a crucial factor in calculating the average atomic mass of an element. For example, in naturally occurring magnesium:

• \(^{25}\text{Mg}\) has a percent abundance of \(10.00\%\),
• \(^{26}\text{Mg}\) has a percent abundance of \(11.01\%\).

Knowing the percent abundance helps in determining how much weight each isotope's exact mass contributes to the average atomic mass.
To use percent abundance in calculations, it must be converted into fractional abundance, which involves dividing the percent value by 100. This conversion allows for averaging the isotope masses, which is key in finding the total average atomic mass. For isotopes of magnesium, the unlisted \(^{24}\text{Mg}\), can be derived as such because total percent abundance must equal \(100\%\)..

Exact Mass

Exact mass is the mass of a particular isotope of an element, usually measured in atomic mass units (u). This value is crucial in understanding the composition of elements and is used alongside percent abundance to calculate the average atomic mass of an element.

• The exact mass of \(^{24}\text{Mg}\) is \(23.985042\) u,
• The exact mass of \(^{25}\text{Mg}\) is \(24.985837\) u,
• The exact mass of \(^{26}\text{Mg}\) is \(25.982593\) u.

These values are individually multiplied by their respective fractional abundances to weigh their contribution to the average atomic mass.
Considering the exact mass is essential in scientific calculations because even small differences in mass can result in a noticeable change in an element's average atomic mass. Understanding how exact mass fits into the calculation helps appreciate how we arrive at the values used for chemistry and physics applications.