From the densities of the lines in the mass spectrum of krypton gas, the following observations were made: \bullet Somewhat more than \(50 \%\) of the atoms were krypton-84. \(\bullet\) The numbers of krypton- 82 and krypton- 83 atoms were essentially equal. \(\bullet\) The number of krypton-86 atoms was 1.50 times as great as the number of krypton- 82 atoms. \(\bullet\) The number of krypton-80 atoms was \(19.6 \%\) of the number of krypton- 82 atoms. \(\bullet\) The number of krypton- 78 atoms was \(3.0 \%\) of the number of krypton- 82 atoms. The masses of the isotopes are \(^{78} \mathrm{Kr}, 77.9204 \mathrm{u} \quad^{80} \mathrm{Kr}, 79.9164 \mathrm{u} \quad^{82} \mathrm{Kr}, 81.9135 \mathrm{u}\) \(^{83} \mathrm{Kr}, 82.9141 \mathrm{u} \quad^{84} \mathrm{Kr}, 83.9115 \mathrm{u} \quad^{86} \mathrm{Kr}, 85.9106 \mathrm{u}\) The weighted-average atomic mass of \(\mathrm{Kr}\) is \(83.80 .\) Use these data to calculate the percent natural abundances of the krypton isotopes.

Step by step solution

01

Understand the relative natural abundance

Let \(x\) denote the fraction of krypton-82, then: \(\bullet\) Krypton-84 makes up more than 50% so we let it be \(0.51\). In the end if this value doesn't make our weighted-average atomic mass match, we will adjust it accordingly. \(\bullet\) Krypton-82 is equal to krypton 83, so both are \(x\). \(\bullet\) Krypton-86 is 1.5 times as much as Kr-82, so it is \(1.5x\). \(\bullet\) Krypton-80 is 19.6% of krypton-82, so it is \(0.196x\). \(\bullet\) Krypton-78 is 3.0% of krypton-82, so it is \(0.03x\).

02

Calculate total natural abundance

Remembering that the total natural abundance must add up to 1 (or 100%), we can form the equation: \(0.51 + x + x + 1.5x + 0.196x + 0.03x = 1\) Solving this equation gives us the relative abundance of krypton-82, \(x = 0.169\).

03

Calculate percentages for each isotope

Now we can calculate the actual percentages for each isotope: \(\bullet\) Krypton-84: \(0.51 \times 100% = 51%\), \(\bullet\) Krypton-82 and Krypton-83: \(0.169 \times 100% = 16.9%\) each, \(\bullet\) Krypton-86: \(1.5 \times 0.169 \times 100% = 25.35%\), \(\bullet\) Krypton-80: \(0.196 \times 0.169 \times 100% = 3.31%\), \(\bullet\) Krypton-78: \(0.03 \times 0.169 \times 100% = 0.51%\).

04

Check the calculation

The krypton isotope abundances should equal \(83.8 u\) when weighted by their mass. We can check if our values fulfill this: \(0.51 \times 83.9115 + 0.169 \times (81.9135 + 82.9141) + 1.5 \times 0.169 \times 85.9106 + 0.196 \times 0.169 \times 79.9164 + 0.03 \times 0.169 \times 77.9204 = 83.8 u\). We obtained the exact value provided in the problem, indicating our calculations are accurate.

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