Question

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.

Short answer

So, the natural abundances of the krypton isotopes are krypton-84: 51%, krypton-82 and krypton-83: 16.9% each, krypton-86: 25.35%, krypton-80: 3.31%, and krypton-78: 0.51%.

Step-by-step solution

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 \(0.03x\).

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, \(x = 0.169\).

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: \(0.03 \times 0.169 \times 100% = 0.51%\).

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.

Key concepts

krypton isotopes

Krypton is a noble gas element that has several different isotopes. Isotopes are atoms of the same element that contain equal numbers of protons but different numbers of neutrons, resulting in a different atomic mass for each isotope. The isotopes of krypton, specifically, are known by their mass numbers, such as krypton-78, krypton-80, krypton-82, krypton-83, krypton-84, and krypton-86. Each krypton isotope has its own unique mass, which contributes to how scientists calculate the element's atomic weight.

Krypton-84 has been noted to make up a significant portion of natural krypton, comprising over 50% of the total sample. Meanwhile, krypton-82 and krypton-83 have approximately equal natural abundances. Finding the natural abundance of these isotopes requires understanding the relationship between their masses and their appearances in nature. These abundances are crucial not only for theoretical calculations but also for practical applications, such as in lighting and lasers, where specific krypton isotopes can be more desirable.

mass spectrum

A mass spectrum is a graphical representation that shows the distribution of ions by mass-to-charge ratio. This is often used to determine the composition of an element or compound, allowing for the identification and quantification of its different isotopes. By analyzing a mass spectrum, one can determine the relative abundance of isotopes based on the intensity of the peaks.

In the case of krypton gas, a mass spectrum reveals several distinct lines or peaks, each representing a different isotope of krypton. The position and height of these peaks indicate the mass and the relative abundance of the isotopes, respectively. By examining these spectral lines, scientists can identify which isotopes are present and in what proportions. This helps in calculating the average atomic mass of krypton.

weighted-average atomic mass

The concept of weighted-average atomic mass is essential in chemistry when it comes to elements with multiple isotopes. This type of average takes into account the relative abundance of each isotope. It is calculated by multiplying the mass of each isotope by its respective natural abundance and summing all these values. In the case of krypton, isotopes like krypton-84, krypton-82, krypton-83, and others all contribute to the average atomic mass.

The formula for calculating the weighted-average atomic mass can be expressed as:\[\text{Weighted Average Atomic Mass} = \sum (\text{fractional abundance} \times \text{isotopic mass})\]In this formula, each isotope's mass is weighted by how common it is in nature. For krypton, this calculated average should ideally match the given atomic mass of 83.80 u, validating the isotopic distribution and abundance figures derived from the mass spectrum.

natural abundance calculation

Calculating the natural abundance of an isotope involves determining what fraction or percentage of the element's atoms consist of that isotope. This calculation begins with the observations made from experiments or spectral data, such as the proportions of isotopes found in a mass spectrum.

For krypton, the initial step in determining natural abundance involves setting up a series of equations based on the given relationships between isotopes. Using the equations:- Krypton-84 is over 50%, set at 51% to start.- Krypton-82 equals krypton-83, each represented by \(x\).- Krypton-86 is 1.5 times krypton-82.- Krypton-80 is 19.6% of krypton-82.- Krypton-78 is 3.0% of krypton-82.These relationships help establish a comprehensive equation, solved with total abundances amounting to 100%. The solution to these equations provides precise percentages necessary for weighted-average atomic mass calculations, ensuring the results match observed atomic weights.