Question

Atomic chlorine has two naturally occurring isotopes, \(^{35} \mathrm{Cl}\) and \(^{37} \mathrm{Cl}\). If the molar abundance of these isotopes are \(75.4 \%\) and \(24.6 \%,\) respectively, what fraction of a mole of molecular chlorine \(\left(\mathrm{Cl}_{2}\right)\) will have one of each isotope? What fraction will contain just the \(^{35} \mathrm{Cl}\) isotope?

Short answer

The fraction of \(\mathrm{Cl}_{2}\) molecules with one \(^{35} \mathrm{Cl}\) isotope and one \(^{37} \mathrm{Cl}\) isotope is 0.372 or 37.2%. The fraction containing just the \(^{35} \mathrm{Cl}\) isotope is 0.568 or 56.8%.

Step-by-step solution

Convert molar abundances to probabilities

To calculate the probability of an atom being a certain isotope, we can divide the molar abundance of the isotope by the total molar abundance (which is always 100%). For \(^{35} \mathrm{Cl}\): Probability = \(\frac{75.4}{100} = 0.754\) For \(^{37} \mathrm{Cl}\): Probability = \(\frac{24.6}{100} = 0.246\)

Calculate probabilities for different Cl₂ combinations

There are three possible combinations of isotopes in a \(\mathrm{Cl}_{2}\) molecule: 1. Two \(^{35} \mathrm{Cl}\) atoms. 2. Two \(^{37} \mathrm{Cl}\) atoms. 3. One \(^{35} \mathrm{Cl}\) atom and one \(^{37} \mathrm{Cl}\) atom. Let's now calculate the probabilities for each combination. 1. Probability of Two \(^{35} \mathrm{Cl}\) atoms: \( P\left(^{35}\mathrm{Cl}-^{35}\mathrm{Cl}\right) = P\left(^{35}\mathrm{Cl}\right) \times P\left(^{35}\mathrm{Cl}\right) = 0.754 \times 0.754 = 0.568 \) 2. Probability of Two \(^{37} \mathrm{Cl}\) atoms: \( P\left(^{37}\mathrm{Cl}-^{37}\mathrm{Cl}\right) = P\left(^{37}\mathrm{Cl}\right) \times P\left(^{37}\mathrm{Cl}\right) = 0.246 \times 0.246 = 0.060 \) 3. Probability of One \(^{35} \mathrm{Cl}\) atom and One \(^{37} \mathrm{Cl}\) atom: \( P\left(^{35}\mathrm{Cl}-^{37}\mathrm{Cl}\right) = P\left(^{35}\mathrm{Cl}\right) \times P\left(^{37}\mathrm{Cl}\right) + P\left(^{37}\mathrm{Cl}\right) \times P\left(^{35}\mathrm{Cl}\right) = 0.754 \times 0.246 + 0.246 \times 0.754 = 0.372 \)

Answer the questions

The fraction of \(\mathrm{Cl}_{2}\) molecules with one \(^{35} \mathrm{Cl}\) isotope and one \(^{37} \mathrm{Cl}\) isotope is 0.372, which is the same as 37.2%. The fraction containing just the \(^{35} \mathrm{Cl}\) isotope is 0.568, which is the same as 56.8%.

Key concepts

Molar Abundance

Molar abundance refers to the proportion of a specific isotope out of the total amount of isotopes present in a sample of an element. It is often expressed as a percentage and is crucial for various calculations in chemistry, particularly when dealing with isotopic mixtures.

For example, to find the molar abundance of chlorine isotopes, we consider the percentage of each isotope present. If chlorine has two isotopes with molar abundances of 75.4% for ³⁵Cl and 24.6% for ³⁷Cl, this indicates that in a large sample, about 75.4 out of every 100 chlorine atoms would be the isotope ³⁵Cl, and 24.6 would be ³⁷Cl.

Molar abundance is used to calculate the average atomic mass of an element listed on the periodic table, which takes into account the relative masses and abundances of all naturally occurring isotopes.

Atomic Isotopes

Atomic isotopes are different forms of an element's atoms that have the same number of protons but different numbers of neutrons. This results in isotopes having different atomic masses.

For instance, chlorine has two stable isotopes: ³⁵Cl with 18 neutrons and ³⁷Cl with 20 neutrons. Both share the same atomic number (17 protons), ensuring they're chemically identical, but their different neutron counts alter their atomic weight. Isotopes have unique physical properties, like their mass and stability, which allows for applications like radiometric dating or medical diagnostics using radioactive isotopes.

Chlorine Isotopes

Chlorine has two main isotopes that are naturally occurring, ³⁵Cl and ³⁷Cl. As we explore their abundances, we find that the vast majority of chlorine atoms in nature are in the form of ³⁵Cl. In our original exercise, ³⁵Cl accounts for 75.4% of chlorine atoms, while ³⁷Cl constitutes the remaining 24.6%.

This information is vital when we mix isotopes to form molecules like Cl₂. The different combinations of isotopes yield molecules with distinct masses, which can affect physical properties like boiling and melting points. Additionally, this isotopic composition is important in industries and sciences that rely on precise measurements, such as pharmaceutical manufacturing and environmental analysis.

Probability Calculations

Probability calculations in the context of isotopic abundance involve determining the likelihood of forming different isotopic molecules based on the abundances of their constituent isotopes. These calculations use the molar abundances as probabilities.

In our exercise, we are calculating the different combinations of chlorine isotopes forming a Cl₂ molecule. To find the fraction of molecules with one isotope of each kind, we multiply the molar abundances (expressed as probabilities) of the two isotopes. We also consider combinations with two atoms of the same isotope. Since a Cl₂ molecule can form from two ³⁵Cl atoms, two ³⁷Cl atoms, or one of each, the probabilities are squared for the first two scenarios and multiplied by 2 in the third case, accounting for both possible arrangements of dissimilar isotopes.