Question

The isotope exchange reaction for \(\mathrm{Cl}_{2}\) is as follows: \(^{35} \mathrm{Cl}^{35} \mathrm{Cl}+^{37} \mathrm{Cl}^{37} \mathrm{Cl} \rightleftharpoons 2^{37} \mathrm{Cl}^{35} \mathrm{Cl} .\) The equilibrium constant for this reaction is \(\sim 4 .\) Furthermore, the equilibrium constant for similar isotope-exchange reactions is also close to this value. Demonstrate why this would be so.

Short answer

The equilibrium constant for the isotope exchange reaction of \(\mathrm{Cl}_2\) is close to 4 due to the probabilities of reactants and products being formed. The probabilities are determined by the fractional abundance of the isotopes involved in the reaction. In this case, the equilibrium constant can be calculated as \(K=\frac{\text{probability of products}}{\text{probability of reactants}}=\frac{\frac{1}{2}}{\frac{3}{4} \times \frac{1}{4}}=4\). Similar isotope-exchange reactions will have comparable probabilities and thus their equilibrium constants will also be close to this value.

Step-by-step solution

Understand the Reaction

Here, we are dealing with an isotope exchange reaction for Cl2 gas. In this reaction, two molecules of Cl2 exchange their isotopes, specifically \(^{35}\mathrm{Cl}\) and \(^{37}\mathrm{Cl}\). The balanced chemical equation for this reaction is: \[^{35} \mathrm{Cl}^{35} \mathrm{Cl}+^{37} \mathrm{Cl}^{37} \mathrm{Cl} \rightleftharpoons 2^{37} \mathrm{Cl}^{35} \mathrm{Cl}\]

Determine probabilities

To understand why the equilibrium constant for this reaction is 4, we first need to determine the probabilities of each reactant and product occurring. We can represent this probability in a fractional form: Probability of \(^{35}\mathrm{Cl}^{35}\mathrm{Cl}\) molecule formation: \[\frac{(\frac{3}{4})^2}{\frac{3}{4}} = \frac{3}{4}\] Here, we use the fact that the probability of getting \(^{35}\mathrm{Cl}\) is \(\frac{3}{4}\) since there are 3 out of 4 total isotopes of Cl are \(^{35}\mathrm{Cl}\) Probability of \(^{37}\mathrm{Cl}^{37}\mathrm{Cl}\) molecule formation: \[\frac{(\frac{1}{4})^2}{\frac{1}{4}}=\frac{1}{4}\] Here, the probability of getting \(^{37}\mathrm{Cl}\) is \(\frac{1}{4}\) since it represents the one remaining possible isotope of Cl Probability of \(^{37}\mathrm{Cl}^{35}\mathrm{Cl}\) molecule formation: \[2 \times \frac{3}{4} \times \frac{1}{4} = \frac{1}{2}\] Here, we multiply by 2 because there are two possible outcomes: \(^{35}\mathrm{Cl}^{37}\mathrm{Cl}\) or \(^{37}\mathrm{Cl}^{35}\mathrm{Cl}\)

Calculate the equilibrium constant

Now, we can calculate the equilibrium constant (\(K\)) using the probabilities derived in the previous step. The ratio of the products (\(2^{37}\mathrm{Cl}^{35}\mathrm{Cl}\)) to the reactants (\(^{35}\mathrm{Cl}^{35}\mathrm{Cl} \times ^{37}\mathrm{Cl}^{37}\mathrm{Cl}\)) is: \[K=\frac{\text{probability of products}}{\text{probability of reactants}}=\frac{\frac{1}{2}}{\frac{3}{4} \times \frac{1}{4}}=4\] This result demonstrates that the equilibrium constant for the isotope exchange reaction is around 4. This can also be observed for similar isotope-exchange reactions, as the probability of reactants and products being formed will follow a similar pattern.

Key concepts

Equilibrium Constant

The equilibrium constant, denoted as \( K \), is a vital concept in physical chemistry that quantifies the balance between reactants and products in a chemical reaction at equilibrium. In simple terms, it gives us a numerical representation of the extent to which a reaction will proceed before reaching a state of dynamic balance, where the rate of the forward reaction equals the rate of the reverse reaction.

For the isotope exchange reaction \( {}^{35}\text{Cl}_2 + {}^{37}\text{Cl}_2 \rightleftharpoons 2 {}^{37}\text{Cl}{}^{35}\text{Cl} \), the equilibrium constant is said to be around 4. This implies that at equilibrium, the ratio of the concentration of products to the concentration of reactants raised to the power of their stoichiometric coefficients is approximately 4. To fully comprehend this, it's essential to delve into the probabilities of each species' formation based on the natural abundance of isotopes, allowing us to better understand why the equilibrium constant for this isotope exchange is consistently around this value.

Chemical Reaction Probabilities

In the realm of physical chemistry, understanding the chemical reaction probabilities can offer insight into the likelihood of various molecular interactions occurring. This is especially pertinent in reactions involving isotopes, such as the isotope exchange reaction involving chlorine isotopes.

By considering the natural abundance of \( {}^{35}\text{Cl} \) and \( {}^{37}\text{Cl} \), we can calculate the probability of the formation of each molecule. For instance, the probability of forming \( {}^{35}\text{Cl}_2 \) is derived from the squared probability of obtaining an \( {}^{35}\text{Cl} \) atom because the molecule consists of two identical atoms. The same logic applies for \( {}^{37}\text{Cl}_2 \). However, for \( {}^{37}\text{Cl}{}^{35}\text{Cl} \), the calculation includes the probabilities of both isotopes and a multiplication factor to account for the different possible combinations. Equipped with this deeper understanding, students can grasp why the equilibrium constant is a predictable value - it's a direct extension of the underlying atomic probabilities.

Physical Chemistry

Physical chemistry merges the principles of physics and chemistry to explain how chemical substances behave on a molecular and atomic level. It provides a quantitative framework for describing the properties and behavior of chemical systems, including reactions like the isotope exchange we are analyzing.

In this type of reaction, physical chemistry helps us decipher the equilibrium constant through the lens of atomic probabilities and molecular behavior. The exercises that tackle isotope exchange reactions are a quintessential example of how the field of physical chemistry applies mathematical principles to elucidate the probabilistic nature of chemical processes. It's the understanding of these probabilities and how they relate to macroscopic properties like equilibrium constants that equip students with the tools to predict and rationalize outcomes in a wide array of chemical reactions.