Problem 80 - Advanced Problems in Physical Chemistry for Competitive Examinations

Question

When excess toluene- \(\alpha\) -d \(\left(\mathrm{C}_{6} \mathrm{H}_{5} \mathrm{CH}_{2} \mathrm{D}\right)\) was photochemically monochlorinated at \(80^{\circ} \mathrm{C}\) with \(0.1\) mole of chlorine, there was obtained \(0.0212\) mole DCl and \(0.0848\) mole HCl. What is the value of the isotope effect, \(K^{\mathrm{H}} / \mathrm{K}^{\mathrm{D}}\) ? (a) \(\frac{1}{4}\) (b) \(\frac{4}{1}\) (c) \(\frac{5}{1}\) (d) \(\frac{1}{5}\)

Step-by-Step Solution

Answer

The value of the isotope effect, K^H / K^D, is 4.

Step 1: Understand the Isotope Effect

The isotope effect is the change in rate of a chemical reaction when one of the atoms in the reactants is replaced by one of its isotopes. For a monochlorination reaction, the isotope effect, noted as K^H / K^D, is the rate constant of the reaction involving the hydrogen isotope (K^H) divided by the rate constant of the reaction involving the deuterium isotope (K^D).

Step 2: Calculate Mole Ratio

Calculate the mole ratio of HCl to DCl produced in the reaction. This is obtained by dividing the moles of HCl produced by the moles of DCl produced. Mole ratio = moles of HCl / moles of DCl = 0.0848 / 0.0212.

Step 3: Determine the Isotope Effect

Given that the initial amount of chlorine is the limiting reagent and it is in excess compared to the toluene derivative, the ratio of H to D incorporation into the product (HCl to DCl) should directly relate to the isotope effect. Since the mole ratio reflects the relative rates of the two reactions, this ratio is equal to K^H / K^D. Therefore, the isotope effect is the same as the calculated mole ratio.

Key Concepts

Physical Chemistry

Physical chemistry explores the principles that govern the behavior of molecules and the chemical processes they undergo. One significant aspect within this field is the study of reaction rates and mechanisms, which belong to a subject called chemical kinetics. Our exercise navigates through a concept called the isotope effect, a phenomenon observed when a reactant atom in a chemical reaction is replaced by its isotope, potentially altering the reaction's speed.

In the given problem, a physical chemist would examine how the substitution of hydrogen with its heavier isotope, deuterium, influences the reaction rate of toluene's monochlorination. The change stems from differences in bond energies and vibrational frequencies between the isotopes, reflecting the underlying quantum mechanical nature of chemical bonds.

Chemical Kinetics

Chemical kinetics deals with studying the rates of chemical processes and the factors affecting them. It is crucial for predicting how fast reactions will occur, which is essential for all branches of chemistry, including industrial processes, biochemistry, and environmental science.

The exercise at hand demonstrates kinetics by comparing the reaction rates of toluene when hydrogen is replaced by deuterium. It is a visual representation of how chemical kinetics can provide insight into the influence of isotopic substitution on the time it takes for a reaction to proceed. In this context, understanding kinetics means connecting the dots between the experimental data (the amounts of HCl and DCl produced) and the intrinsic reaction rates of the isotopic variants.

Rate Constant

The rate constant is a proportionality factor that connects the rate of a reaction to the concentrations of the reactants. It is an essential term in the rate equation that describes how the rate depends on the molarity of the reactants. In a simple reaction, the rate equation might look like this: rate = k [A], where 'k' is the rate constant, and [A] is the concentration of reactant A.

In the exercise, we are essentially comparing two rate constants, K^H and K^D, associated with H- and D-substituted toluenes respectively. These constants are crucial for understanding the reaction's dynamics and how the presence of different isotopes can affect the speed at which products are formed.

Isotopic Substitution

Isotopic substitution is when an atom in a molecule is replaced by one of its isotopes—atoms with the same number of protons but a different number of neutrons. This can significantly impact a molecule’s physical properties, such as melting point and density, as well as chemical properties like reaction rates.

In our textbook problem, isotopic substitution involves replacing a hydrogen atom (H) with a deuterium atom (D) in toluene. This minor change leads to a measurable difference in the rate at which the monochlorination reaction occurs, which is quantified by the isotope effect, K^H/K^D. Understanding this concept is vital as it highlights the sensitive balance of chemical reactions and the possible effects that even a single neutron can impose.