Question

Starting with the definition of partial rate factor, derive an expression relating \(\mathrm{p}^{\mathrm{G}} \mathrm{f}\) to the rate of substitution para to \(\mathrm{G}\) in \(\mathrm{C}_{6} \mathrm{H}_{5} \mathrm{G}\).

Short answer

In summary, the partial rate factor \( p^{\mathrm{G}} \) for a group G is defined as the ratio of the rate constant of a specific reaction involving G to the rate constant of the same reaction involving a reference group (usually hydrogen). The expression relating partial rate factor to the rate of substitution is given by: \( p^{\mathrm{G}} \propto \frac{r_\mathrm{G}}{r_\mathrm{H}} \) where \(r_G\) is the rate of substitution of group G and \(r_H\) is the rate of substitution of the reference group, hydrogen. This shows that the partial rate factor is proportional to the ratio of the rate of substitution of G to the rate of substitution of H.

Step-by-step solution

Understanding the concepts of partial rate factor and rate of substitution

The partial rate factor (p^G) is a measure of the reactivity of a given group G when it is substituted for another group in an organic molecule. It is defined as the ratio of the rate constant of a specific reaction involving G to the rate constant of the same reaction involving a reference group, usually hydrogen (H). The rate of substitution is the rate at which a given group (in this case, G) takes the place of another group (in this case, H) in an organic compound (here, C6H5G).

Deriving the expression for p^G in terms of rates of substitution

Let k_G be the rate constant for the reaction involving G and k_H be the rate constant for the reaction involving H (as a reference group). The definition of partial rate factor p^G is: \( p^{\mathrm{G}} = \frac{k_\mathrm{G}}{k_\mathrm{H}} \) Now let r_G and r_H be the rates of substitution of group G and H, respectively. According to the definition, these rates are proportional to the rate constants: \( r_\mathrm{G} \propto k_\mathrm{G} \) \( r_\mathrm{H} \propto k_\mathrm{H} \) We can introduce proportionality constants, a and b, to make these relationships equalities: \( r_\mathrm{G} = a k_\mathrm{G} \) \( r_\mathrm{H} = b k_\mathrm{H} \) Now we need to eliminate k_G and k_H from the equation for p^G using the expressions for r_G and r_H. We can do that by rearranging the expressions for r_G and r_H: \( k_\mathrm{G} = \frac{r_\mathrm{G}}{a} \) \( k_\mathrm{H} = \frac{r_\mathrm{H}}{b} \) Substitute these expressions into the equation for p^G: \( p^{\mathrm{G}} = \frac{\frac{r_\mathrm{G}}{a}}{\frac{r_\mathrm{H}}{b}} = \frac{r_\mathrm{G}}{r_\mathrm{H}} \cdot \frac{b}{a} \) Since b/a is a constant independent of G and H, we can write the final expression as: \( p^{\mathrm{G}} \propto \frac{r_\mathrm{G}}{r_\mathrm{H}} \) This expression shows that the partial rate factor p^G is proportional to the ratio of the rate of substitution of G to the rate of substitution of H.

Key concepts

Understanding the Rate of Substitution

The rate of substitution in organic chemistry is a fundamental concept that deals with the frequency at which one atom or functional group in a molecule is replaced by another. Factors affecting this rate include the nature of the leaving group, the conditions of the reaction, and the stability of the intermediate states.

For instance, in electrophilic aromatic substitution reactions, certain groups can either activate or deactivate the aromatic ring towards the substitution. Activating groups increase the rate of substitution, while deactivating groups decrease it. Through the introduction of specific substituents, the electronic and steric effects can be altered, thus changing the rate at which the substitution process occurs. This is crucial for the synthesis of various organic compounds, as precise control over substitution rates allows chemists to design and execute complex reaction pathways.

Role and Calculation of the Rate Constant

The rate constant, symbolized as k, is a major player in the dynamics of a chemical reaction. It represents the proportionality factor between the reaction rate and the concentrations of reactants raised to the power of their stoichiometric coefficients in the rate equation. In simple terms, it is a measure of how quickly a reaction proceeds.

For a given reaction at a particular temperature, the rate constant is constant and only changes with temperature or if a catalyst is introduced. In mathematical terms, for a reaction of the form A → B, the rate can be expressed as rate = k[A], where [A] denotes the concentration of reactant A. The rate constant is usually determined experimentally and offers vital information regarding the reaction mechanism and potential energy surface of the reaction.

Fundamentals of Organic Chemistry Reaction Kinetics

Organic chemistry reaction kinetics delve into the rates of chemical reactions involving organic compounds. This branch of chemistry analyzes factors that influence reaction rates, such as reactant concentrations, temperature, solvent, and catalyst presence.

By studying the kinetics of a reaction, one can extract valuable information about the reaction mechanism, which describes the step-by-step process of bond breaking and formation. Kinetics also aids in understanding the stability and reactivity of transition states and intermediates formed during a reaction. Reaction kinetics is essential for predicting reaction outcomes, optimizing reaction conditions, and designing efficient synthetic routes in organic synthesis.

Proportionality in Reaction Rates

The concept of proportionality in reaction rates plays an integral role in interpreting chemical reactions. It helps us understand how changes in one variable affect the speed of the reaction. Generally, if a particular reactant's concentration is doubled, and the reaction rate also doubles, we say that the rate is directly proportional to the concentration of that reactant.

This proportional relationship is present in the rate laws that define the kinetics of a reaction. These laws are mathematical equations that relate the rate of a reaction to the concentration of the reactants. This proportionality can be linear or exponential, depending on the order of the reaction concerning each reactant. Analyzing these relationships provides chemists with insights into the mechanistic details and efficiency of the reactions.