Question

For a gaseous reaction: \(\mathrm{A}(\mathrm{g}) \rightarrow\) Products (order \(=n\) ), the rate may be expressed as: (i) \(-\frac{\mathrm{d} P_{\mathrm{A}}}{\mathrm{d} t}=K_{1} \cdot P_{\mathrm{A}}^{n}\) (ii) \(-\frac{1}{V} \frac{\mathrm{d} n_{\mathrm{A}}}{\mathrm{d} t}=K_{2} \cdot C_{\mathrm{A}}^{n}\) The rate constants, \(K_{1}\) and \(K_{2}\) are related as \(\left(P_{A}\right.\) and \(C_{A}\) are the partial pressure and molar concentration of \(\mathrm{A}\) at time ' \(t^{\prime}\), respectively) (a) \(K_{1}=K_{2}\) (b) \(K_{2}=K_{1} \cdot(R T)^{n}\) (c) \(K_{2}=K_{1} \cdot(R T)^{1-n}\) (d) \(K_{2}=K_{1} \cdot(R T)^{n-1}\)

Short answer

\(K_{2} = K_{1} \cdot (RT)^{n}\)

Step-by-step solution

Understand the relationship between partial pressure and molar concentration

The partial pressure of a gas is related to its molar concentration by the ideal gas law equation: \(P = \frac{n}{V}RT\), or \(P_A = C_A RT\), where \(P_A\) is the partial pressure of A, \(n\) is the number of moles, \(V\) is the volume of the gas, \(C_A\) is the molar concentration of A, \(R\) is the ideal gas constant, and \(T\) is the temperature.

Relate the rate of the reaction in terms of partial pressure and molar concentration

According to the given equations (i) and (ii), the rates of the reaction can be expressed in two forms that are proportional to the rate constants \(K_1\) and \(K_2\). Notice that these rates are equivalent since they describe the same reaction, so we can set them equal to each other after expressing \(P_A\) in terms of \(C_A\).

Express the rate of the reaction with partial pressure in terms of molar concentration

Using the relationship between partial pressure and molar concentration (\(P_A = C_A RT\)), equation (i) can be written as \(-\frac{\mathrm{d} P_{\mathrm{A}}}{\mathrm{d} t}=K_{1} \cdot (C_A RT)^{n}\).

Compare the two expressions for the reaction rate

Now we set the two expressions equal to each other: \(K_{1} \cdot (C_A RT)^{n} = K_{2} \cdot C_{A}^{n}\). This yields the equation \(K_{1} \cdot (RT)^{n} = K_{2}\) after cancelling out the common term \(C_{A}^{n}\).

Solve for the relationship between \(K_1\) and \(K_2\)

Rearranging the equation from the previous step gives us \(K_{2} = K_{1} \cdot (RT)^{n}\). Thus, we find the correct relation between \(K_1\) and \(K_2\), which matches option (b).

Key concepts

Kinetics of Gaseous Reactions

Understanding the kinetics of gaseous reactions is critical in the study of reaction dynamics. In kinetics, we explore how different factors such as temperature, pressure, and concentration affect the speed or rate at which a reaction proceeds. For gaseous reactions in particular, the behavior of the gas — described by the ideal gas law — plays a significant role in determining the reaction rate. The study of kinetics provides insight into the mechanism of reactions and the calculation of the rate at which reactants convert to products.
Kinetics also involves figuring out the rate laws, which show how the rate of reaction is related to the concentration of the reactants. The order of reaction, denoted as 'n' in the rate law, tells us the power to which the concentration of a reactant is raised, indicative of how changes in concentration influence the reaction rate.

Ideal Gas Law

The ideal gas law is a fundamental equation in the study of gases, serving as an important bridge between the microscopic world of molecules and the macroscopic properties of gases that we can measure. It states that the pressure (P), volume (V), and temperature (T) of an ideal gas are related to the number of moles (n) present by the formula PV = nRT. Here, R is the universal gas constant. This law assumes that gases consist of many small particles moving in random directions, with no interactions other than perfectly elastic collisions.
In the context of reaction kinetics, the ideal gas law allows us to switch between different measures of a gas's concentration. We can use this law to relate the partial pressure of a gas to its molar concentration, which can be crucial when comparing reaction rates based on different units.

Rate of Reaction

The rate of reaction is an expression of the speed at which reactants transform into products. It’s generally defined as the change in concentration of a reactant or product per unit time. In the case of a gaseous reactant, we can measure this rate as a change in molar concentration or as a change in partial pressure over time. Hence, two different rate constants, K₁ and K₂, are used depending on whether we're considering pressure or concentration.
A major point to note is that these rate constants, although related, can have different units. It’s not just the numerical value that matters, but also how they are connected through the ideal gas law that ultimately allows us to equate rates expressed in terms of pressure with those in terms of concentration.

Partial Pressure and Molar Concentration Relationship

The relationship between partial pressure and molar concentration can be tricky to grasp. The partial pressure of a gas is the pressure it would exert if it alone occupied the entire volume of the mixture at the same temperature. This is closely tied to its molar concentration through the ideal gas law. When we say that P_A = C_A RT, we’re stating that the partial pressure of gas A (P_A) is directly proportional to its molar concentration (C_A) in the gas phase, where T is the temperature and R is the ideal gas constant.
Under constant temperature and volume, any change in the concentration directly affects the partial pressure and vice versa. Such understanding is instrumental when dealing with the kinetic equations of gaseous reactions, especially when converting between rate expressions that use pressure and those that use concentration, a common situation in chemical kinetics.