Question

At \(325^{\circ} \mathrm{C}, \quad 1, \quad 3\) -butadiene dimerizes according to the equation: \(2 \mathrm{C}_{4} \mathrm{H}_{6}(\mathrm{~g})\) \(\rightarrow \mathrm{C}_{8} \mathrm{H}_{12}(\mathrm{~g}) .\) It is found that the partial pressure, \(P\), of the reactant obeys the law: \(\frac{1}{P}=k t+\frac{1}{P_{0}}\). Order of the reaction is (a) 1 (b) \(0.5\) (c) 2 (d) \(1.5\)

Short answer

The order of the reaction is (a) 1.

Step-by-step solution

- Understanding the Problem

First, we need to understand the given information. The problem states that 1,3-butadiene dimerizes according to a given reaction and that the partial pressure of the reactant changes over time according to a given equation. Our goal is to determine the order of the reaction, which is related to how the rate depends on the concentration of reactants.

- Analyzing the Rate Law

The equation provided, \(\frac{1}{P}=k t+\frac{1}{P_{0}}\), can be recognized as a rate law. Here, \(P\) is the partial pressure of the reactant, \(P_0\) is the initial partial pressure of the reactant, \(k\) is the rate constant, and \(t\) is time. The rate law equation is a linear equation in the form of \(y=mx+c\), which corresponds to a first-order reaction. This is because for a first-order reaction, the rate is directly proportional to the concentration (or in this case, the partial pressure) of one reactant.

- Determining Order from Rate Law

The linear form of the rate law with respect to time implies that it is a first-order reaction. For zero-order reactions, we would expect the rate to be independent of the concentration of the reactants. For a second-order reaction, the rate would be proportional to the square of the reactant concentration. Therefore, in this scenario, since the rate of the chemical reaction depends linearly on the inverse of the partial pressure, it is a first-order reaction.

Key concepts

Chemical Kinetics

Chemical kinetics is the branch of chemistry that deals with the rates at which chemical reactions occur and the factors that influence these rates. Understanding kinetics can help us predict how quickly a reaction will proceed under given conditions and how changes in temperature, concentration, or pressure will affect the reaction's progression.

For the reaction where 1,3-butadiene (C₄H₆) dimerizes to form C₈H₁₂, the rate at which this process occurs is essential. By measuring the change in partial pressure over time, we can glean information about the reaction's kinetics, particularly the reaction order, which tells us how the rate is dependent on reactant concentration. A first-order reaction, as this exercise suggests, implies that the rate of the reaction is directly proportional to the concentration of 1,3-butadiene.

Rate Law

The rate law is an equation that links the rate of a reaction to the concentration of the reactants. It can be expressed in the general form: \[ \text{rate} = k[ \text{A} ]^n \] where \(k\) is the rate constant, \([A]\) is the concentration of reactant A, and \(n\) is the reaction order with respect to A.

When we analyze the provided rate law \( \frac{1}{P} = kt + \frac{1}{P_0} \), it mirrors the form of a first-order reaction's integrated rate law. In case of a first-order reaction, the rate law simplifies to \(\text{rate} = k [A]\), which shows that the rate is directly proportional to the reactant's concentration—or in our case, its partial pressure. The constant \(k\) holds vital information about the reaction's kinetics as it can vary with temperature and the presence of catalysts. Understanding the rate law for a given reaction is key to predicting the speed of the reaction and how it changes under different conditions.

Partial Pressure

Partial pressure is a measure of the pressure contributed by a single gas in a mixture of gases and is directly proportional to the concentration of that gas. It plays a crucial role in chemical kinetics, especially for reactions involving gases. The relationship between the partial pressure and the concentration of a gas is given by the ideal gas law: \[ PV = nRT \] where \(P\) is the partial pressure, \(V\) is the volume, \(n\) is the number of moles, \(R\) is the universal gas constant, and \(T\) is the temperature.

When dealing with the reaction of 1,3-butadiene, the rate law given in terms of partial pressure allows us to analyze the reaction kinetics without directly measuring concentrations. Since the volume and temperature are often constant for a given set-up, any changes in partial pressure reflect changes in the amount of reactant gas present. Thus, the partial pressure provides a convenient methodology to track the reaction's progress and determine its order.