Question

The rate expression for the reaction: \(\mathrm{A}(\mathrm{g})+\mathrm{B}(\mathrm{g}) \rightarrow \mathrm{C}(\mathrm{g})\) is rate \(=K \cdot C_{\mathrm{A}}^{2} \cdot C_{\mathrm{B}}^{1 / 2} .\) What changes in the initial concentration of \(A\) and \(B\) will cause the rate of reaction increase by a factor of eight? (a) \(C_{\mathrm{A}} \times 2 ; C_{\mathrm{B}} \times 2\) (b) \(C_{\mathrm{A}} \times 2 ; C_{\mathrm{B}} \times 4\) (c) \(C_{\mathrm{A}} \times 1, C_{\mathrm{B}} \times 4\) (d) \(C_{\mathrm{A}} \times 4, C_{\mathrm{B}} \times 1\)

Short answer

The initial concentration of A needs to be doubled and the initial concentration of B needs to be quadrupled to cause the rate of reaction to increase by a factor of eight. Therefore, the correct answer is (b).

Step-by-step solution

- Understand the Rate Law

The given rate law is rate = K * C_{A}^{2} * C_{B}^{1/2}. This equation states that the rate of the reaction is directly proportional to the square of the concentration of species A and to the square root of the concentration of species B.

- Influence of Concentrations on Rate

We need to determine the increase in the concentrations of A and B that would result in an eightfold increase in the rate of reaction. Since rate is proportional to C_{A}^{2} and C_{B}^{1/2}, we can calculate how changes in concentrations will affect the rate.

- Calculating the Impact on Rate for Option (a)

If C_{A} is doubled, the factor affecting the rate for A becomes (2 * C_{A})^{2} = 4C_{A}^{2}. If C_{B} is doubled, the factor affecting the rate for B becomes (2 * C_{B})^{1/2} = 2^{1/2}C_{B}^{1/2}. The overall rate change is 4 * 2^{1/2}, which is not equal to 8. So, option (a) is not correct.

- Calculating the Impact on Rate for Option (b)

If C_{A} is doubled, the factor affecting the rate for A remains as 4C_{A}^{2}. If C_{B} is quadrupled, the factor affecting the rate for B becomes (4 * C_{B})^{1/2} = 2C_{B}^{1/2}. The overall rate change is 4 * 2 = 8. Therefore, the rate of reaction will increase by a factor of eight for option (b).

- Verifying Other Options

Since we have already determined that option (b) results in the desired eightfold increase in rate, there is no need to calculate the impact on rate for options (c) and (d). However, note that both these options do not provide the needed change in reaction rate; option (c) increases by a factor of 2, and option (d) increases by a factor of 16.

Key concepts

Chemical Kinetics

Chemical kinetics is the study of rates of chemical processes and the factors affecting them. It tells us how fast a reaction occurs and provides insights into the reaction mechanism. In essence, kinetics deals with the speed at which reactant molecules transform into products. The rate of a chemical reaction can be expressed in terms of the change in the concentration of reactants or products over time. Factors such as temperature, pressure, concentration of reactants, and the presence of a catalyst can significantly influence the rate.
Understanding kinetics is crucial in various industries, from pharmaceuticals, where reaction speed affects the time to create drugs, to manufacturing, where it's essential for controlling production rates. In academic settings, kinetics helps students understand why certain reactions happen quickly, like an explosion, whereas others, like rusting, take years.

Concentration Effect on Reaction Rate

The concentration of reactants profoundly impacts the rate of reaction. Generally, a higher concentration of reactants leads to an increased reaction rate. This happens because more reactant molecules are likely to collide with enough energy to undergo a successful chemical change.
For instance, if you add more sugar to a cup of tea, the rate at which sugar dissolves increases up to a saturation point. This principle applies to gas reactions as well as liquids and solids. However, not all reactions show the same dependency on concentration. The effect of concentration on reaction rate is quantitatively described by the rate law and its specific rate constant.
The 'rate law' or 'rate equation' tells us exactly how the rate depends on the concentration. In our exercise, the rate of the reaction increases when the concentrations of the gases are increased, yet each reactant has a different impact on the rate, depending on its order in the reaction, as shown in the rate law expression.

Rate Law Expression

The rate law expression is a mathematical equation that relates the rate of a reaction to the concentration of reactants. Each reactant in the rate law comes with an exponent, which can be an integer or fraction, known as the 'order' of the reaction with respect to that reactant. The sum of the orders gives the overall reaction order.
In the given exercise, the rate of the reaction is dependent on the concentrations of gases A and B to the power of 2 and 1/2, respectively. This tells us the reaction is second-order with respect to A and half-order with respect to B. So, if the concentration of A is doubled (C_A \times 2), the rate is quadrupled (\({2}^{2}\)). Similarly, if the concentration of B is quadrupled (C_B \times 4), the reaction rate doubles (\({4}^{1/2}\)). Therefore, to increase the reaction rate by a factor of eight, we must adjust the concentrations of A and B according to the interplay of their orders in the rate equation.
Understanding rate laws is not only essential for solving textbook problems but also plays a pivotal role in optimizing reactions in research and industry, allowing chemists to control the speed of reactions and achieve desired outcomes efficiently.