Question

For the general reaction \(\mathrm{A}+\mathrm{B} \rightarrow \mathrm{C}+\mathrm{D}\), what is the effect on the number of collisions between \(\mathrm{A}\) and \(\mathrm{B}\) of (a) tripling the concentration of each, (b) quadrupling the concentration of each?

Short answer

(a) When the concentrations of both A and B are tripled, the number of collisions between them is 9 times the initial number of collisions. (b) When the concentrations of both A and B are quadrupled, the number of collisions between them is 16 times the initial number of collisions.

Step-by-step solution

The collision theory states that the reaction rate is proportional to the number of collisions between the reacting molecules. Mathematically, this can be expressed as: \[ Rate \propto [A][B] \] Here, [A] and [B] represent the concentrations of the reactants A and B, respectively. #Step 2: Determine the initial reaction rate and number of collisions#

Initially, we will assume the reaction rate and the number of collisions between A and B to be equal. This is because there is no information given about the specific rate constant, so we will consider the proportionality constant to be 1. \[ Rate = [A][B] \] #Step 3: Calculate the change in reaction rate and number of collisions when concentrations are tripled#

Since we are tripling the concentration of both A and B, the new concentrations are: \[ [A'] = 3[A] \] \[ [B'] = 3[B] \] To find the new reaction rate and number of collisions, substitute these new concentrations into the reaction rate formula: \[ Rate' = [A'][B'] = (3[A])(3[B]) = 9[A][B] \] For the case when the concentrations are tripled, the number of collisions between A and B is 9 times the initial number of collisions. #Step 4: Calculate the change in reaction rate and number of collisions when concentrations are quadrupled#

Now, we will consider when the concentrations of both reactants are quadrupled: \[ [A''] = 4[A] \] \[ [B''] = 4[B] \] Again, substitute these new concentrations into the reaction rate formula to find the new reaction rate and number of collisions: \[ Rate'' = [A''][B''] = (4[A])(4[B]) = 16[A][B] \] For the case when the concentrations are quadrupled, the number of collisions between A and B is 16 times the initial number of collisions. To summarize: (a) Tripling the concentrations of both A and B results in nine times the initial number of collisions between A and B. (b) Quadrupling the concentrations of both A and B results in sixteen times the initial number of collisions between A and B.

Key concepts

Reaction Rate

The reaction rate refers to how quickly reactants are converted into products in a chemical reaction. When examining the reaction \( \text{A} + \text{B} \rightarrow \text{C} + \text{D} \), the speed of the process, or rate, hinges upon the frequency and energy of collisions between reactant molecules. According to collision theory, only those collisions with sufficient energy, called the activation energy, will result in a successful conversion to products. This means:- The more frequent the collisions, the higher the rate.- The more energy in the collisions, the better the chance of overcoming the activation energy barrier.

In mathematical terms, the reaction rate \( R \) is proportional to the concentrations of the reacting substances, such as \( [A] \) and \( [B] \). The relationship can be expressed as:\[ R \propto [A][B] \]The formula shows that as the concentration of reactants increases, the reaction rate increases. This is because higher concentrations lead to more molecules colliding more often.

By understanding this relationship, we can predict how changes in conditions affect the reaction speed.

Concentration Effect

Concentration effect refers to how changing the concentration of reactants can impact the reaction rate. When concentrations increase, there are more reactant particles in a given volume, which increases the frequency of collisions. Let's see this in action using the reaction \( \text{A} + \text{B} \rightarrow \text{C} + \text{D} \):

- **Tripling the concentration**: Increasing the concentration of \( A \) and \( B \) by a factor of three means more collisions. Mathematically, substituting new concentrations \( [A'] = 3[A] \) and \( [B'] = 3[B] \) into the rate equation results in: \[ R' = (3[A])(3[B]) = 9[A][B] \] Hence, the reaction rate, and number of collisions, becomes nine times greater.
- **Quadrupling the concentration**: Similarly, multiplying concentrations by four leads to: \[ R'' = (4[A])(4[B]) = 16[A][B] \] Here, the rate and collisions increase sixteenfold.
Through these examples, we see the direct results of varying reactant concentrations. They demonstrate fundamental principles of the collision theory in practical terms.

Chemical Kinetics

Chemical kinetics is the branch of chemistry that deals with reaction rates and the factors affecting them. It helps us understand how different conditions change how fast a reaction proceeds. The key aspects of chemical kinetics include:

- **Reaction rate laws** govern how concentration affects the rate of reaction. These laws help write mathematical predictions about how changes alter the speed of reactions.- **Activation energy** is the minimum energy barrier reactants must overcome to form products. Temperature and catalysts can influence this energy requirement.- **Inhibitors and Catalysts**: These substances can affect rates by increasing (catalysts) or decreasing (inhibitors) them without being consumed in the reaction.

Chemical kinetics offers valuable insights into reaction mechanisms, allowing chemists to \( \text{predict} \) and \( \text{control} \) the rate of chemical changes. By understanding these principles, scientists and engineers can better design and optimize industrial processes, improve safety in chemical manufacturing, and deepen our comprehension of biochemical processes in living organisms.