Question

The rate constant is given by the equation: \(K=P \cdot A \cdot e^{-E_{a} / R T}\). Which factor should register a decrease for the reaction to proceed more rapidly? (a) \(T\) (b) \(A\) (c) \(E_{\text {a }}\) (d) \(P\)

Short answer

(c) Decreasing the activation energy \(E_{a}\) will cause the reaction to proceed more rapidly.

Step-by-step solution

Understand the Arrhenius Equation

The given rate constant equation is a form of the Arrhenius equation: \(K = P \cdot A \cdot e^{-E_{a} / RT}\), where \(K\) is the rate constant, \(P\) is the pre-exponential factor, \(A\) is the frequency factor, \(E_{a}\) is the activation energy, \(R\) is the universal gas constant, and \(T\) is the temperature in Kelvin. According to this equation, the rate constant \(K\) increases as the exponent \(-E_{a} / RT\) increases.

Analyze the Effect of Decreasing Each Factor

To increase the rate constant and make the reaction proceed more rapidly, we need the exponent \(-E_{a} / RT\) to increase. This can happen in two ways: by decreasing the activation energy \(E_{a}\) or by increasing the temperature \(T\). Conversely, decreasing \(T\) would decrease the rate constant. Decreasing \(P\) or \(A\) would also directly decrease \(K\) because they are directly multiplied by the exponential term.

Determine the Correct Factor to Decrease

Only the decrease in activation energy \(E_{a}\) will result in an increase in the exponent value, thus increasing the rate constant and making the reaction proceed more rapidly. Therefore, the factor that should register a decrease to make the reaction proceed more rapidly is the activation energy, \(E_{a}\).

Key concepts

Activation Energy

Activation energy, denoted as \(E_{a}\), is the minimum amount of energy required for reactants to transform into products during a chemical reaction. It serves as an energy barrier that must be overcome for reactants to be converted into products. Imagine pushing a ball up a hill before it can roll down on the other side. Activation energy is akin to this uphill effort in the landscape of chemical reactions.

In the given exercise, we learned that decreasing the activation energy of a reaction will increase its rate. This is depicted in the Arrhenius equation where a smaller \(E_{a}\) value makes the exponent \(-E_{a} / RT\) less negative, which results in a larger rate constant \(K\). It’s similar to making the hill smaller so that it’s easier for the ball to roll over it. Catalysts frequently accomplish this in chemistry by providing an alternative pathway with a lower activation energy. As a result, reactions occur faster, even at lower temperatures.

Rate Constant

The rate constant, represented as \(K\), is a coefficient that quantifies the speed of a chemical reaction. It ties together the concentrations of the reactants with the rate at which they transform into products. The larger the value of \(K\), the faster the reaction proceeds. The Arrhenius equation provides us with a mathematical understanding of how temperature, activation energy, and other factors affect the rate constant.

The key to grasping this concept is to realise that the rate constant is an indicator of reactivity. If one could adjust it at will, it would be like having a speed dial for chemical reactions. Factors such as temperature, presence of a catalyst, and concentration of reactants all play pivotal roles in either cranking up or dialing down this speed dial. The exercise demonstrates that lowering activation energy effectively turns up the speed, thus hastening the reaction.

Chemical Kinetics

Chemical kinetics is the study of the rates of chemical processes and the factors that affect them. It explores the road map of a reaction, charting how quickly reactants turn into products and which highways or back alleys the reaction takes. In this scientific exploration, the Arrhenius equation is a crucial GPS that guides us through the understanding of reaction rates.

In kinetic terms, the exercise showcases that by changing certain variables, such as decreasing the activation energy, we can directly influence the pace at which a reaction proceeds. Chemical kinetics embraces an array of such variables including concentration, surface area, temperature, and even the presence of light. The interplay of these factors dictates whether a reaction will be a sprint or a marathon, allowing chemists to predict and control reaction speeds for practical applications like industry, environmental science, and medicine.