Problem 102 - Chemistry Matter and Change

Question

The rate law is rate \(=k[\) phenolphthalein]. If the rate constant for the reaction is \(1.0 \times 10^{-2} \mathrm{s}^{-1}\) , what is the instantaneous rate of reaction when the concentra- tion of phenolphthalein is 0.0025\(M ?\)

Step-by-Step Solution

Answer

The instantaneous rate of reaction is \(2.5 \times 10^{-5} \ \mathrm{M/s}\).

Step 1: Identify the Rate Law

The rate law for the reaction is given as \( \text{rate} = k [\text{phenolphthalein}] \). This indicates a first-order reaction with respect to phenolphthalein.

Step 2: Identify Known Values

We are given the rate constant \( k = 1.0 \times 10^{-2} \ \mathrm{s}^{-1} \) and the concentration of phenolphthalein \( [\text{phenolphthalein}] = 0.0025 \ M \).

Step 3: Substitute Values into Rate Law

Substitute the known values into the rate law: \( \text{rate} = (1.0 \times 10^{-2} \ \mathrm{s}^{-1})(0.0025 \ M) \).

Step 4: Calculate the Instantaneous Rate

Perform the multiplication: \( \text{rate} = 1.0 \times 10^{-2} \times 0.0025 = 2.5 \times 10^{-5} \ \mathrm{M/s} \).

Key Concepts

Instantaneous Rate in Chemical Reactions

The instantaneous rate of a chemical reaction is a snapshot of how quickly a reaction is proceeding at a specific moment in time. Unlike average rate, which considers the change over a larger time interval, instantaneous rate gives us a more precise understanding. In a first-order reaction, the instantaneous rate is directly proportional to the concentration of the reactant. For example, if the rate law is given by \( \text{rate} = k [\text{phenolphthalein}] \), the instantaneous rate can be calculated by substituting the current concentration of phenolphthalein and the rate constant into this equation. Calculating the instantaneous rate helps us predict how fast a product forms or how fast a reactant is consumed, guiding decisions in chemical processes and experiments.
To find the instantaneous rate, simply multiply the given rate constant by the concentration of the reactant at that moment.

Understanding First-Order Reactions

First-order reactions are characterized by having a rate that depends linearly on the concentration of a single reactant. In simpler terms, the rate will double if the concentration of the reactant doubles. This is essential for predicting the behavior of reactions over time.
The general form of the rate law for a first-order reaction is \( \text{rate} = k [A] \), where \(k\) is the rate constant and \([A]\) is the concentration of the reactant. Such reactions are common in various chemical systems and often involve either a single step or a series of steps that effectively involve one molecule in the rate-determining step of the mechanism.

• First-order reactions can be easily monitored over time as they often result in predictable, straight line plots when logarithmic concentration data is plotted against time.
• Understanding first-order reactions aids in determining reaction mechanisms and is pivotal in kinetics studies.

Recognizing these reactions is key to successfully controlling conditions in labs and industrial settings.

Role of Rate Constant in Reactions

The rate constant \(k\) is a crucial aspect of chemical kinetics and provides insight into the speed of a reaction. It is a proportionality constant in the rate equation that links the reaction rate with the concentrations of reactants.

• In a first-order reaction, the unit of the rate constant is \( \mathrm{s}^{-1} \), indicating per second, meaning the rate is fast in consuming the reactant.
• The value of \(k\) is determined experimentally and can vary dramatically between reactions and with changes in temperature.

Slight changes in \(k\) can denote vastly different reaction characteristics. This constant thereby informs us about reaction rates under specified conditions and helps predict how long a reaction will take to reach completion. Moreover, \(k\) is invaluable in calculating the instantaneous rate when given the concentrations of reactants.

Importance of Concentration in Reaction Rates

Concentration refers to how much of a substance is present in a given volume. In the context of reaction rates, understanding concentration is vital as it affects how quickly reactions happen.
For many reactions, including the first-order reactions, the rate directly depends on the concentration of reactants. If we look at the rate law \( \text{rate} = k [\text{phenolphthalein}] \), it unequivocally shows that the concentration of phenolphthalein dictates how fast the reaction proceeds.
Key Takeaways:

• Increasing the concentration generally increases the reaction rate because more reactant molecules result in more collisions and successful reaction events per unit time.
• Conversely, lower concentrations reduce the rate, often seen as a slower reaction.

Understanding the role of concentration helps in formulating the correct stoichiometry in laboratory experiments and industrial processes, ensuring reactions proceed at a desired level of efficiency.