Question

Write an equation relating the concentration of a reactant \(\mathrm{A}\) at \(t=0\) to that at \(t=t\) for a first-order reaction. Define all the terms and give their units.

Short answer

The equation relating the concentration of a reactant A at t=0 to that at t=t is \[-\ln\left(\frac{[A]}{[A]_0}\right) = kt\]. The terms defined are: \([A]\) which is the concentration of the reactant at time \(t\) (M), \([A]_0\) is the initial concentration of the reactant (M), \(k\) is the rate constant (s\(^{-1}\) or min\(^{-1}\)), and \(t\) is time (s or min).

Step-by-step solution

Write the general rate law for a first order reaction

The concentration of a reactant A at any given time in a first-order reaction can be expressed using the first-order rate law equation as: \[-\ln\left(\frac{[A]}{[A]_0}\right) = kt\] Where: \([A]\) is the concentration of the reactant at time \(t\) (Molarity, M), \([A]_0\) is the initial concentration of the reactant (Molarity, M), \(k\) is the rate constant (s\(^{-1}\) or min\(^{-1}\)), and \(t\) is time(s or min).

Define the terms in the equation

Here are the definitions of the terms in the equation: \([A]\): Concentration of A at time \(t\) - It's the amount of substance A per unit volume at time \(t\). It's measured in moles per liter (M or mol/L). \([A]_0\): Initial concentration of A - This is the amount of substance A per unit volume at the beginning of the reaction (at \(t = 0\)). It's also measured in moles per liter (M or mol/L). \(k\): Rate constant - This a constant that offers a measure of the speed or rate of the reaction. It's specific to the reaction at a given temperature and depends on the nature of the reacting species. It's units for a first-order reaction is either seconds\(^{-1}\) (s\(^{-1}\)) or minutes\(^{-1}\) (min\(^{-1}\)), depending on the time scale used. \(t\): Time - The point in time at which the concentration of A is measured. It can be in any unit of time (second, minute, hour, etc.) but should remain consistent with the unit of rate constant.

Key concepts

Rate Law

The rate law is a fundamental equation that links the rate of a chemical reaction to the concentration of its reactants. For a first-order reaction, which is our focus here, the rate of reaction is directly proportional to the concentration of one reactant. Mathematically, this can be written as rate = k[A], where rate is the speed of the reaction, k is the rate constant, and [A] is the concentration of the reactant A. This formula tells us that as the concentration of A changes over time, the rate of the reaction will change accordingly.

Understanding the rate law is crucial for predicting how fast a reaction will occur under different concentrations, which is valuable in both academic and industrial chemical processes. Simplifying this concept, we can liken the rate law to a recipe: the amount of an ingredient (concentration of A) affects the outcome of the baking process (rate of the reaction), and the recipe specifics (rate constant k) tell us how to relate these factors.

Reaction Kinetics

Reaction kinetics involves the study of the rates of chemical reactions and the mechanisms by which they occur. It tells us not just how fast a reaction proceeds, but it also provides insights into the 'pathway' taken from reactants to products. This encompasses various types of reactions, including first-order reactions, where the rate only depends on the concentration of one reactant.

In the context of our exercise, reaction kinetics helps us understand the relationship between the reactant concentration and time for a first-order reaction. It gives us the tools to predict concentrations at any point in time, and to analyze how factors like temperature or catalysts would alter the reaction's rate. By examining the kinetics of a reaction, chemists can optimize conditions to achieve more efficient processes, whether it's brewing beer or synthesizing pharmaceuticals.

Chemical Concentration

Chemical concentration refers to the amount of a substance within a given volume of solution. It's commonly expressed in moles per liter (Molarity, M). In a scenario such as a first-order reaction, the concentration of the reactant is a variable that changes over time as the reactant is being consumed to form products.

Understanding how concentration varies is key to controlling and predicting the outcome of a chemical reaction. Going back to our cooking analogy, think of concentration as the proportion of spices in a dish; too much or too little can significantly affect the flavor. Similarly, in chemical reactions, the 'flavor' or outcome can depend on reactant concentrations, and accurately measuring these levels is essential for successful reactions. Through monitoring chemical concentration, it is possible to observe how reactants are consumed and how products are formed throughout the reaction.

Rate Constant

The rate constant, symbolized by k, is a proportionality factor that is unique to each chemical reaction at a given temperature. For a first-order reaction, it's expressed in reciprocal seconds \( (s^{-1}) \)or reciprocal minutes \( (min^{-1}) \). The rate constant reveals how quickly the reaction proceeds; a higher value of k suggests a faster reaction.

Importantly, the rate constant is not influenced by the concentration of reactants but is affected by environmental factors like temperature. In terms of our cooking example, the rate constant would be akin to the intensity of the heat; a higher flame (rate constant) cooks food faster. The rate constant is thus a valuable indicator, helping chemists to deduce the speed of a reaction and to compare the kinetics of different reactions under standardized conditions.