Question

Write integrated rate laws for zero-order, first-order, and secondorder reactions of the form \(\mathrm{A} \longrightarrow\) products.

Short answer

Zero-order: \[A\] = \[A\text{_{0}}\] - kt. First-order: ln(\text{[A]}/\[A\text{_{0}}\]) = -kt. Second-order: 1/[[A]] - 1/[[A]]_0 = kt.

Step-by-step solution

Zero-Order Reaction

A zero-order reaction has a rate that is independent of the concentration of the reactant. Its rate law can be written as: rate = k, where k is the rate constant. The integrated rate law for a zero-order reaction can be derived from the rate law by separating variables and integrating from the initial concentration of A, \[A\text{_{0}}\], to the concentration of A at time t, \[A\], and from 0 to t. This yields the equation: \[[A\] - \[A\text{_{0}}\] = -kt.

First-Order Reaction

The rate of a first-order reaction is directly proportional to the concentration of one reactant. The rate law for a first-order reaction is: rate = k\text{[A]}. By separating variables and integrating, the integrated rate law for a first-order reaction is derived as: ln(\text{[A]}/\[A\text{_{0}}\]) = -kt.

Second-Order Reaction

For a second-order reaction, the rate is proportional to the square of the concentration of one reactant. The rate law for a second-order reaction is: rate = k[[A]]^2. The integrated rate law is found by separating variables and integrating, which gives: 1/[[A]] - 1/[[A]]_0 = kt.

Key concepts

Understanding Zero-Order Reactions

When exploring the kinetics of chemical reactions, a zero-order reaction is one where the rate is constant and does not depend on the concentration of the reactant. Imagine a bustling kitchen where chefs are cooking meals at a steady rate, regardless of how many ingredients are on hand. This is akin to a zero-order reaction, where the rate law is simply expressed as 'rate = k', with 'k' being the constant rate of the reaction.

Delving into the mathematics, the integrated rate law for a zero-order reaction can be represented as \[A\] - \[A_{0}\] = -kt. This equation tells us that the concentration of the reactant A decreases linearly over time, similar to a clock ticking down steadily. In other words, for each unit of time that passes, a fixed amount of the reactant A is transformed into products, until it is depleted. This type of reaction is quite rare and generally occurs under conditions where the reactant is in excess or when a catalyst surface is saturated with the reactant.

• Rate Constant (k): The speed at which the reaction proceeds.
• Initial Concentration (\[A_{0}\]): The starting concentration of reactant A.
• Concentration of A at time t (\[A\]): The concentration after time t has elapsed.

When you solve problems involving zero-order reactions, it’s important to remember that once the reactant is exhausted, the reaction stops—like a kitchen running out of ingredients and unable to cook more meals.

First-Order Reactions Simplified

First-order reactions are like a game of musical chairs—each participant (reactant molecule) has a chance to find a chair (transform into product) that is directly proportional to the number of players (concentration of the reactant). In chemical terms, the rate of a first-order reaction is directly proportional to the concentration of the reactant. It follows the rate law 'rate = k[A]', signifying that the rate changes with varying concentrations of reactant A.

The integrated rate law for a first-order reaction is a bit more complex but equally revealing, given by the natural logarithm ln(\[A\]/\[A_{0}\]) = -kt. This logarithmic relationship indicates that as time ticks away, the concentration of the reactant A decreases exponentially. Unlike the steady cookware in a zero-order kitchen, the concentration here affects the cooking speed. You can think of k as the pace at which the chefs work—the higher it is, the faster the ingredients are used up.

• Natural Logarithm (ln): Showcases the exponential decline of reactant concentration over time.
• Dependence on Concentration: Indicates the reaction speeds up or slows down based on reactant availability.

Understanding first-order kinetics is crucial for processes like radioactive decay and simple enzyme-catalysed reactions, where the amount of reactant present can drastically impact the reaction rate.

Deciphering Second-Order Reactions

Second-order reactions take us into a different dynamic, akin to a team sport where cooperation is key. The rate of reaction depends on the interaction between two reactant molecules or two times the concentration of a single reactant as described by the rate law 'rate = k\[[A\]]^2'. Imagine partners in a dance competition moving with a speed that's influenced by the level of coordination between them. Here the coordination correlates to the concentration of reactant A.

The integrated rate law for a second-order reaction draws a picture of how these interactions change over time, described by the equation 1/\[[A\]] - 1/\[[A_{0}\]] = kt. This highlights the fact that as the reactant concentration decreases, the reaction rate also decreases, but much more dramatically compared to first-order reactions. Unlike the linear or exponential relationships seen in zero- and first-order reactions, second-order reactions form a curve that gets steeper as time progresses.

• Rate Law and Squared Concentration: Underscores the idea that the reaction rate is tied to the square of the reactant’s concentration.
• Integrated Rate Law: Allows us to predict concentrations at various times and deduce the reaction kinetics.

Common examples of second-order reactions might include bimolecular reactions where two reactant molecules collide and react. By understanding this relationship, students can appreciate that not all reactions proceed at the same pace and that the concentration of reactants can have a substantial impact on the reaction rate.