Explain the difference between the rate law for a reaction and the integrated rate law for a reaction. What relationship does each kind of rate law express?

The chemical reaction rate is a measure of how fast a reaction proceeds. It's crucial to understand that this rate can be affected by several factors, including reactant concentrations, temperature, presence of catalysts, and surface area of solid reactants. The reaction rate is often given as the change in concentration of a reactant or product per unit time.

For instance, if a reaction produces gas, the rate could be determined by measuring the volume of gas produced over time. In liquid solutions, the concentration of a dissolved substance can be measured. Remember, the rate is not constant; it decreases as reactants are used up unless the reaction is zero-order, where the rate remains constant.

Reaction kinetics is the study of the rates of chemical processes. It involves looking at how different conditions, like temperature and concentration, impact the speed of the reaction. Kinetics also explores the mechanisms of reactions, helping scientists to understand the actual steps that take place during a reaction. A key idea in kinetics is the activation energy, the minimum energy that reactant molecules must have to start the reaction.

Kinetics differs from thermodynamics, which tells us the potential for a reaction to occur but not how quickly it will happen. Thermodynamics says if the reaction can happen; kinetics tells us how fast.

First-order reactions are a category of reactions where the rate depends linearly on the concentration of one reactant. This means that if you double the concentration of the reactant, the rate of the reaction will also double. For these reactions, the rate law can be written as \( \text{Rate} = k[A] \), where 'A' is the concentration of the reactant and 'k' is the rate constant.

The integrated rate law for a first-order reaction is \( \text{ln}[A] = -kt + \text{ln}[A]_0 \). This equation allows us to calculate the concentration of reactant A at any time 't', giving us a clearer picture of the reaction's progression over time, not just at a single moment.

Reaction order represents the power to which the concentration of a reactant is raised in the rate law. It indicates how the rate of a reaction is affected by the concentration of a reactant. The order is determined experimentically; it’s not based on the stoichiometry of the balanced chemical equation.

The overall order of a reaction is the sum of the orders with respect to each reactant. For example, a reaction with rate law \( \text{Rate} = k[A]^2[B] \) is third-order overall, second-order with respect to 'A', and first-order with respect to 'B'. Understanding the reaction order is critical for controlling the conditions needed to optimize the reaction rate in industrial and laboratory settings.