Question

Explain how the rate law for a chemical reaction is used to determine the instantaneous rate of the reaction.

Short answer

The rate law is an equation that relates the reaction rate to the concentration of reactants in a chemical reaction, given by the formula $Rate=k[A{]}^m[B{]}^n$, where k is the rate constant, and m and n are the reaction orders. To determine the instantaneous rate of the reaction, first, find the reaction orders (m and n) for each reactant through experimental data or given information. Then, calculate the rate constant (k) using a specific experiment. Finally, plug in the concentrations of the reactants at the desired moment into the rate law equation to find the instantaneous rate. For example, if at a specific point, [A] = 0.075 mol/L and [B] = 0.150 mol/L with k = 0.50 L/mol·s, m=1, and n=1, the instantaneous rate would be $Rate=(0.50L/mol·s)(0.075mol/L)(0.150mol/L)=0.005625mol/L·s$.

Step-by-step solution

1. Understanding the rate law for a chemical reaction

The rate law is an equation that relates the reaction rate to the concentration of reactants in a chemical reaction. It has the following general form: $$Rate=k[A{]}^m[B{]}^n$$ where "Rate" is the reaction rate, k is the rate constant, [A] and [B] are the concentrations of reactants A and B, respectively, and m and n are the reaction order with respect to each reactant.

2. Definition of instantaneous rate

Instantaneous rate refers to the rate of change of concentration of a specific reactant or product at a particular moment or point in time during the reaction. It is different from the average rate, as the average rate considers the change in concentration over a defined period.

3. Determining the order for each reactant

In order to use the rate law to determine the instantaneous rate of the reaction, we need to know the reaction order with respect to each of the reactants (m and n). This can be found through a series of experiments where the concentrations of the reactants are systematically varied and the reaction rate is measured. Alternatively, the orders can be obtained from given information or context provided in the exercise.

4. Calculating the rate constant, k

Once the reaction orders have been determined, a specific experiment can be chosen to calculate the rate constant, k. Use the measured reaction rate and the concentrations of reactants from the chosen experiment to solve the rate law equation for k. For example, if the experimental data shows that the reaction rate is 0.010 mol/L·s when [A] = 0.100 mol/L and [B] = 0.200 mol/L, and m and n are both 1, then: $$0.010=k(0.100{)}^1(0.200{)}^1$$ Solving for k, we find that k = 0.50 L/mol·s.

5. Using the rate law to determine the instantaneous rate of the reaction

Now, with the reaction orders and rate constant known, we can use the rate law to find the instantaneous rate for any given set of reactant concentrations. Simply plug the concentrations of the reactants at the specific moment or point in time into the rate law equation. For example, if at a particular moment during the reaction, [A] = 0.075 mol/L and [B] = 0.150 mol/L, the instantaneous rate can be determined as follows: $$Rate=(0.50L/mol·s)(0.075mol/L{)}^1(0.150mol/L{)}^1$$ After calculating, the instantaneous rate is found to be 0.005625 mol/L·s.

Key concepts

Reaction Rate

The reaction rate in chemistry is a fundamental concept that measures how fast a chemical reaction occurs. It is defined as the rate at which the concentration of a reactant decreases or the concentration of a product increases over time. Understanding the reaction rate is crucial for predicting how long a reaction will take to reach completion and for designing chemical processes that are both efficient and safe.

Reaction rates can be affected by various factors, including the concentration of reactants, temperature, presence of catalysts, and the physical state of the reactants. Higher concentrations often lead to a higher reaction rate because more reactant molecules are available to collide and react. Similarly, an increase in temperature typically accelerates the reaction rate due to a rise in the energy and frequency of molecular collisions. Catalysts can also speed up a reaction without being consumed by lowering the activation energy required for the reaction to proceed.

Instantaneous Rate

The instantaneous rate is essentially a snapshot of the reaction rate at a particular moment in time, much like how an instantaneous speed tells you how fast a car is moving at a specific instant. Unlike average reaction rates, which look at reactant changes over larger time intervals, instantaneous rates are more precise and offer insight into the reaction's behavior at an exact point.

To measure the instantaneous rate, one would typically take the derivative of the concentration with respect to time, which in practical terms can often be approximated by measuring the change in concentration over a very short time interval. This level of detail is particularly important when monitoring reactions that have varying rates over the course of the process, such as when a reactant is nearly used up, or when a new product starts to influence the reaction dynamics.

Rate Constant

The rate constant, symbolized by the letter 'k', is a proportionality constant that appears in the rate law equation for a chemical reaction. It is a measure of the intrinsic reactivity of the reaction, independent of the concentration of reactants. The value of the rate constant is determined by experimental data and changes with temperature but remains constant for a given reaction at a fixed temperature.

A higher rate constant signifies a faster reaction under the same conditions. When calculating the rate constant from experimental data, a known reaction rate and reactant concentrations are used. The rate constant provides critical information for predicting how a reaction proceeds and is essential when scaling up reactions for industrial purposes, where consistent product yield and quality are crucial.

Reaction Order

Reaction order is a term that quantifies the dependency of the reaction rate on the concentration of each reactant. The overall order of the reaction is the sum of the exponents of the concentration terms in the rate law equation. Determining the order of a reaction is pivotal in understanding the step-by-step mechanism of the reaction.

For example, in a reaction with a rate law expressed as $Rate=k[A{]}^m[B{]}^n$, 'm' and 'n' are the orders with respect to reactants A and B, respectively. If 'm' is 2, the reaction is said to be second-order with respect to A, meaning the rate of reaction is proportional to the square of the concentration of A. Similarly, if 'n' is 1, the reaction is first-order with respect to B. If 'm' or 'n' is zero, the rate is independent of that particular reactant. Understanding reaction orders helps chemists tailor conditions to optimize reaction rates for desired outcomes in both research and industrial applications.