Question

The concentrations in a chemical reaction are expressed in moles per liter and time is expressed in seconds. If the overall rate law is third-order, what are the units for the rate and the rate constant?

Short answer

The units for the rate and the rate constant in a third-order chemical reaction are as follows: - Rate: moles per liter per second (M/s) - Rate constant: \(1 / (M^2 s)\)

Step-by-step solution

Understand the third-order rate law

A third-order rate law means that the rate of the reaction depends on the concentration of the reactants raised to the power of their stoichiometric coefficients, which add up to 3. Mathematically, the rate law can be expressed as: Rate = k[A]^x[B]^y where: - Rate is the rate of the reaction - k is the rate constant - [A] and [B] represent the concentrations of reactants A and B - x and y are the orders of the reaction with respect to reactants A and B respectively, and x + y = 3

Express the concentrations and time in appropriate units

We know that the concentrations are given in moles per liter (M) and time is given in seconds (s). Therefore, we can express the rate of the reaction in terms of moles per liter per second (M/s). Rate = M/s

Determine the units of the rate constant

We can rewrite the rate law with the appropriate units: (M/s) = k (M)^x (M)^y Since x + y = 3, we get: (M/s) = k (M)^3 Now, to find the units for the rate constant k, we'll isolate k by dividing both sides of the equation by (M)^3: k = (M/s) / (M)^3 Simplifying, we get the units for the rate constant k: k = 1 / (M^2 s)

Present the final answer

The units for the rate and the rate constant are as follows: - Rate: moles per liter per second (M/s) - Rate constant: 1 / (moles² per liter² per second) (1 / M^2 s)

Key concepts

Understanding Chemical Reaction Rates

Chemical reaction rates refer to the speed at which reactants are transformed into products during a chemical reaction. They are vital for understanding how fast a reaction occurs under certain conditions and are influenced by various factors including temperature, concentration of reactants, presence of catalysts, and surface area of reactants.

In the context of a reaction, the rate is usually expressed as the change in concentration of a reactant or product per unit time. For instance, if the concentration of a reactant decreases over time, the rate of reaction can be described using the equation \( \text{Rate} = -\frac{d[A]}{dt} \) where \( [A] \) is the concentration of the reactant A and \( t \) is time. The negative sign indicates that the concentration of A decreases as the reaction proceeds.

To measure the rate, we often use moles per liter per second (M/s) as the standard units, indicating how many moles of a substance are converted per liter of solution per second. Understanding the rate of reaction helps in controlling processes, for instance, in industrial synthesis where making a product efficiently is crucial.

Deciphering the Rate Constant

The rate constant, denoted by \( k \) plays a crucial role in the rate law, which is a mathematical equation that relates the rate of a chemical reaction to the concentration of its reactants. Unlike the reaction rate, the rate constant is not affected by the concentration of reactants; instead, it's a unique value for each reaction at a given temperature.

The rate constant essentially sets the timescale of the reaction—if \( k \) is large, the reaction will proceed quickly; if \( k \) is small, the reaction will be slower. For a reaction represented by the rate law \( \text{Rate} = k[A]^x[B]^y \) where \( [A] \) and \( [B] \) are the concentrations of reactants A and B, and \( x \) and \( y \) are their respective reaction orders, the units of \( k \) can be determined analytically to ensure dimensional consistency.

It is important to recognize that for different order reactions, the units of \( k \) will also differ. Natural log units are used in first-order reactions, while no units are required for zero-order reactions. Second-order reactions involve inverse concentration unit, just as third-order reactions require the inverse square of the concentration unit, always ensuring that the rate itself is expressed in consistent units, typically \( M/s \).

Exploring Third-Order Reactions

A third-order reaction is a chemical process where the rate is proportional to the cube of the concentration of one reactant or the product of the concentrations of three different reactants. This means that the sum of the stoichiometric coefficients related to the reactant's concentration in the rate law equation equals three.

Mathematically, for a third-order reaction with two reactants A and B, the rate law can be represented by \( \text{Rate} = k[A]^x[B]^y \) with \( x + y = 3 \) . This indicates that the overall order of the reaction is the sum of the exponents of the concentration terms.

Understanding third-order reactions is important when studying complex chemical processes, especially in the fields of kinetics and catalysis. Since the rate depends steeply on the concentration, even small changes can cause significant variations in the rate, which can be critical for the design of chemical reactors and safety procedures in industrial settings.

The units for the rate constant \( k \) in a third-order reaction, as derived from the example in the exercise, turn out to be \( 1 / M^2 s \) which reflects the intricate relationship between the reactant concentrations and time for such reactions.