Question

For a reaction \(A \rightarrow\) products which is first order with respect to \(\mathrm{A},\) show that the units of the rate constant are \(s^{-1}\). Similarly, confirm that the units of the second order rate constant are \(\mathrm{dm}^{3} \mathrm{mol}^{-1} \mathrm{s}^{-1}\).

Short answer

First-order rate constant: \(s^{-1}\); Second-order rate constant: \(\text{dm}^3 \text{mol}^{-1} \text{s}^{-1}\).

Step-by-step solution

Define the Rate Expression for First Order Reaction

In a first-order reaction, the rate of reaction is directly proportional to the concentration of the reactant. Thus, the rate for the reaction \(A \rightarrow\) products is given by \(\text{Rate} = k[A]\), where \(k\) is the rate constant and \([A]\) is the concentration of \(A\) in molarity (M).

Determine Units for First Order Rate Constant

For a first-order reaction, the rate is expressed in concentration per unit time (M/s). So, substituting units, we have \(M/s = k \times M\). Simplifying gives \(k = \frac{M/s}{M} = s^{-1}\). This shows that the units for the rate constant \(k\) in a first-order reaction are \(s^{-1}\).

Define the Rate Expression for Second Order Reaction

In a second-order reaction, the rate of reaction is proportional to the square of the concentration of the reactant, or the product of the concentrations of two reactants. The rate is expressed as \(\text{Rate} = k[A]^2\) or \(k[A][B]\).

Determine Units for Second Order Rate Constant

In a second-order reaction involving a single reactant, the rate is \(M/s = k \times M^2\). Solving for \(k\), \(k = \frac{M/s}{M^2} = \frac{1}{M \cdot s} = \text{dm}^3 \text{mol}^{-1} \text{s}^{-1}\) since 1 M (molarity) is equivalent to \( \text{mol} \cdot \text{dm}^{-3}\). In the case of two reactants, the derivation is similar.

Key concepts

First-Order Reaction

In a first-order reaction, the rate at which reactants convert to products is directly proportional to the concentration of a single reactant. This means that as the concentration of the reactant decreases, the reaction rate also decreases. Such reactions are described by the equation:

• Rate = \( k[A] \)

Here, \( k \) is the rate constant, and \( [A] \) represents the concentration of the reactant \( A \) in terms of molarity (M).
Examples from real life include the radioactive decay of isotopes where the amount of un-decayed material decreases over time. A first-order reaction graph typically shows an exponential decrease in concentration over time when plotted. This simple model makes it easier to predict how long a process will take or how much reactant will remain after a certain period.

Second-Order Reaction

Second-order reactions have their reaction rates depend on the concentrations of either two different reactants or the square of the concentration of a single reactant. For these reactions, the rate equations can be expressed as:

• Rate = \( k[A]^2 \)
• Rate = \( k[A][B] \)

For a reaction where one piece of zinc metal reacts with two moles of acid, the rate would fit the second formulation. In this scenario, any change in the concentration of either reactant significantly affects the reaction rate. Because of this direct dependency, second-order reactions are more sensitive to concentration changes than first-order reactions.
Graphically, when you plot the inverse of the concentration against time, you'll often get a linear relationship. Scientists use these graphical methods to infer the order of the reaction and further understand the underlying kinetics.

Rate Constant Units

The rate constant \( k \) has different units depending on the order of the reaction. It acts as a bridge, relating the concentration of reactants to the rate of the reaction.
In first-order reactions, the units of the rate constant \( k \) are \( s^{-1} \). This is because when units are substituted into the rate equation \( \text{Rate} = k[A] \), and you simplify \( M/s = k imes M \), you get \( k = s^{-1} \).
For second-order reactions, the units become more complex as more variables are involved. With units substituted into \( \text{Rate} = k[A]^2 \), you derive \( k \) units as \( \mathrm{dm}^3 \, \mathrm{mol}^{-1} \, \mathrm{s}^{-1} \). This unit signifies the broader interaction of reactants in these types of reactions, underscoring the complexity of second-order processes.

Reaction Rate

The reaction rate of a chemical reaction measures how quickly reactants are transformed into products. This can be crucial for controlling industrial processes, predicting reaction behaviors, and ensuring safety in experimental settings.
The rate can be influenced by several factors including:

• Concentration of reactants: Higher concentrations can lead to more collisions, thus increasing the rate.
• Temperature: Higher temperatures generally increase kinetic energy, leading to more effective collisions.
• Catalysts: These substances can change the rate without being consumed in the process.

Understanding reaction rates not only helps in the control of processes but also provides deeper insights into the reaction mechanism itself, which is crucial for scientists and engineers.

Molarity

Molarity serves as a convenient concentration measurement within the realm of chemistry, representing the number of moles of solute per liter of solution. This standard measure, denoted as \( M \), allows chemists to express concentrations in a uniform manner, facilitating calculations and comparisons.

• For example, a 1 M solution contains 1 mole of solute per liter of solution.
• In chemical kinetics, molarity allows for the easy calculation of reaction rates and the relationship of concentration with other factors, like rate constants.

Molarity becomes especially important when studying reactions because slight changes in concentration can significantly impact reaction rates. It effectively provides a straightforward way to observe how changes in concentration can affect the kinetics of a reaction.