Question

The rate law for the reaction $$ \begin{array}{c}{\mathrm{Cl}_{2}(g)+\mathrm{CHCl}_{3}(g) \longrightarrow \mathrm{HCl}(g)+\mathrm{CCl}_{4}(g)} \\ {\text { Rate }=k\left[\mathrm{Cl}_{2}\right]^{1 / 2}\left[\mathrm{CHCl}_{3}\right]}\end{array} $$ What are the units for \(k,\) assuming time in seconds and concentration in mol/L?

Short answer

The units for the rate constant \(k\) in the given rate law are: \(k \ [\text{units}] = \frac{\text{L}^{1/2}}{\text{mol}^{1/2}\cdot\text{s}}\).

Step-by-step solution

The given rate law for the reaction is: $$ \text{Rate} = k[\text{Cl}_2]^{1/2}[\text{CHCl}_3] $$ Here, k is the rate constant, and the concentrations of Cl₂ and CHCl₃ are raised to the power of 1/2 and 1, respectively. #Step 2: Write the units for each term#

We know that the rate has the unit of concentration per unit time, which in this case is mol/L·s. The concentrations in the rate law have the unit of mol/L. Rewrite the rate law with units: $$ \frac{\text{mol}}{\text{L}\cdot \text{s}} = k \left(\frac{\text{mol}}{\text{L}}\right)^{1/2} \left(\frac{\text{mol}}{\text{L}}\right) $$ #Step 3: Solve for the units of k#

Now, we will solve for the units of k by isolating k in the equation and cancelling out the units: $$ k = \frac{\frac{\text{mol}}{\text{L}\cdot \text{s}}}{\left(\frac{\text{mol}}{\text{L}}\right)^{1/2}\left(\frac{\text{mol}}{\text{L}}\right)} $$ $$ k = \frac{\frac{1}{\text{s}}}{\left(\frac{\text{mol}}{\text{L}}\right)^{1/2}} $$ $$ k = \frac{\text{L}^{1/2}}{\text{mol}^{1/2}\cdot\text{s}} $$ #Step 4: Present the final units for k#

The units for the rate constant k in the given rate law are: $$ k \ [\text{units}] = \frac{\text{L}^{1/2}}{\text{mol}^{1/2}\cdot\text{s}} $$

Key concepts

reaction rate law

The reaction rate law is an equation that links the speed or rate of a chemical reaction to the concentration of the reactants. Understanding the rate law helps chemists predict how changes in concentration can affect the reaction rate. This law is typically expressed in terms of the rate constant, denoted as \( k \), and the concentrations of reactants raised to a power, often termed the order of reaction.

For example, the rate law for a reaction between chlorine gas \( \text{Cl}_2 \) and chloroform \( \text{CHCl}_3 \) is given by:

• \( \text{Rate} = k[\text{Cl}_2]^{1/2}[\text{CHCl}_3] \)

Here, the exponents \( 1/2 \) and \( 1 \) are reaction orders that show how sensitive the rate is concerning each reactant's concentration.

This means that the rate of reaction is directly proportional to the concentration of \( \text{Cl}_2 \) raised to the power of 1/2, and \( \text{CHCl}_3 \) raised to the power of 1. By understanding these relationships, we can determine how fast a reaction will occur under various conditions.

chemical kinetics

Chemical kinetics is a fundamental concept in chemistry that deals with the speed or rate at which chemical reactions occur and the mechanisms through which they proceed. It's essential for understanding how reactions happen and how they can be controlled.

Different reactions proceed at different speeds. Some may happen almost immediately, like the combustion of gasoline, while others may take years, such as the rusting of iron. By studying chemical kinetics, scientists can determine the rate of a reaction and understand the various factors affecting it, such as:

• Concentration of reactants
• Temperature
• Presence of a catalyst
• Physical state of reactants

Each of these factors can influence the energy required for the reaction and the frequency with which reactant molecules collide. Through kinetics, chemists can optimize reactions for industrial use, making processes faster and more efficient.

The rate constant \( k \) is a key player here. Its value can indicate how quickly a reaction will proceed under a given set of conditions. So, understanding chemical kinetics is crucial for effectively using and predicting the behaviors of chemicals in both lab and real-world applications.

units of measurement

In chemical kinetics, each term in the rate law must have consistent units to ensure that the equation correctly represents the rate of a reaction. The units of the rate constant \( k \) are particularly important, as they vary depending on the overall order of the reaction.

For the reaction

• \( \text{Cl}_2(g) + \text{CHCl}_3(g) \rightarrow \text{HCl}(g) + \text{CCl}_4(g) \)

The rate law is \( \text{Rate} = k[\text{Cl}_2]^{1/2}[\text{CHCl}_3] \).

Here are how the units are determined:

• Rate: Usually expressed in mol/L·s, indicating how much product appears per time unit.
• Concentration: mol/L = these units are used for reactants \([\text{Cl}_2] \) and \([\text{CHCl}_3] \).

To find the units for \( k \), divide the units of rate by the units of concentration terms:\[ k = \frac{\text{mol/L}\cdot\text{s}}{\left(\text{mol/L}\right)^{1/2}\left(\text{mol/L}\right)} = \frac{\text{L}^{1/2}}{\text{mol}^{1/2}\cdot\text{s}}\]This means that the rate constant \( k \) for this reaction has the units L\(^{1/2}\)/mol\(^{1/2} \cdot \)s, which balances out the equation to be dimensionally consistent. Understanding these units is key to interpreting the rate law and making accurate calculations and predictions.