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Measurements of the initial rate of reaction between two compounds, triphenylmethyl hexachloroantimonate (substance I) and bis-(9-ethyl-3-carbazolyl)methane (substance II), in 1,2 -dichloroethane at \(40^{\circ} \mathrm{C}\) yielded these data: \begin{tabular}{lrl} \hline Initial Concentration \(\times 10^{5}\) \((\mathrm{~mol} / \mathrm{L})\) & & \\ \hline\([[]]\) & [1]] & Initial Rate \(\times 10^{9}\) \(\left(\mathrm{~mol} \mathrm{~L}^{-1} \mathrm{~s}^{-1}\right)\) \\ \hline 1.65 & 10.6 & 1.50 \\ 14.9 & 10.6 & 17.7 \\ 14.9 & 7.10 & 11.2 \\ 14.9 & 3.52 & 6.30 \\ 14.9 & 1.76 & 3.10 \\ 4.97 & 10.6 & 4.52 \\ 2.48 & 10.6 & 2.70 \\ \hline \end{tabular} (a) Determine the order of the reaction with respect to substance I and substance II. (b) Derive the rate law for this reaction. (c) Calculate the rate constant \(k\) and express it in appropriate units. (d) Calculate the initial rate of reaction when \([\mathbf{I}]=\) \(8.3 \times 10^{-5} \mathrm{~mol} / \mathrm{L}\) and \([\mathrm{II}]=6.78 \times 10^{-5} \mathrm{~mol} / \mathrm{L}\)

Step by step solution

01

Determining the Order with Respect to Substance I

To find the order with respect to substance I, compare initial rates when the concentration of II is constant.\Using experiments 1 and 2: \(\frac{\text{Rate 2}}{\text{Rate 1}} = \frac{17.7}{1.50} = 11.8\).\\(\frac{[\text{I}]_2}{[\text{I}]_1} = \frac{14.9}{1.65} = 9.03\).\\(11.8 = (9.03)^m\).\Solving \(m \approx 1\). Thus, order with respect to I is 1.

02

Determining the Order with Respect to Substance II

To find the order with respect to substance II, use data where the concentration of I is constant.\Using experiments 2 and 4: \(\frac{\text{Rate 4}}{\text{Rate 2}} = \frac{6.30}{17.7} = 0.356\).\\(\frac{[\text{II}]_4}{[\text{II}]_2} = \frac{3.52}{10.6} = 0.332\).\\(0.356 = (0.332)^n\).\Solving \(n \approx 1\). Thus, order with respect to II is 1.

03

Deriving the Rate Law

Since the reaction is first order with respect to both I and II, the rate law is:\[ \text{Rate} = k \cdot [\text{I}] \cdot [\text{II}] \]

04

Calculating the Rate Constant k

Use the rate law and experimental data from any trial.\Using experiment 1:\[ 1.50 \times 10^{-9} = k \cdot (1.65 \times 10^{-5}) \cdot (10.6 \times 10^{-5}) \]\[ k = \frac{1.50 \times 10^{-9}}{1.65 \times 10^{-5} \cdot 10.6 \times 10^{-5}} = 8.54 \times 10^{1} \, \text{L} \cdot \text{mol}^{-1} \cdot \text{s}^{-1} \]

05

Calculating the Initial Rate for Given Concentrations

Using \([\text{I}] = 8.3 \times 10^{-5} \) mol/L and \([\text{II}] = 6.78 \times 10^{-5} \) mol/L:\[ \text{Rate} = 8.54 \times 10^{1} \cdot (8.3 \times 10^{-5}) \cdot (6.78 \times 10^{-5}) \]Calculate:\[ \text{Rate} \approx 4.81 \times 10^{-9} \, \text{mol} \, \text{L}^{-1} \, \text{s}^{-1} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Rate of Reaction

The rate of reaction is a crucial concept in chemical kinetics. It refers to the speed at which reactants are converted into products. This speed can be quantified as the change in concentration of a reactant or product per unit time. In simpler terms, it tells us how fast the reaction is occurring.

In the exercise provided, we measure the initial rate of reaction between two substances, triphenylmethyl hexachloroantimonate and bis-(9-ethyl-3-carbazolyl)methane, by observing how their concentrations change over time at a certain temperature. By analyzing the data, we can establish a mathematical relationship known as the rate law. This helps in understanding how factors like concentration affect the rate and allows for predictions on how altering these factors might change the reaction speed.

To determine the rate, one often uses specific experiments where the initial concentrations and rates are carefully measured. These measurements assist in drawing conclusions about the orders of reaction and other kinetics parameters.

Order of Reaction

The order of reaction provides insight into how the concentration of a substance affects the rate of reaction. It is determined experimentally and can vary for each reactant involved in the reaction. Essentially, the order tells us the power to which the concentration of a reactant is raised in the rate law.

In the given exercise, the reaction was found to be first order with respect to both substances I and II. This means that the rate of reaction is directly proportional to the concentration of each substance. For instance, doubling the concentration of substance I will double the rate, given substance II鈥檚 concentration remains constant.

To find these orders, scientists observe changes in rate while varying concentrations of one reactant and keeping others constant. Then, they compare the experimental ratio of rate changes to the ratio of concentration changes, solving for a power, often leading to integers or simple fractions. Understanding these orders is vital for developing accurate models for the rates of chemical reactions.

Rate Law

The rate law is an equation that relates the rate of a chemical reaction to the concentration of the reactants. It is formulated based on the order of reaction and the rate constant. The general form of a rate law for a reaction involving reactants A and B might look like this:

• Rate = k [A]^m [B]^n

where "k" is the rate constant, and "m" and "n" are the orders of the reaction with respect to A and B.

In the exercise, the reaction between substances I and II is first order with respect to both, leading to a rate law:

• Rate = k [I] [II]

This linear relationship indicates that the rate is simply dependent on the product of the concentrations of I and II, each raised to the power of one. The rate constant, k, which has units of L mol鈦宦� s鈦宦� in this particular context, requires calculations from experimental data to determine.

The rate law is a powerful tool in predicting how changes in concentrations affect the rate, thus helping in controlling reaction conditions for desired outcomes in industrial and laboratory settings.