Question

The tabulated data were collected for this reaction at \(500^{\circ} \mathrm{C}\) : $$ \mathrm{CH}_{3} \mathrm{CN}(g) \longrightarrow \mathrm{CH}_{3} \mathrm{NC}(g) $$ $$ \begin{array}{cc} \text { Time (h) } & {\left[\mathrm{CH}_{3} \mathrm{CN]}\right. \text { (M) }} \\\ 0.0 & 1.000 \\ \hline 5.0 & 0.794 \\ \hline 10.0 & 0.631 \\ \hline 15.0 & 0.501 \\ \hline 20.0 & 0.398 \\ \hline 25.0 & 0.316 \\ \hline \end{array} $$ a. Determine the order of the reaction and the value of the rate constant at this temperature. b. What is the half-life for this reaction (at the initial concentration)? c. How long will it take for \(90 \%\) of the \(\mathrm{CH}_{3} \mathrm{CN}\) to convert to \(\mathrm{CH}_{3} \mathrm{NC} ?\)

Short answer

The reaction is first-order with respect to CH3CN. Rate constant k is determined from the slope of the \( \ln([CH_3CN]) \) versus time plot, and the half-life is given by \( t_{1/2} = \frac{0.693}{k} \). To calculate the time for 90% conversion, use the first-order integrated rate law with the final concentration being 10% of the initial concentration.

Step-by-step solution

- Inspect The Change In Concentration

Plot the concentration of CH3CN versus time to see if the reaction fits zero, first, or second order kinetics. If concentration halves over equal time intervals, it's a first order reaction.

- Use A Kinetic Model To Determine Reaction Order

For a first-order reaction, the rate is given by the equation \( -\frac{{d[CH_3CN]}}{{dt}} = k[CH_3CN] \).Apply the integrated rate law for the first order reaction to the data: \( \ln([CH_3CN]_0) - \ln([CH_3CN]_t) = kt \). Test the linearity of \( \ln([CH_3CN]) \) versus time to confirm the order.

- Calculate The Rate Constant, k

Use the initial concentration and the concentration at known times to find the rate constant k from the plot of \( \ln([CH_3CN]) \) versus time. The slope of this line gives the rate constant k.

- Determine The Half-Life

For a first-order reaction, half-life \( t_{1/2} \) is calculated using the equation \( t_{1/2} = \frac{0.693}{k} \). Use the calculated rate constant k to determine the half-life.

- Calculate Time For 90% Conversion

To find the time for 90% conversion, use \( [CH_3CN] = 0.1 \times [CH_3CN]_0 \). Apply the first-order integrated rate law to solve for time when 10% of the original concentration is left.

Key concepts

Reaction Order

Understanding reaction order is crucial for predicting how the concentration of reactants will influence the rate of a chemical reaction. It's essentially an exponent that describes how the rate is affected by the concentration of a reactant. For example, if a reaction has a first order with respect to a reactant A, it means that the rate of the reaction is directly proportional to the concentration of A. On the other hand, if it’s second order, the rate is proportional to the square of the concentration of A.

Determining the reaction order can be done experimentally by measuring concentration changes over time and analyzing the data. In the given exercise, plotting the concentration of CH3CN against time and seeing that the concentration decreases by half over equal time intervals suggests a first-order reaction. This is because only first-order reactions showcase this unique halving behavior in equal time intervals.

Rate Constant

The rate constant, denoted as k, is a proportionality factor that provides the relationship between the rate of a chemical reaction and the concentrations of reactants. For a given reaction at a specific temperature, the rate constant is a fixed value. This value is crucial because it allows us to calculate how fast a reaction proceeds when the concentration of the reactants is known.

In the context of the provided exercise, once the order of the reaction is confirmed to be first, the rate constant can be determined. The step-by-step solution involves plotting \( \ln([CH_3CN]) \) against time and then finding the slope of the resulting line, which directly gives us the rate constant, k.

Half-Life

Half-life, represented as \(t_{1/2}\), is a term commonly used in chemical kinetics to describe the time it takes for half of a reactant to be used up in a reaction. In a first-order reaction, half-life is independent of the initial concentration, meaning it stays constant regardless of the starting amount of the reactant.

Using the rate constant obtained from the experiment's data, we can easily calculate the half-life for a first-order reaction by applying the formula \(t_{1/2} = \frac{0.693}{k}\). This demonstrates the direct relationship between the half-life and rate constant, illustrating the integral role the rate constant plays in reaction timing.

First-Order Reactions

First-order reactions are a fundamental concept in chemical kinetics where the rate is directly proportional to the concentration of one reactant. Specifically, the rate of reaction changes linearly with changes in concentration. In mathematical terms, this is expressed as \( -\frac{d[CH_3CN]}{dt} = k[CH_3CN] \), where the negative sign indicates that the concentration of the reactant, CH3CN in this case, decreases over time.

The characteristic property of a first-order reaction is that its half-life remains constant over the course of the reaction. This can be extremely helpful in practical applications, such as determining the shelf-life of pharmaceuticals or predicting the rate at which pollutants degrade in the environment.

Integrated Rate Law

The integrated rate law for first-order reactions gives us a direct connection between the concentration of reactants and time. Unlike the rate law, which tells us the reaction rate at a moment in time, the integrated rate law describes how concentrations change over time.

For a first-order reaction, the integrated rate law is \( \ln([CH_3CN]_0) - \ln([CH_3CN]_t) = kt \), where \( [CH_3CN]_0 \) is the initial concentration and \( [CH_3CN]_t \) is the concentration at time t. This equation can be used to calculate the time needed for the reactant concentration to reach any given value, such as in the exercise where we determine the time necessary for a 90% conversion of the reactant. By rearranging the equation and solving for time, t, you obtain how long it will take for the reaction to proceed to a certain extent.