Question

A certain substance, initially present at \(0.0800 M,\) decomposes by zero-order kinetics with a rate constant of \(2.50 \times 10^{-2} \mathrm{mol} / \mathrm{L}\) . s. Calculate the time (in seconds required for the system to reach a concentration of 0.0210\(M .\)

Short answer

The time required for the system to reach a concentration of 0.0210 M is 2360 seconds.

Step-by-step solution

Write down the given information

We are given the following information: - Initial concentration \([A]_0 = 0.0800 \mathrm{M}\) - Rate constant \(k = 2.50 \times 10^{-2} \mathrm{mol}/\mathrm{L} \cdot s\) - Final concentration \([A] = 0.0210 \mathrm{M}\)

Use the integrated rate law equation for zero-order reactions

The integrated rate law for a zero-order reaction is given by the following formula: \[[A] = [A]_0 - kt\] We need to solve this equation for the time \(t\).

Plug in the given values and solve for time \(t\)

Substitute the given concentrations and rate constant into the equation: \begin{align*} 0.0210 &= 0.0800 - (2.50 \times 10^{-2})t \\ \end{align*} Now, solve for \(t\): \begin{align*} (2.50 \times 10^{-2})t &= 0.0800 - 0.0210 \\ t &= \frac{0.0800 - 0.0210}{2.50 \times 10^{-2}} \\ t &= \frac{0.0590}{2.50 \times 10^{-2}} \\ t &= 2360\,\mathrm{s} \end{align*}

Present the result

The time required for the system to reach a concentration of 0.0210 M is 2360 seconds.

Key concepts

Integrated Rate Law

The integrated rate law is a powerful tool in understanding the progression of chemical reactions over time. For zero-order kinetics, this law is especially straightforward. In such reactions, the rate at which the concentration of a reactant decreases is constant. This means that the change in concentration does not depend on the concentration itself.

For zero-order reactions, the integrated rate equation is formulated as:

• \[ [A] = [A]_0 - kt \]

Where:

• \([A]\) is the concentration of the reactant at time \(t\).
• \([A]_0\) is the initial concentration of the reactant.
• \(k\) is the zero-order rate constant.
• \(t\) is the time elapsed.

As you can see, the integrated rate law expresses the concentration at any given time based on the initial concentration, the rate constant, and time. This equation is linear, reflecting the fixed rate of decrease in concentration characteristic of zero-order reactions.

Reaction Rate Constant

The reaction rate constant, denoted as \( k \), is crucial for determining the speed of a reaction. It is a proportionality constant that links the reaction rate to the concentrations of the reactants.In zero-order reactions, the rate constant tells us how much the concentration decreases per unit of time regardless of the concentration itself.

Its units reflect the order of the reaction. For zero-order processes, the units are typically \( \mathrm{mol/L \,s} \), as shown in our example where the rate constant is \(2.50 \times 10^{-2} \mathrm{mol/L \,s}\). When solving problems, this constant allows us to calculate how long a reaction will take to reach a certain concentration. This understanding opens up the ability to predict changes in concentration with time, an invaluable piece of information in both laboratory and industrial settings. Remember, the value of \(k\) is characteristic of the specific reaction and conditions, defining how fast or slow the conversion proceeds.

Chemical Concentration Change

In every chemical reaction, understanding how concentration changes over time is essential. This is especially true in reactions following zero-order kinetics, where the concentration decreases linearly with time.

Initially, the reaction starts with a known concentration, which in our original exercise was \(0.0800 \mathrm{M}\). Over time, as the reaction proceeds, the concentration decreases at a constant rate. To predict the concentration at any future time, or to determine how long it takes to reach a specific concentration, we use the integrated rate law formula as discussed earlier. This approach is helpful for determining reaction duration and conversion rates.In our example, we calculated the time taken to reach a concentration of \(0.0210 \mathrm{M}\), which was 2360 seconds. This demonstrates how effectively you can model and predict concentration changes in zero-order kinetics by using rate equations.