Problem 35 - General Chemistry: Principles and Modern Applications

Question

The decomposition of \(\mathrm{HI}(\mathrm{g})\) at \(700 \mathrm{K}\) is followed for \(400 \mathrm{s},\) yielding the following data: at \(t=0,[\mathrm{HI}]=\) \(1.00 \mathrm{M} ;\) at \(t=100 \mathrm{s},[\mathrm{HI}]=0.90 \mathrm{M} ;\) at \(t=200 \mathrm{s}, [\mathrm{HI}]=0.81 \mathrm{M} ; t=300 \mathrm{s},[\mathrm{HI}]=0.74 \mathrm{M} ;\) at \(t=400 \mathrm{s}, [\mathrm{HI}]=0.68 \mathrm{M} .\) What are the reaction order and the rate constant for the reaction: $$\mathrm{HI}(\mathrm{g}) \longrightarrow \frac{1}{2} \mathrm{H}_{2}(\mathrm{g})+\frac{1}{2} \mathrm{I}_{2}(\mathrm{g}) ?$$ Write the rate law for the reaction at 700 K.

Step-by-Step Solution

Answer

The reaction order is 1 (first order). The rate law for the reaction at 700 K is \[Rate = k[HI]\].

Step 1: Determine the reaction order

To deduce the reaction order, we observe the behavior of the concentration of \(HI\) over time. We can see that the concentration of \(HI\) is decreasing by a factor of 0.9 for every 100 s time interval (from 1.00 M to 0.90 M, then 0.90 M to 0.81 M and so on), suggesting a first-order reaction.

Step 2: Calculate the rate constant using the first-order reaction formula

For a first-order reaction, the rate constant \(k\) can be calculated using the formula: \[k = \frac{1}{t} ln\frac{[HI]_0} {[HI]_t}\] where \([HI]_0\) is the initial concentration of \(HI\), \([HI]_t\) is the concentration of \(HI\) at a certain time \(t\). Given that \([HI]_0\) = 1.00 M, \([HI]_t\) = 0.90 M, and \(t\) = 100 s, we can substitute these values into the formula: \[k = \frac{1}{100 s} ln\frac{1.00 M}{0.90 M}\]

Step 3: Write the rate law of the reaction

Once we have determined the order of the reaction and the rate constant, we can write the rate law of the reaction. For a first-order reaction involving a reactant \(HI\), the rate law is: \[Rate = k[HI]\] where \(k\) is the rate constant we calculated and \([HI]\) is the concentration of \(HI\).

Key Concepts

Rate Law

Understanding the rate law of a chemical reaction is key to predicting how the concentration of reactants changes over time. For any reaction, the rate law expresses the rate of the reaction as a function of the concentration of reactants. It typically takes the form:

• Rate = k[A]^m[B]^n

Here, the rate is dependent on the concentration of reactants such as [A] and [B], and the exponents m and n indicate the order of the reaction with respect to each reactant. The rate constant, denoted as k, is a crucial part of the rate law, determining the speed of the reaction.
Knowing the rate law helps in understanding how altering concentrations or the conditions can impact the course of the reaction. For reactions involving gases like HI, the concentration is often measured in molarity (M). Recognizing this allows chemists to carefully manipulate and control reactions.

First-order Reaction

A first-order reaction is a common type of reaction where the rate depends linearly on the concentration of one reactant. In our example with the decomposition of HI, the observed concentration changes over even time intervals suggest it follows first-order kinetics.

• Characteristic Behavior: In a first-order reaction, if you plot the natural logarithm of the concentration (ln[HI]) versus time, it yields a straight line.
• Equation: The integrated rate law for a first-order reaction is:\[ ln[HI]_t = ln[HI]_0 - kt \]where \([HI]_0\) is the initial concentration, and \([HI]_t\) is the concentration at time t.

This helps in predicting how the concentration of HI decreases over time, and it also provides a method to calculate the rate constant, which is a pivotal aspect of understanding first-order reactions.

Rate Constant Calculation

Once you establish that a reaction is first-order, the next step is to calculate the rate constant, k. This constant is vital as it quantifies the reaction rate.

• Using the Formula: For first-order reactions, the rate constant k can be calculated using:\[ k = \frac{1}{t} \ln \left(\frac{[HI]_0}{[HI]_t}\right) \]where t is the time elapsed from the start of the reaction.
• Example Calculation: Given \([HI]_0 = 1.00 \, \text{M}\), \([HI]_{100s} = 0.90 \, \text{M}\), and \(t = 100 \, \text{s}\), substituting these values gives:\[ k = \frac{1}{100} \ln \left(\frac{1.00}{0.90}\right) \]Calculating this yields a specific value for k, indicating how fast the HI concentration decreases at 700 K.

Understanding and calculating the rate constant allows chemists to compare reaction speeds under different conditions or with different reactions.