Problem 95 - Chemistry: The Central Science in SI Units, Global Edition

Question

The dimerization of \(\mathrm{C}_{2} \mathrm{~F}_{4}\) to \(\mathrm{C}_{4} \mathrm{~F}_{8}\) has a rate constant \(k=0.045 \mathrm{M}^{-1} \mathrm{~s}^{-1}\) at \(450 \mathrm{~K} .\) (a) Based on the unit of \(k\) what is the reaction order in \(\mathrm{C}_{2} \mathrm{~F}_{4} ?(\mathbf{b})\) If the initial concentration of \(\mathrm{C}_{2} \mathrm{~F}_{4}\) is \(0.100 \mathrm{M}\), how long would it take for the concentration to decrease to \(0.020 \mathrm{M}\) at \(450 \mathrm{~K}\) ?

Step-by-Step Solution

Answer

The reaction order in C2F4 is second-order, as the rate constant, k, has units of \(M^{-1} . s^{-1}\). To calculate the time it takes for the concentration of C2F4 to decrease from 0.100 M to 0.020 M at 450 K, the integrated rate law for a second-order reaction is used: \(\frac{1}{[A]_{t}}-\frac{1}{[A]_{0}} = kt\). By plugging in the given values, the time (t) required is approximately 888.89 seconds.

Step 1: Determine the reaction order

We know that the units of the rate constant vary depending on the reaction order. The reaction orders and their corresponding rate constant units are as follows: - Zero-order: \(M.s^{-1}\) - First-order: \(s^{-1}\) - Second-order: \(M^{-1}.s^{-1}\) The given rate constant, k has the unit \(M^{-1} . s^{-1}\). Comparing this unit to the units listed above, we can determine that the reaction order in C2F4 is second-order.

Step 2: Write the integrated rate law

Since we know that the reaction is second-order, we can write the integrated rate law for a second-order reaction: \[\frac{1}{[A]_{t}}-\frac{1}{[A]_{0}} = kt\] Here, \([A]_{0}\) represents the initial concentration, \([A]_{t}\) represents the final concentration of C2F4, k is the rate constant, and t is the time.

Step 3: Solve for time

We are given the initial concentration \([A]_{0} = 0.100 M\), the final concentration \([A]_{t} = 0.020 M\), and the rate constant \(k = 0.045 M^{-1}.s^{-1}\). We can plug these values into the integrated rate law equation and solve for time (t): \[\frac{1}{0.020} - \frac{1}{0.100} = 0.045 \cdot t\] Now, calculate t: \(t = \frac{\frac{1}{0.020} - \frac{1}{0.100}}{0.045} \) \(t = \frac{50 - 10}{0.045}\) \(t = \frac{40}{0.045}\) \(t \approx 888.89 s\)

Step 4: Result

The time it takes for the concentration of C2F4 to decrease from 0.100 M to 0.020 M at 450 K is approximately 888.89 seconds.

Key Concepts

Reaction Order

In chemical kinetics, the reaction order is a fundamental concept that tells us how the concentration of reactants affects the rate of reaction. Reaction orders can be zero, first, second, or even fractional and complex, but in simpler terms: it relates the changing concentration of reactants to the speed at which a chemical reaction proceeds.

For example, if a reaction is designated as being of 'second-order' with respect to a reactant, it means that the rate of reaction is proportional to the square of the concentration of that reactant. The total reaction order can be determined by adding up the individual orders with respect to all reactants in the rate equation.

The way we understand reaction orders is often through the units of the rate constant (k) given in the problem. The units themselves change depending on what the order is. Recognizing these units helps chemists deduce the reaction order from experimental data easily. In this particular example, the unit of the rate constant given is \(M^{-1}\,s^{-1}\), which correlates directly to a second-order reaction. This means the rate of the reaction here is directly related to the square of the concentration.

Rate Constant

The rate constant (k) is a crucial parameter in the study of chemical kinetics. It appears in the rate equation of a chemical reaction and ties together the rate of reaction with the concentrations of reactants, adjusted by the reaction order.

The size of the rate constant indicates the speed of the reaction: a larger rate constant means a faster reaction at given concentrations of reactants. However, it's not solely the value that provides insight, but also the units in which it is expressed. That's because the units of k vary with the reaction order, acting like a fingerprint that gives clues about how the reaction completes itself.

For example, in this exercise, since the dimerization reaction has a rate constant given as \(0.045 \, M^{-1}\,s^{-1}\), it confirms that our reaction order is second-order. It's through the use of the rate constant in kinetic calculations, along with knowledge of initial concentrations, that we solve practical problems, such as determining how long a reaction will take.

Second-order Reactions

Second-order reactions are a type of chemical reaction where the rate is proportional to either the concentration of two reactants or the square of the concentration of a single reactant.

The integrated rate law for a second-order reaction with respect to a single reactant is given by the expression: \[\frac{1}{[A]_t} - \frac{1}{[A]_0} = kt\]where \([A]_0\) is the initial concentration, \([A]_t\) is the concentration at time \(t\), and \(k\) is the rate constant. This equation is handy because it allows us to calculate the time it takes for a reactant concentration to decrease to a certain point.

Understanding this concept helps predict how substances in a reaction mixture will evolve over time. In our given problem, the second-order nature allowed us to use the given concentrations and the rate constant to find out that it takes about 888.89 seconds for \([C_2 F_4]\) to decrease from 0.100 M to 0.020 M at 450 K, which showcases the power of the integrated rate law in practical kinetics.