Problem 120 - Chemistry: An Atoms-Focused Approach

Question

On the basis of the frequency factors and activation energy values of the following two reactions, determine which one will have the larger rate constant at room temperature \((298 \mathrm{K})\). \(\mathrm{O}_{3}(g)+\mathrm{Cl}(g) \rightarrow \mathrm{ClO}(g)+\mathrm{O}_{2}(g)\) \(A=2.9 \times 10^{-11} \mathrm{cm}^{3} /(\text { molecules } \cdot \mathrm{s}) \quad E_{2}=2.16 \mathrm{kJ} / \mathrm{mol}\) \(\mathrm{O}_{3}(g)+\mathrm{NO}(g) \rightarrow \mathrm{NO}_{2}(g)+\mathrm{O}_{2}(g)\) \(A=2.0 \times 10^{-12} \mathrm{cm}^{3} /(\text { molecules } \cdot \mathrm{s}) \quad E_{\mathrm{a}}=11.6 \mathrm{kJ} / \mathrm{mol}\)

Step-by-Step Solution

Answer

Use the Arrhenius equation, given that the frequency factors (A) and activation energy values (Ea) for the reactions are, respectively, 2.9x10^-11 cm^3/(molecule·s) and 2.16 kJ/mol for the first reaction, and 2.0x10^-12 cm^³/(molecule·s) and 11.6 kJ/mol for the second reaction. Answer: The reaction O3 + Cl -> ClO + O2 has a larger rate constant at room temperature (298 K).

Step 1: Write down the Arrhenius equation

The Arrhenius equation is given by: \(k = A \times e^{-Ea / (R \cdot T)}\) where - \(k\) is the rate constant - \(A\) is the frequency factor - \(Ea\) is the activation energy - \(R\) is the gas constant (8.314 J/(mol·K)) - \(T\) is the temperature (here, 298 K)

Step 2: Calculate the rate constant for the first reaction (O3 + Cl -> ClO + O2)

For the first reaction, we have \(A = 2.9 \times 10^{-11} \mathrm{cm}^{3}/(\mathrm{molecules} \cdot \mathrm{s})\) and \(Ea = 2.16 \,\mathrm{kJ}/\mathrm{mol}\). We must first convert Ea to J/mol: \(Ea = 2.16 \,\mathrm{kJ}/\mathrm{mol} \cdot 1000\,\mathrm{J}/\mathrm{kJ} = 2160\,\mathrm{J}/\mathrm{mol}\) Then, we can plug these values into the Arrhenius equation to find the rate constant \(k_1\): \(k_1 = 2.9 \times 10^{-11} \cdot e^{-2160 / (8.314 \cdot 298)}\) Now, calculate \(k_1\): \(k_1 \approx 3.09 \times 10^{-12} \, \mathrm{cm}^{3}/(\mathrm{molecules} \cdot \mathrm{s})\)

Step 3: Calculate the rate constant for the second reaction (O3 + NO -> NO2 + O2)

For the second reaction, we have \(A= 2.0 \times 10^{-12} \mathrm{cm}^{3}/(\mathrm{molecules} \cdot \mathrm{s})\) and \(Ea = 11.6 \, \mathrm{kJ}/\mathrm{mol}\). We must also convert Ea to J/mol for this reaction: \(Ea = 11.6 \, \mathrm{kJ}/\mathrm{mol} \cdot 1000\,\mathrm{J}/\mathrm{kJ} = 11600\, \mathrm{J}/\mathrm{mol}\) Then, we can plug these values into the Arrhenius equation to find the rate constant \(k_2\): \(k_2 = 2.0 \times 10^{-12} \cdot e^{-11600 / (8.314 \cdot 298)}\) Now, calculate \(k_2\): \(k_2 \approx 8.40 \times 10^{-17} \,\mathrm{cm}^{3}/(\mathrm{molecules} \cdot \mathrm{s})\)

Step 4: Compare the rate constants of the two reactions

Now we have: \(k_1 \approx 3.09 \times 10^{-12} \,\mathrm{cm}^{3}/(\mathrm{molecules} \cdot \mathrm{s})\) \(k_2 \approx 8.40 \times 10^{-17} \,\mathrm{cm}^{3}/(\mathrm{molecules} \cdot \mathrm{s})\) Since \(k_1 > k_2\), the reaction \(\mathrm{O}_{3}(g)+\mathrm{Cl}(g) \rightarrow \mathrm{ClO}(g)+\mathrm{O}_{2}(g)\) will have a larger rate constant at room temperature (298 K).

Key Concepts

Rate Constant

The rate constant, symbolized by \( k \), is an essential part of the Arrhenius equation. This constant gives us a glimpse into the speed of a chemical reaction under specific conditions like temperature and the presence of a catalyst. In simpler terms, it helps determine how fast or slow a reaction will proceed.

• The larger the rate constant, the faster the reaction.
• The formula for the rate constant is given by the Arrhenius equation: \[ k = A \times e^{-Ea / (R \cdot T)} \] where \( A \) is the frequency factor, \( e \) is the base of the natural logarithm, and \( Ea \), \( R \), and \( T \) are the activation energy, the gas constant, and temperature respectively.

Understanding the rate constant can give insights into reaction efficiency, which is vital for industries relying on chemical processes. This explains why reactions with higher rate constants are preferred, as they achieve desired results sooner. Calculations for rate constants often involve exponentials due to the nature of reaction dynamics, hence the equation's reliance on \( e \).

Activation Energy

Activation energy, denoted as \( Ea \), is the energy barrier that must be surpassed for a chemical reaction to occur. It's essentially the "hill" that reactants need to climb to transform into products. The Arrhenius equation indicates the role of activation energy, as it's a determinant of the rate constant.
When \( Ea \) is small:

• Reactions tend to proceed faster.
• The molecules require less energy to overcome obstacles.

When \( Ea \) is large:

• Reactions are slower because molecules must acquire more energy to proceed.

The unit used for activation energy is typically joules per mole (J/mol) or kilojoules per mole (kJ/mol). In our example, the first reaction had an activation energy of 2.16 kJ/mol, enabling it to proceed much faster than the second reaction, which had a hefty 11.6 kJ/mol. Lower activation energies generally lead to higher rate constants, making the reaction more favorable at the given temperature. This is the reason the \( \mathrm{O}_{3} + \mathrm{Cl} \) reaction had a higher rate constant compared to \( \mathrm{O}_{3}+ \mathrm{NO} \).

Frequency Factor

The frequency factor, represented by \( A \) in the Arrhenius equation, measures how often molecules collide with the proper orientation for a reaction to occur. It is an indicator of the collision rate and probability that collisions will result in a reaction. This factor is usually determined empirically, meaning through experimental observation rather than theoretical calculations.
In general:

• A higher frequency factor suggests that conditions are favourable for more frequent successful reactions.
• The unit of \( A \) is typically in volume per time, such as \( \mathrm{cm}^{3}/ \text{(molecules} \cdot \mathrm{s)} \).

In our exercise, the first reaction had a higher frequency factor of \( 2.9 \times 10^{-11} \), which also contributed to a higher rate constant compared to the second reaction with \( 2.0 \times 10^{-12} \). Therefore, with both lower activation energy and a higher frequency factor, the first reaction proceeds faster at the same temperature, confirming the theoretical predictions of the Arrhenius equation. Understanding the frequency factor helps in designing chemicals and processes that maximize reaction efficiency.