Question

The decomposition of \(\mathrm{NO}_{2}(g)\) occurs by the following bimolecular elementary reaction: $$ 2 \mathrm{NO}_{2}(g) \longrightarrow 2 \mathrm{NO}(g)+\mathrm{O}_{2}(g) $$ The rate constant at 273 \(\mathrm{K}\) is \(2.3 \times 10^{-12} \mathrm{L} / \mathrm{mol} \cdot \mathrm{s}\) , and the activation energy is 111 \(\mathrm{kJ} / \mathrm{mol}\) . How long will it take for the concentration of \(\mathrm{NO}_{2}(g)\) to decrease from an initial partial pressure of 2.5 \(\mathrm{atm}\) to 1.5 \(\mathrm{atm}\) at \(500 . \mathrm{K}\) ? Assume ideal gas behavior.

Short answer

To find the time it takes for the concentration of \(\mathrm{NO}_{2}(g)\) to decrease from an initial partial pressure of 2.5 \(\mathrm{atm}\) to 1.5 \(\mathrm{atm}\) at 500K, we can follow these steps: 1. Use the Arrhenius equation to find the rate constant, \(k_{500}\), at 500K. 2. Calculate the reaction rate using the rate law and given reaction order, and convert the given partial pressures to concentrations using the ideal gas law. 3. Calculate the average concentration between the initial and final concentrations. 4. Use the expressions for reaction rate to find the time it takes for the reaction. After following these steps and plugging in the given values, we can calculate the time, t, which is how long it will take for the concentration of \(\mathrm{NO}_{2}(g)\) to decrease from an initial partial pressure of 2.5 \(\mathrm{atm}\) to 1.5 \(\mathrm{atm}\) at 500K.

Step-by-step solution

1. Using the Arrhenius equation

We can use the Arrhenius equation to find the rate constant, k, at 500K. The equation is: \(k = k_0 e^{(-E_a / R T)}\) where: \(k\): rate constant at temperature T \(k_0\): rate constant at a reference temperature (273K in this case) \(E_a\): activation energy \(R\): gas constant (8.314 J/molK) \(T\): temperature in Kelvin First, convert the activation energy from kJ/mol to J/mol and the rate constant to L/mol·s: \(E_a = 111,000 \, J / mol\) Now, we need to rearrange the Arrhenius equation to solve for \(k_0\): \(k_0 = k / e^{(-E_a / R T)}\) Plug in the values and calculate \(k_0\): \(k_0 = (2.3 \times 10^{-12}) / e^{(-111,000 / (8.314 \times 273))}\) Now we have the value for \(k_0\), so we can plug it back into the Arrhenius equation to find the rate constant, \(k_{500}\), at 500K: \(k_{500} = k_0 e^{(-E_a / R T)}\) \(k_{500} = k_0 e^{(-111,000 / (8.314 \times 500))}\)

2. Calculate the reaction rate

We are given that the reaction is second-order with respect to \(\mathrm{NO}_2\). The rate law for this reaction is: \(rate = k_{500}[NO_2]^2\) Since we are given the initial and final partial pressures, we can convert them to concentrations using the ideal gas law: \(PV = nRT \Rightarrow [NO_2] = \frac{n}{V} = \frac{P}{RT}\) Therefore, the initial and final concentrations of \(NO_2\) are: \([NO_2]_i = \frac{2.5 \,atm}{(0.0821 \, L/atm \cdot mol \cdot K)(500 \, K)}\) \([NO_2]_f = \frac{1.5 \,atm}{(0.0821 \, L/atm \cdot mol \cdot K)(500 \, K)}\) Now we can find the average concentration between the initial and final concentrations: \([NO_2]_{avg} = \frac{[NO_2]_i + [NO_2]_f}{2}\)

3. Calculate the time taken for the reaction

Now that we have the average concentration and the rate constant at 500K, we can find the reaction rate: \(rate = k_{500}[NO_2]_{avg}^2\) We also know that the rate is equal to the change in concentration divided by the change in time: \(rate = \frac{[NO_2]_i - [NO_2]_f}{t}\) By equating both expressions for rate, we get: \(k_{500}[NO_2]_{avg}^2 = \frac{[NO_2]_i - [NO_2]_f}{t}\) Rearrange the equation to solve for time, t: \(t = \frac{[NO_2]_i - [NO_2]_f}{k_{500}[NO_2]_{avg}^2}\) Plug in the values for \([NO_2]_i, [NO_2]_f, k_{500}\), and \([NO_2]_{avg}\), then calculate the time, t. This is how long it will take for the concentration of \(\mathrm{NO}_{2}(g)\) to decrease from an initial partial pressure of 2.5 \(\mathrm{atm}\) to 1.5 \(\mathrm{atm}\) at 500K.

Key concepts

Understanding Reaction Rate

The reaction rate tells us how quickly a reaction proceeds. It's a way to measure how fast reactants are consumed and products are formed over time. Reaction rates depend on several factors:

• Concentration of reactants: Higher concentrations usually increase reaction rates.
• Temperature: Increasing temperature often speeds up reactions.
• Presence of a catalyst: Catalysts can lower the activation energy, speeding up the reaction.

In the exercise, we are dealing with a second-order reaction where the rate depends on the square of the concentration of \( \text{NO}_2 \). The rate law is expressed as:

\( \text{rate} = k [\text{NO}_2]^2 \)

Understanding how concentrations of reactants change and influence reaction speed is crucial in kinetics, providing insight into how long reactions will take under specific conditions.

Activation Energy's Role

Activation energy (\(E_a\)) is the minimum energy required for a reaction to occur. It's like an energy barrier that reactants must overcome to transform into products.

In the context of our exercise, we've been given an activation energy of 111 kJ/mol. This value indicates the energy needed to initiate the decomposition of \( \text{NO}_2 \).

Factors affecting activation energy include:

• Nature of reactants: Different substances require different energy to react.
• Catalysts: These lower the activation energy, making reactions easier to occur.

By understanding activation energy, we can predict the ease of a reaction and how changes in temperature or catalysts may influence the rate of reaction. It's an essential concept for managing and controlling chemical processes.

Exploring the Arrhenius Equation

The Arrhenius equation is a formula that relates the rate constant (\(k\)) of a reaction to temperature and activation energy:

\( k = A e^{-E_a / (RT)} \)

Where:

• \(A\): Frequency factor, representing the frequency of collision with the correct orientation.
• \(E_a\): Activation energy.
• \(R\): Universal gas constant, 8.314 J/mol·K.
• \(T\): Temperature in Kelvin.

This equation helps us understand how temperature affects the reaction rate. As temperature increases, the exponential term \( e^{-E_a / (RT)} \) rises, boosting the rate constant \(k\).

In our exercise, we used the Arrhenius equation to find the rate constant at 500K, which allowed us to calculate the time needed for the \( \text{NO}_2 \) concentration to decrease. It's a powerful tool for predicting the effects of temperature changes on reaction speed.