Question

The decomposition of nitrogen dioxide at a high temperature $$\mathrm{NO}_{2}(\mathrm{g}) \rightarrow \mathrm{NO}(\mathrm{g})+1 / 2 \mathrm{O}_{2}(\mathrm{g})$$ is second order in this reactant. The rate constant for this reaction is \(3.40 \mathrm{L} / \mathrm{mol} \cdot\) min. Determine the time needed for the concentration of \(\mathrm{NO}_{2}\) to decrease from \(2.00 \mathrm{mol} / \mathrm{L}\) to \(1.50 \mathrm{mol} / \mathrm{L}\)

Short answer

The time needed is approximately 0.049 minutes.

Step-by-step solution

Identify the formula for second-order reactions

For a second-order reaction, the relationship between concentration and time is given by the formula: \[ \frac{1}{[A]_t} - \frac{1}{[A]_0} = kt \] where \([A]_t\) is the concentration at time \(t\), \([A]_0\) is the initial concentration, and \(k\) is the rate constant.

Substitute the known values into the formula

We know that the initial concentration, \([A]_0\), is 2.00 mol/L, the final concentration \([A]_t\) is 1.50 mol/L, and \(k\) is the rate constant 3.40 L/mol·min. Substitute these values into the equation: \[ \frac{1}{1.50} - \frac{1}{2.00} = 3.40 \times t \]

Perform the calculations

Calculate \(\frac{1}{1.50}\) and \(\frac{1}{2.00}\):- \(\frac{1}{1.50} = 0.6667\)- \(\frac{1}{2.00} = 0.5000\)Substitute these findings into the equation: \[ 0.6667 - 0.5000 = 3.40 \times t \] Solve for \(t\):\[ 0.1667 = 3.40 \times t \] \[ t = \frac{0.1667}{3.40} \approx 0.0490 \]

State the time required

The time required for the concentration of \(\mathrm{NO}_2\) to decrease from 2.00 mol/L to 1.50 mol/L is approximately 0.049 minutes.

Key concepts

Rate Constant

The rate constant, often denoted by the symbol \( k \), is a crucial factor in chemical kinetics. It quantifies the rate at which a chemical reaction occurs. The rate constant offers insight into how quickly or slowly a reaction proceeds. Importantly, its value is influence by the temperature and the nature of the reactants involved in the reaction.

For a second-order reaction, like the decomposition of nitrogen dioxide (\( \text{NO}_2 \)), the rate constant has specific units: \( \text{L/mol} \cdot \text{time} \). In our example, the rate constant is given as \( 3.40 \; \text{L/mol} \cdot \text{min} \).

• A larger rate constant suggests a faster reaction.
• This value is critical when calculating how concentrations change over time.
• It assists in predicting how quickly reactants are consumed or products are formed.

Understanding the rate constant is fundamental to mastering chemical kinetics, especially when working with reaction rate equations.

Chemical Kinetics

Chemical kinetics is the branch of chemistry concerned with the speeds or rates at which chemical reactions occur. It helps us understand how different variables affect the reaction rate. This includes factors like concentration, temperature, and the presence of catalysts.

For second-order reactions, such as the decomposition of \( \text{NO}_2 \) we are dealing with, the rate of reaction depends on the square of the concentration of a single reactant, \( \text{NO}_2 \).

Key aspects of chemical kinetics include:

• **Reaction Rate:** This is the change in concentration of reactants or products per unit time.
• **Influencing Factors:** Concentration, temperature, and catalysts can all alter the reaction rate.
• **Order of Reaction:** The reaction order provides clues on how reactant concentration influences the rate.

Chemical kinetics offers tools to model how reactions progress, making it essential for industries relying on efficient chemical processes.

Reaction Rate Equation

The reaction rate equation for a second-order reaction is designed to relate the concentrations of reactants over time. It provides a way to predict the progress of the reaction and the time required to reach a certain concentration.

In the context of our nitrogen dioxide reaction, the second-order rate equation is:\[\frac{1}{[A]_t} - \frac{1}{[A]_0} = kt\]where:

• \([A]_t\) is the concentration at time \(t\).
• \([A]_0\) is the initial concentration.
• \(k\) is the rate constant.

This equation allows calculation of the time taken for the concentration of \(\text{NO}_2\) to change from 2.00 mol/L to 1.50 mol/L. Simply substitute the known values into the formula to find the duration required for such a change.

This equation highlights the relationship between the concentration of reactants and time, illustrating how concentration decreases as the reaction progresses due to reactant consumption.