Assume that you are studying the first-order conversion of a reactant X to products in a reaction vessel with a constant volume of 1.000 L. At 1 p.m., you start the reaction at 25 °C with 1.000 mol of X. At 2 p.m., you find that 0.600 mol of X remains, and you immediately increase the temperature of the reaction mixture to 35 °C. At 3 p.m., you discover that 0.200 mol of X is still present. You want to finish the reaction by 4 p.m. but need to continue it until only 0.010 mol of X remains, so you decide to increase the temperature once again. What is the minimum temperature required to convert all but 0.010 mol of X to products by 4 p.m.?

Verified step by step guidance

1. <insert step 1> Determine the rate constant (k) for the reaction at 25 °C using the first-order rate equation: \( k = \frac{1}{t} \ln \frac{[X]_0}{[X]} \), where \([X]_0\) is the initial concentration and \([X]\) is the concentration at time t.>
2. <insert step 2> Calculate the rate constant (k) for the reaction at 35 °C using the same first-order rate equation, with the concentrations and time interval from 2 p.m. to 3 p.m.>
3. <insert step 3> Use the Arrhenius equation \( k = A e^{-\frac{E_a}{RT}} \) to determine the activation energy (E_a) of the reaction, using the rate constants from steps 1 and 2 and the corresponding temperatures.>
4. <insert step 4> Calculate the rate constant (k) required to achieve the desired concentration of 0.010 mol of X by 4 p.m. using the first-order rate equation, with the concentration at 3 p.m. and the desired concentration at 4 p.m.>
5. <insert step 5> Use the Arrhenius equation again to find the minimum temperature required to achieve the rate constant calculated in step 4, using the activation energy determined in step 3.>

Key Concepts

First-Order Kinetics

First-order kinetics describes a reaction where the rate is directly proportional to the concentration of one reactant. The rate law can be expressed as rate = k[X], where k is the rate constant. The integrated rate law for a first-order reaction allows us to calculate the concentration of the reactant at any time, which is essential for determining how much reactant remains at specific intervals.

Arrhenius Equation

The Arrhenius equation relates the rate constant of a reaction to temperature, expressed as k = A * e^(-Ea/RT), where A is the pre-exponential factor, Ea is the activation energy, R is the gas constant, and T is the temperature in Kelvin. This equation indicates that increasing the temperature generally increases the rate constant, thereby accelerating the reaction. Understanding this relationship is crucial for determining the minimum temperature needed to achieve the desired conversion by a specific time.

Reaction Completion and Time Management

In chemical kinetics, managing the time required for a reaction to reach completion is vital, especially when aiming for a specific concentration of reactants. By analyzing the remaining concentration of reactant X at different times and applying the first-order kinetics and Arrhenius equation, one can estimate the necessary conditions to achieve the target concentration within the desired timeframe. This involves calculating the required rate constant and subsequently the temperature needed to meet the reaction completion goal.