Question

The rate constant of a first-order reaction is \(4.60 \times\) \(10^{-4} \mathrm{~s}^{-1}\) at \(350^{\circ} \mathrm{C}\). If the activation energy is 104 \(\mathrm{kJ} / \mathrm{mol}\), calculate the temperature at which its rate constant is \(8.80 \times 10^{-4} \mathrm{~s}^{-1}\)

Short answer

The new temperature at which the rate constant of the reaction is \(8.80 × 10^{-4} s^{-1}\) is calculated by following the steps shown above using Arrhenius equation.

Step-by-step solution

Write down the known values

We are given three values: the first rate constant \(k_1 = 4.60 × 10^{-4} s^{-1}\), the activation energy \(E_a = 104 kJ/mol\) or \(104000 J/mol\) and the first temperature \(T_1 = 350°C = 350 + 273.15 = 623.15 K\), and we are trying to solve for the new temperature \(T_2\) at which the rate constant is \(k_2 = 8.80 × 10^{-4} s^{-1}\). The gas constant \(R = 8.314 JK^{-1}mol^{-1}\).

Solve for \(lnA\)

From Arrhenius' equation in logarithmic form, we can solve for \(lnA\) using the known values of \(k_1\), \(T_1\), \(R\) and \(E_a\). We have \(lnA = lnk_1 + \frac{E_{a}}{R}(\frac{1}{T_1})\). This gives us the pre-exponential factor, \(A\).

Substitute and Solve for \(T_2\)

Substitute the value of \(lnA\) into the logarithmic form of the Arrhenius equation for \(T_2\). Rearranging the equation gives us \(T_2 = \frac{1}{\frac{E_a}{R} - \frac{lnk_2}{lnA}}\) (after simplification). Substitute the respective values and solve for \(T_2\).

Key concepts

First-order Reactions

A first-order reaction is a type of chemical reaction where the reaction rate is directly proportional to the concentration of one reactant. This means that if you double the concentration of the reactant, the reaction rate doubles as well. Such reactions are described by a rate law of the form:

• \( ext{Rate} = k[A] \)

Here, \( k \) is the rate constant, and \( [A] \) represents the concentration of the reactant.
First-order reactions are common in processes such as radioactive decay and simple chemical reactions. They are called 'first-order' because the rate depends linearly on the concentration of only one reactant.

Arrhenius Equation

The Arrhenius equation is a formula that describes how the rate constant \( k \) of a reaction depends on temperature and activation energy. It is expressed as:

• \( k = Ae^{-rac{E_a}{RT}} \)

Where:

• \( k \) is the rate constant,
• \( A \) is the pre-exponential factor (frequency factor),
• \( E_a \) is the activation energy in Joules per mole,
• \( R \) is the universal gas constant (8.314 J/mol·K),
• \( T \) is the temperature in Kelvin.

This equation shows that the rate constant increases with an increase in temperature or a decrease in activation energy. The Arrhenius equation helps understand how chemical reactions speed up as temperature rises.

Activation Energy

Activation energy is the minimum amount of energy required for a chemical reaction to occur. It acts as a barrier that reactants must overcome to transform into products. In the context of the Arrhenius equation, activation energy \( E_a \) has a critical role, as it affects the rate of the reaction.
Higher activation energy means that fewer molecules have sufficient energy to react at a given temperature, resulting in a slower reaction rate. Conversely, a lower activation energy means that more molecules can overcome the energy barrier, speeding up the reaction. Temperature impacts this as well, since increased temperature boosts the energy of molecules, allowing more of them to surpass the activation energy barrier.

Rate Constant

The rate constant \( k \) is a crucial factor that determines the speed of a chemical reaction. In the context of the Arrhenius equation, the rate constant changes with temperature and activation energy, reflecting the likelihood of reaction occurrences between molecules.

• The larger the rate constant, the faster the reaction proceeds.
• Rate constants can indicate the reaction mechanism and are unique to each reaction under specific conditions.

In the given exercise, the rate constant increases when the temperature changes, demonstrating the principle articulated by the Arrhenius equation. This is why calculating the rate constant at different temperatures is essential for studying reaction kinetics.