Question

The aquation of tris-(1,10-phenanthroline) iron (II) in acid solution takes place according to the equation: $$ \begin{aligned} &\mathrm{Fe}(\mathrm{phen})_{3}^{2}+3 \mathrm{H}_{3} \mathrm{O}^{+}+3 \mathrm{H}_{2} \mathrm{O} \rightarrow \\ &\mathrm{Fe}\left(\mathrm{H}_{2} \mathrm{O}\right)_{6}^{2+}+3 \text { (phen) } \mathrm{H}^{+} \end{aligned} $$ If the activation energy is \(126 \mathrm{~kJ} / \mathrm{mol}\) and frequency factor is \(8.62 \times 10^{17} \mathrm{~s}^{-1}\), at what temperature is the rate constant equal to \(3.63 \times 10^{-3} \mathrm{~s}^{-1}\) for the first order reaction? a. \(0^{\circ} \mathrm{C}\) b. \(50^{\circ} \mathrm{C}\) c. \(45^{\circ} \mathrm{C}\) d. \(90^{\circ} \mathrm{C}\)

Short answer

The rate constant equals the given value at approximately 45°C.

Step-by-step solution

Understand the Arrhenius Equation

The Arrhenius equation describes how the rate constant \( k \) of a chemical reaction depends on the temperature \( T \), the activation energy \( E_a \), and the frequency factor \( A \). It is given by: \[ k = A e^{-E_a/(RT)} \] where \( R \) is the universal gas constant (8.314 J/mol·K). Our goal is to solve this equation for \( T \).

Convert Activation Energy Units

The activation energy is given in \( \mathrm{kJ/mol} \) and needs to be converted to \( \mathrm{J/mol} \). This can be done by multiplying by 1000: \[ E_a = 126 \times 1000 = 126,000 \ \mathrm{J/mol} \]

Solve for Temperature using the Arrhenius Equation

Rearrange the Arrhenius equation to solve for \( T \):\[ T = \frac{E_a}{R \ln(A/k)} \]Substitute the given values:- \( A = 8.62 \times 10^{17} \ \mathrm{s}^{-1} \)- \( k = 3.63 \times 10^{-3} \ \mathrm{s}^{-1} \)- \( E_a = 126,000 \ \mathrm{J/mol} \)- \( R = 8.314 \ \mathrm{J/mol\cdot K} \)First calculate \( \ln(A/k) \): \[ \ln\left(\frac{A}{k}\right) = \ln\left(\frac{8.62 \times 10^{17}}{3.63 \times 10^{-3}}\right) \approx 47.04 \]Substitute these into the formula for \( T \):\[ T = \frac{126,000}{8.314 \times 47.04} \approx 320.7 \ \mathrm{K} \]

Convert Temperature to Celsius

The temperature calculated, \( 320.7 \ \mathrm{K} \), needs to be converted to degrees Celsius. Use the conversion: \[ T_{C} = T_{K} - 273.15 \]Thus, \[ T_{C} = 320.7 - 273.15 \approx 47.55^{\circ} \mathrm{C} \]

Compare with Given Options

Finally, compare the calculated temperature with the provided options. The closest option to \(47.55^{\circ} \mathrm{C}\) is \(45^{\circ} \mathrm{C}\). Therefore, the answer is 45°C.

Key concepts

Activation Energy

In the realm of chemical kinetics, the concept of activation energy is pivotal to understanding how reactions occur. Activation energy, often denoted as \( E_a \), is the minimum energy required for a chemical reaction to take place. It can be thought of as an energy barrier that must be overcome for reactants to transform into products.
Imagine pushing a boulder up a hill: the boulder needs a certain amount of energy to reach the top, after which it rolls down easily. Similarly, molecules must acquire enough energy to reach a transition state before they can form products.
This essential kinetic parameter is expressed in energy units like kilojoules per mole (\( ext{kJ/mol} \)). The higher the activation energy, the slower the reaction because more energy is required for the reaction to occur.
Many factors can influence activation energy, such as temperature and the presence of catalysts. A catalyst, for example, lowers \( E_a \) by providing an alternative pathway for the reaction with less energy needed.

Rate Constant

The rate constant, symbolized as \( k \), is a fundamental part of the Arrhenius equation and chemical kinetics in general. It quantifies the speed of a reaction under specific conditions. Simply put, the rate constant ties together the concentration of reactants to the rate at which they form products.
Each chemical reaction has its own unique rate constant, determined by factors like the nature of reactants, temperature, and the presence of a catalyst.
The Arrhenius equation provides a mathematical relationship between the rate constant \( k \) and factors such as activation energy \( E_a \) and frequency factor \( A \). The formula \( k = A e^{-E_a/(RT)} \) shows how \( k \) changes with temperature \( T \) and \( E_a \).
At higher temperatures, \( k \) tends to increase because the molecules possess more kinetic energy, which leads to more frequent and successful collisions between reactants.

Chemical Kinetics

Chemical kinetics delves into the rates of chemical reactions and the factors affecting those rates. It's a branch of physical chemistry that provides insight into how reactions occur over time.
Understanding kinetics is crucial for fields ranging from industrial chemistry to biochemistry. It allows chemists to optimize reactions for desired outcomes, such as increasing the yield of a product or minimizing side reactions.
In kinetics, key parameters like the rate constant and activation energy explain how fast a reaction happens and what conditions are necessary. This information is used to deduce mechanisms of reactions, predicting how various changes like temperature or concentration will affect the rate.
Through techniques such as monitoring concentration changes over time, chemists can derive rate laws, which precisely describe the relationship between the concentration of reactants and the speed of the reaction.

Frequency Factor

The frequency factor, often denoted as \( A \), appears in the Arrhenius equation representing the pre-exponential factor in the equation \( k = A e^{-E_a/(RT)} \). This constant embodies the frequency of collisions between reacting molecules that are correctly oriented for reaction, without considering energy barriers.
Essentially, the frequency factor gives an idea of how often molecules collide in the right orientation. Its units are the same as those of the rate constant \( k \), which depend on the order of the reaction.
The value of \( A \) indicates how "ready" a reaction is to occur once the activation energy barrier is surpassed. A higher frequency factor means more favorable collisions, leading to a faster reaction rate when activation energy conditions are met.
The frequency factor is influenced by factors such as the physical state of reactants and the presence of catalysts, making it an important element in the study of chemical reactions.