Question

The equation tris(1,10-phenanthroline) iron(II) in acid solution takes place according to the equation: \(\mathrm{Fe}(\text { phen })_{3}^{2+}+3 \mathrm{H}_{3} \mathrm{O}^{+}+3 \mathrm{H}_{2} \mathrm{O} \rightarrow\) $$ \mathrm{Fe}\left(\mathrm{H}_{2} \mathrm{O}\right)_{6}^{2+}+3 \text { (Phen) } \mathrm{H}^{+} $$ If the activation energy (Ea) is \(126 \mathrm{~kJ} / \mathrm{mol}\) and the rate constant at \(30^{\circ} \mathrm{C}\) is \(9.8 \times 10^{-3} \mathrm{~min}^{-1}\), what is the frequency factor (A)? a. \(9.5 \times 10^{18} \mathrm{~min}^{-1}\) b. \(2.5 \times 10^{19} \mathrm{~min}^{-1}\) c. \(55 \times 10^{19} \mathrm{~min}^{-1}\) d. \(5.0 \times 10^{19} \mathrm{~min}^{-1}\)

Short answer

The frequency factor, \( A \), is \(5.0 \times 10^{19} \text{ min}^{-1}\) (option d).

Step-by-step solution

Understanding the Problem

We need to find the frequency factor (A) in the Arrhenius equation given the activation energy and the rate constant at a specific temperature. The Arrhenius equation is \( k = A e^{-\frac{E_a}{RT}} \), where \( k \) is the rate constant, \( E_a \) is the activation energy, \( R \) is the gas constant, and \( T \) is the temperature in Kelvin.

Convert Temperature to Kelvin

The given temperature is \( 30^{\circ} C \). To convert Celsius to Kelvin, use the formula \( T(K) = T(^{\circ} C) + 273.15 \). Therefore, at \( 30^{\circ} C \), \( T \) in Kelvin is \( 30 + 273.15 = 303.15 \) K.

Use Arrhenius Equation

The Arrhenius equation is \( k = A e^{-\frac{E_a}{RT}} \). Rearrange this to solve for \( A \): \( A = \frac{k}{e^{-\frac{E_a}{RT}}} \).

Calculate Exponential Factor

Plug in the values: \( E_a = 126 \text{ kJ/mol} = 126,000 \text{ J/mol} \), \( R = 8.314 \text{ J/(mol·K)} \), and \( T = 303.15 \text{ K} \). Calculate the exponential factor: \( e^{-\frac{126,000}{8.314 \times 303.15}} \).

Calculate the Frequency Factor A

Calculate \( A \) using the rate constant \( k = 9.8 \times 10^{-3} \) min\(^{-1}\) and the exponential factor derived in the previous step. Solve \( A = \frac{9.8 \times 10^{-3}}{e^{-\frac{126,000}{8.314 \times 303.15}}} \).

Select the Correct Answer

After calculating \( A \), compare it with the given multiple-choice options: a. \(9.5 \times 10^{18}\), b. \(2.5 \times 10^{19}\), c. \(55 \times 10^{19}\), d. \(5.0 \times 10^{19}\).

Key concepts

Activation Energy

Activation energy is a crucial concept in understanding chemical reactions, especially when utilizing the Arrhenius equation. It represents the minimum energy that reacting molecules must possess for a reaction to occur. This energy barrier must be overcome for reactants to transform into products. In the context of the given exercise, the activation energy (\(E_a\) ) is specified as 126 kJ/mol.

With higher activation energy, fewer molecules have sufficient energy at a given temperature to surpass this threshold, which consequently results in a slower reaction rate. Conversely, a lower activation energy means more molecules can participate in the reaction, leading to a faster reaction. Knowing the activation energy helps chemists determine how temperature or catalysts can impact the rate of a reaction.

Frequency Factor

The frequency factor, denoted by \(A\), in the Arrhenius equation, is also known as the pre-exponential factor. It provides insight into how frequently molecules collide with the correct orientation to react. While the activation energy determines if a collision can lead to a reaction, the frequency factor tells us about the collision rate and effectiveness.

In the equation \(k = A e^{-\frac{E_a}{RT}}\), \(A\) is significant because it considers the physical probability of a reaction occurring. Thus, it reflects factors such as molecular orientations and collision frequencies. Calculating the frequency factor involves rearranging the Arrhenius equation to solve for \(A\), which involves utilizing the known values of \(k\), \(E_a\), \(R\), and \(T\).

Often, this factor can be determined experimentally and can vary significantly between different reactions, usually being larger than the rate constant.

Rate Constant

The rate constant, \(k\), is an integral part of the Arrhenius equation and represents the proportionality constant that links the reaction rate to the concentration of reactants raised to the power of their stoichiometric coefficients in the rate law. It is a critical value that can provide insight into the speed of a reaction under specific conditions.

In our problem, the rate constant is given as \(9.8 \times 10^{-3} \text{ min}^{-1}\) at \(30^{\circ} C\). The Arrhenius equation shows that \(k\) depends strongly on temperature, as even small changes in temperature can lead to significant variations in the rate constant value. This relationship highlights the sensitivity of chemical reactions to temperature changes and aids in predicting how reactions will behave under different thermal conditions.

Temperature Conversion

Temperature conversion is often necessary when using the Arrhenius equation, since it requires the temperature to be in Kelvin (K) for correct application. The Kelvin scale is an absolute temperature scale used in scientific calculations because it directly relates to the energy of particles.

In the exercise, a temperature conversion from Celsius to Kelvin is essential. The conversion formula is simple:

• \(T(K) = T(^{\circ}C) + 273.15\)

For example, at \(30^{\circ} C\), the temperature in Kelvin becomes \(303.15\) K. By ensuring temperatures are in Kelvin, we maintain the consistency needed for accurate calculation in scientific reactions and remain consistent with the SI unit system. Consistency in units allows for reliable comparison and compatibility across different scientific studies and equations.