Question

The activation energy of a reaction is \(56.8 \mathrm{~kJ} / \mathrm{mol}\), and the frequency factor is \(1.5 \times 10^{11} / \mathrm{s}\). Calculate the rate constant of the reaction at \(25^{\circ} \mathrm{C}\).

Short answer

\(k \approx (1.5 \times 10^{11} / s) \times e^{-22.924}\). Use a calculator to find the numerical value.

Step-by-step solution

Convert Temperature to Kelvin

The given temperature is in degrees Celsius which must be converted to Kelvin since the Arrhenius equation requires temperature in Kelvin. The conversion is done by adding 273.15 to the Celsius temperature. So, the temperature in Kelvin, T, is: \(T = 25 + 273.15 = 298.15~K\).

Identify the Arrhenius Equation

The Arrhenius equation is used to calculate the rate constant, \(k\), and is given by: \[k = Ae^{-\frac{E_a}{RT}}\], where \(A\) is the frequency factor, \(E_a\) is the activation energy in joules per mole, \(R\) is the gas constant (8.314 J/(mol·K)), and \(T\) is the temperature in Kelvin.

Convert Activation Energy to Joules

The activation energy is given in kJ/mol and must be converted to J/mol. Multiply the given activation energy by 1000 to convert it to joules: \(E_a = 56.8 \times 1000 = 56800~J/mol\).

Calculate the Rate Constant

Plug the known values into the Arrhenius equation and solve for the rate constant (\(k\)): \[k = (1.5 \times 10^{11} / s) \times e^{-\frac{56800 J/mol}{(8.314 J/(mol\cdot K))(298.15 K)}}\].

Evaluate the Exponential Term

Calculate the exponent by dividing the activation energy by the product of the gas constant and temperature. Then, take the Euler's number 'e' to the power of this negative quotient to get the exponential factor: \[e^{-\frac{56800}{(8.314)(298.15)}} \approx e^{-22.924}\].

Compute the Rate Constant

Next, use a calculator to evaluate the exponential term and then multiply that result by the frequency factor to find the rate constant \(k\): \[k \approx (1.5 \times 10^{11} / s) \times e^{-22.924}\].

Key concepts

Understanding Activation Energy

Activation energy, denoted as \(E_a\), is a crucial concept in chemical kinetics that represents the energy barrier that reactant molecules must overcome to undergo a chemical reaction. It's a measure of the minimum energy required for a reaction to proceed. If the activation energy is high, only a small fraction of the reactant molecules have enough energy to react, resulting in a slower reaction rate. Conversely, if the activation energy is low, many more molecules can engage in the reaction, leading to a faster reaction rate.

When students work through textbook problems involving activation energy, they should focus on visualizing it as a hill that reactants must climb to transform into products. It's not just a number to plug into an equation; it's an indicator of how 'difficult' a reaction is to get started. Understanding this concept helps students appreciate why certain reactions occur spontaneously while others require the addition of heat or catalysts to proceed at a noticeable rate.

Rate Constant Calculation

The rate constant \(k\) is a numerically important factor in the Arrhenius equation as it determines the speed at which reactions occur. It is sensitive to temperature and the nature of the reacting substances, including their activation energy. To calculate the rate constant using the Arrhenius equation \(k = Ae^{-\frac{E_a}{RT}}\), students need to understand how each variable in the equation affects the outcome.

The frequency factor (\(A\)) represents the number of times that reactants approach the activation barrier per unit of time. The exponential factor \(e^{-\frac{E_a}{RT}}\) decreases sharply with increasing \(E_a\), but increases with rising temperature (\(T\)). This is because a higher temperature means that the reactant molecules are more energetic on average, which makes it more likely for them to have enough energy to overcome the activation barrier. Students should also note that all energies must be in consistent units, typically J/mol when using the gas constant \(R\) in J/(mol·K).

Temperature Conversion to Kelvin

The importance of converting temperature to Kelvin in chemical kinetics cannot be understated. Kelvin is the SI unit for thermodynamic temperature and is used in scientific equations to ensure uniformity and accuracy. In the case of the Arrhenius equation, temperature must be in Kelvin to properly evaluate the rate constant. To convert Celsius to Kelvin, students should remember the simple conversion formula: \(T(K) = T(°C) + 273.15\).

This step is fundamental as it sets the stage for performing accurate calculations. Temperature influences reaction rates profoundly; therefore, it needs to be precisely measured and converted. When students encounter temperatures in Celsius in their problems, they should instinctively convert to Kelvin as one of their first steps. This detail might seem minor but is essential for obtaining correct results in their calculations.