Question

A first-order reaction has rate constants of \(4.6 \times 10^{-2} \mathrm{s}^{-1}\) and \(8.1 \times 10^{-2} \mathrm{s}^{-1}\) at \(0^{\circ} \mathrm{C}\) and \(20 .^{\circ} \mathrm{C},\) respectively. What is the value of the activation energy?

Short answer

The activation energy for this first-order reaction can be determined using the Arrhenius equation and the given rate constants at two different temperatures. After converting the temperatures to Kelvin and solving for the activation energy (Ea), we find that the activation energy is approximately \(41.28 \frac{\textrm{kJ}}{\textrm{mol}}\).

Step-by-step solution

Convert temperatures to Kelvin

In order for the Arrhenius equation to be used, the temperatures need to be in Kelvin (K). Celsius can be converted to Kelvin by simply adding 273.15: \(T_1 = 0^{\circ} C + 273.15 K = 273.15 K\) \(T_2 = 20^{\circ} C + 273.15 K = 293.15 K\) Now we have the two temperatures, \(T_1\) and \(T_2\), in Kelvin: 273.15 K and 293.15 K.

Write down the Arrhenius equation

The Arrhenius equation relates the temperature, rate constant, and activation energy as follows: \(k = A \cdot e^{-\frac{Ea}{RT}}\) where k is the rate constant, A is the pre-exponential factor, Ea is the activation energy, R is the gas constant (8.314 J/mol·K), and T is the temperature (in Kelvin). Since we have two sets of values for k and T, we can set up two equations using the Arrhenius equation: \(k_1 = A \cdot e^{-\frac{Ea}{R \cdot T_1}}\) \(k_2 = A \cdot e^{-\frac{Ea}{R \cdot T_2}}\)

Solve for the activation energy (Ea)

We can find Ea by dividing the two equations (k1 and k2) and eliminating the pre-exponential factor A: \(\frac{k_1}{k_2} = \frac{A \cdot e^{-\frac{Ea}{R \cdot T_1}}}{A \cdot e^{-\frac{Ea}{R \cdot T_2}}}\) The A terms will cancel out, and we can rearrange the equation to solve for Ea: \(Ea = R \cdot (\frac{T_1 \cdot T_2 \cdot ln(\frac{k_1}{k_2})}{T_2 - T_1})\) Now, we can plug in the values for k1, k2, T1, and T2: \(Ea = 8.314 \frac{\textrm{J}}{\textrm{mol} \cdot \textrm{K}} \cdot (\frac{273.15 \textrm{K} \cdot 293.15 \textrm{K} \cdot ln(\frac{4.6 \times 10^{-2} \mathrm{s}^{-1}}{8.1 \times 10^{-2} \mathrm{s}^{-1}})}{293.15 \textrm{K} - 273.15 \textrm{K}})\)

Compute the activation energy

\( Ea = 8.314 \frac{\textrm{J}}{\textrm{mol} \cdot \textrm{K}} \cdot (\frac{273.15 \textrm{K} \cdot 293.15 \textrm{K} \cdot ln(0.568)}{20 \textrm{K}})\) Now, calculate the activation energy: \(Ea = 41283.96 \frac{\textrm{J}}{\textrm{mol}}\) Convert the activation energy to kilojoules per mole: \(Ea = 41.28 \frac{\textrm{kJ}}{\textrm{mol}}\) The activation energy for this first-order reaction is approximately 41.28 kJ/mol.

Key concepts

First-Order Reaction

In the world of chemistry, a first-order reaction is one where the rate of reaction is directly proportional to the concentration of a single reactant. This means that if the concentration of the reactant doubles, the rate of reaction also doubles. Such reactions can be nicely expressed with a simple rate equation: \[\text{Rate} = k \cdot [A]\]Where:- \(k\) is the rate constant, which remains constant at a given temperature.- \([A]\) is the concentration of the reactant.One of the distinctive features of a first-order reaction is its characteristic half-life, which remains constant throughout the reaction. The half-life is the time it takes for the concentration of the reactant to decrease to half its initial value. This is given by the formula:\[ t_{1/2} = \frac{0.693}{k}\]Understanding first-order reactions is essential for predicting how reactions progress over time, especially in fields like pharmacokinetics, where drug concentration over time is crucial.

Arrhenius Equation

The Arrhenius equation is a fundamental formula that connects the dots between the rate of a chemical reaction and the temperature at which it occurs. Its purpose is to express how the rate constant \(k\) changes with temperature \(T\), the activation energy \(Ea\), and a pre-exponential factor \(A\). The equation is:\[k = A \cdot e^{-\frac{Ea}{RT}} \]Where:- \(k\) is the rate constant.- \(A\) is the pre-exponential factor, which is a constant for each chemical reaction.- \(Ea\) is the activation energy, the energy needed for the reaction to occur.- \(R\) is the universal gas constant.- \(T\) is the temperature in Kelvin.The Arrhenius equation beautifully shows that higher temperatures decrease the exponential term, thereby increasing the rate constant, making the reaction go faster. This relationship helps us understand how and why reactions speed up with temperature, which is crucial in many industrial and laboratory processes.

Rate Constant

The rate constant, often denoted as \(k\), is an integral part of understanding how quickly a reaction proceeds. This constant is specific for each reaction at a given temperature and reflects the inherent characteristics of the reaction.

• The units of \(k\) depend on the order of the reaction. For first-order reactions, \(k\)'s units are typically \( s^{-1} \).
• It represents the speed at which a reaction approaches completion.
• Changes in temperature can significantly affect the rate constant, as shown by the Arrhenius equation.

Explaining the role of the rate constant can deepen our understanding of the kinetics of a reaction, helping chemists control and predict the outcomes of chemical processes.

Temperature Conversion

When working with chemical kinetics, it is vital to use Kelvin for temperature measurements. The Kelvin scale is the standard in scientific calculations because it allows for absolute temperature measurement without negative numbers. Conversion from Celsius to Kelvin is simple:\[T(\text{K}) = T(\degree C) + 273.15\]This small adjustment ensures that all the kinetic formulas, like the Arrhenius equation, work correctly because the equations often involve ratios or exponentials of temperature - which can give erroneous results with the Celsius scale.Using Kelvin is especially important in reactions where precise temperature measurements can dramatically affect outcomes. Therefore, it's a crucial step in ensuring accuracy in chemical calculations and experiments.