Question

The rate constant \((k)\) for a reaction is measured as a function of tem- perature. A plot of ln \(k\) versus 1\(/ T(\) in \(\mathrm{K})\) is linear and has a slope of \(-1.01 \times 10^{4} \mathrm{K}\) . Calculare the activation energy for the reaction.

Short answer

The activation energy for the reaction is\( 8.39 \times 10^{4} \text{ J/mol} \).

Step-by-step solution

Understanding the Arrhenius Equation

The Arrhenius equation relates the rate constant \(k\) of a chemical reaction to the temperature \(T\) and the activation energy \(E_a\) as follows: \[ \ln(k) = -\frac{E_a}{R}\left(\frac{1}{T}\right) + \ln(A) \] where \(A\) is the pre-exponential factor and \(R\) is the universal gas constant, approximately equal to 8.314 J/(mol K). The slope of the plot of \(\ln(k)\) versus \(1/T\) is equal to \(\frac{-E_a}{R}\).

Identify the Slope

From the given information, the slope of the plot \(\ln(k)\) versus \(1/T\) is \( -1.01 \times 10^{4} \mathrm{K} \). This slope corresponds to \(\frac{-E_a}{R}\).

Calculate the Activation Energy

We can rearrange the slope equation to solve for the activation energy, \(E_a\): \[ E_a = -\text{slope} \times R \] Substituting the values, we get: \[ E_a = -(-1.01 \times 10^{4} \mathrm{K}) \times 8.314 \text{ J/(mol K)} \E_a = 1.01 \times 10^{4} \mathrm{K} \times 8.314 \text{ J/(mol K)} \E_a = 8.39 \times 10^{4} \text{ J/mol} \] Note that the negative sign cancels because the slope is already negative.

Key concepts

Arrhenius Equation

Understanding the Arrhenius Equation is crucial for students studying chemical kinetics. This mathematical formula elegantly connects the rate at which a chemical reaction proceeds to temperature and a quantity known as activation energy. In simple terms, the Arrhenius Equation can be expressed as:

\[ \text{ln}(k) = -\frac{E_a}{R}\times\frac{1}{T} + \text{ln}(A) \]
Here, \( k \) is the rate constant of the reaction, \( E_a \) represents the activation energy, \( R \) signifies the universal gas constant, \( T \) is the temperature (in Kelvin), and \( A \) is the pre-exponential factor, sometimes referred to as the frequency factor.

Relevance of the Arrhenius Equation

The beauty of this equation is that it not only predicts how changes in temperature affect reaction rates but also offers insight into the intrinsic energy barriers (activation energy) that must be overcome for reactants to transform into products.

When graphing \( \text{ln}(k) \) against \( \frac{1}{T} \), a straight line indicates a consistent reaction mechanism over the temperature range. The slope of this line, crucially linked to the activation energy, allows us to extract valuable kinetic information about the reaction.

Rate Constant

Diving into the Rate Constant concept, students will find that it's a fundamental parameter in the study of chemical reactions. The rate constant \( (k) \) is a measure of the speed of a chemical reaction, and for a single-step reaction, it's related to the probability of the reactive collision between molecules.

Quantitative Aspects of the Rate Constant

In quantitative terms, the value of the rate constant can dictate whether a reaction is fast or slow at a given temperature. Now, what makes the rate constant particularly interesting is its temperature dependence, which is described by the Arrhenius Equation.

The value of \( k \) increases with temperature, generally making reactions faster as thermal energy helps molecules to overcome the activation energy barrier more readily.

The rate constant is also influenced by the presence of catalysts, which effectively lower the activation energy and hence increase \( k \), allowing the reaction to proceed at a faster rate without the need for a temperature increase.

Chemical Kinetics

Chemical Kinetics is a branch of chemistry that deals with understanding how fast chemical reactions occur, which is crucial for everything from industrial chemical processes to metabolic pathways in our bodies.

Key considerations in chemical kinetics include the speed (rate) of a reaction, how that rate changes with varying concentrations of reactants or products, and the effect of temperature, as encapsulated in the rate constant \( k \).

Applying Chemical Kinetics

By studying reaction rates, we can deduce information about the reaction mechanism—the step-by-step sequence of elementary reactions that leads to the formation of products. Kinetics also allows chemists to design experiments to measure and analyze reaction rates to better understand these mechanisms and the factors that influence them, such as temperature, catalysts, and the molecular orientation of the reacting species.

Furthermore, through kinetics, we can predict the outcomes of reactions under different conditions, making it a vital tool in fields such as pharmacology, environmental science, and materials engineering.

Universal Gas Constant

Lastly, the Universal Gas Constant \( (R) \) is a fundamental constant in chemistry that appears in various key equations, including the ideal gas law and, as we discussed, the Arrhenius Equation. The value of the constant is approximately 8.314 J/(mol·K).

Understanding \( R \) is vital because it bridges macroscopic and molecular-level phenomena. As the proportionality factor in the ideal gas law, it relates the pressure, volume, and temperature of an ideal gas to the amount of substance.

Role in Activation Energy Calculation

In the context of activation energy calculation, \( R \) allows us to convert the slope obtained from a plot of \( \text{ln}(k) \) versus \( \frac{1}{T} \) into an energy quantity, expressing the sensitivity of the reaction rate to temperature change. This conversion is pivotal for chemists to understand the energy requirements for reactions and to design processes that are energy-efficient.