Question

The plot of \(\ln k\) versus \(1 / T\) is linear with slope of: (a) \(-E_{a} / R\) (b) \(E_{a} / R\) (c) \(E_{a} / 2.303 R\) (d) \(-E_{a} / 2.303 R\)

Short answer

The slope of the plot of \(\ln k\) versus \(1 / T\) is (a) \(-E_{a} / R\).

Step-by-step solution

Identify the Relevant Equation

Recognize that the exercise is referring to the Arrhenius equation in its linearized form, which, when taking natural logarithms, becomes \[\begin{equation}\ln k = \ln A - \frac{E_a}{RT}\end{equation}\]Here, \(k\) is the rate constant, \(A\) is the pre-exponential factor, \(E_a\) is the activation energy, \(R\) is the gas constant, and \(T\) is the temperature in Kelvins.

Rewrite the Equation in Linear Form

To create a linear equation comparable to the form \(y = mx + b\), rewrite the above equation as \[\begin{equation}\ln k = -\frac{E_a}{R}\left(\frac{1}{T}\right) + \ln A\end{equation}\]Now, it resembles a straight line with slope \(m\) and y-intercept \(b\).

Compare to the Required Format

Compare the rewritten formula to the general form of a straight line, where the slope of the plot \(\ln k\) versus \(1/T\) can be directly observed as the coefficient of \(1/T\).

Key concepts

Chemical Kinetics

Chemical kinetics is the study of the rates at which chemical reactions proceed and the factors affecting these rates. The objective is to understand the sequence of chemical reactions, known as the reaction mechanism, and to identify the speed, or rate, at which the entire process unfolds.

In the context of the Arrhenius equation, kinetics helps us relate the frequency of molecular collisions and the energy with which they collide to the rate at which reactions progress. By analyzing the rate at which reactants transform into products, chemists can determine the rate law for a reaction, which reflects how the reaction rate is affected by the concentration of reactants.

Understanding kinetics is crucial in a wide range of scientific and industrial processes, such as figuring out the best conditions for synthesizing new materials or the time it would take for a reactant to be consumed in a chemical reactor.

Activation Energy

Activation energy, denoted by the symbol \(E_a\), is a critical concept in chemical kinetics that represents the minimum amount of energy required for reactants to engage in a chemical reaction that leads to products. In essence, it's the energy barrier that molecules must overcome to transform states.

From the Arrhenius equation, we see that a higher activation energy implies that fewer molecules will have the necessary energy to undergo a successful collision, thus resulting in a lower reaction rate. When relating to the slope of the \(\ln k\) versus \(1/T\) plot, activation energy can be deciphered graphically; a steep slope indicates a high activation energy, causing the rate constant to be highly sensitive to changes in temperature.

For instance, a reaction with high activation energy may proceed rapidly at higher temperatures but be almost negligible at low temperatures due to the exponential decrease in reaction rate with temperature.

Rate Constant

The rate constant, represented by \(k\), is a proportionality factor that links the reaction rate with the concentrations of reactants raised to certain powers according to the rate law of the reaction. It is a pivotal component in the Arrhenius equation and varies with temperature, represented in the form \[\[\begin{equation}\ln k = -\frac{E_a}{R}\left(\frac{1}{T}\right) + \ln A\end{equation}\]\]

Within the Arrhenius framework, \(k\) increases exponentially with temperature, highlighting the sensitivity of reactions to thermal conditions. The rate constant is also influenced by the presence of catalysts, which lower the activation energy and consequently increase the reaction rate without themselves being consumed in the reaction.

Its significance is such that understanding \(k\) allows chemists to predict how fast a reaction will proceed under specified conditions, which is especially useful for controlling industrial chemical processes and developing pharmaceuticals.