Animated Solution for Chemistry - Chemical Kinetics: The rate constant ($k$) of a reaction is measured at different temperature ($T$), and the data are plotted in the given figure. The activation energy of the reaction in ${\text{kJ mol}}^{-1}$ is ($R$ is gas constant)

The Arrhenius equation is one of the most beautiful relationships in chemical kinetics, elegantly connecting the rate of a reaction to its temperature. But when this equation is presented as a graph, it can sometimes feel like a puzzle. Let's break down this puzzle step-by-step and uncover the activation energy hidden within the slope.

We start with the fundamental Arrhenius equation:

To make sense of the given graph, we need to map this equation to the standard equation of a straight line, which is $y=mx+c$.

Looking at the graph, our y-axis represents $\ln k$. But what about the x-axis? It's not just $1/T$; it's scaled as $10^3/T$.

To align our equation with the graph's axes, we must introduce this $10^3$ factor into our math. We can rewrite the Arrhenius equation by multiplying and dividing the temperature term by $10^3$:

Now, the mapping is perfect! Our y-variable is $\ln k$. Our x-variable is $\frac{10^3}{T}$. The y-intercept $c$ is $\ln A$. And most importantly, the slope $m$ is $-\frac{E_a}{R\times 10^3}$.

The next step is to extract the numerical value of the slope directly from the graph. We need two clear points on the line.

Observing the intercepts, the line starts exactly on the y-axis at $10$, giving us our first point: $(0,10)$.

It ends exactly on the x-axis at $5$, giving us our second point: $(5,0)$.

Using the slope formula $m=\frac{y_2-y_1}{x_2-x_1}$:

Now, we equate this numerical slope to our theoretical slope:

Units

Solving for the activation energy $E_a$, the negative signs cancel out:

Here is where many students make a critical error. The universal gas constant $R$ is typically used in Joules, meaning our calculated $E_a$ is in Joules per mole.

However, the question specifically asks for the answer in kiloJoules per mole (${\text{kJ mol}}^{-1}$). To convert Joules to kiloJoules, we simply divide by $1000$ (or $10^3$).

And there we have it! By carefully aligning the axes and watching our units, we've successfully extracted the activation energy.