Animated Solution for Chemistry - Chemical Kinetics: Consider the given plots for a reaction obeying Arrhenius equation ($0^{\circ }\mathrm{C}<T<300^{\circ }\mathrm{C}$) : ($k$ and $E_a$ are rate constant and activation energy, respectively)

Imagine you are trying to push a heavy boulder over a hill. The height of the hill is your activation energy ($E_a$), and the energy you have to push it depends on the temperature ($T$). The Arrhenius equation beautifully captures this physical reality in a mathematical form:

Here, $k$ is the rate constant, representing how fast the reaction happens. $A$ is the frequency factor, $R$ is the universal gas constant, and $T$ is the absolute temperature in Kelvin.

The Energy Barrier

Let's look at the first plot, which shows the relationship between the rate constant $k$ and the activation energy $E_a$. Imagine keeping the temperature constant. What happens if we increase the height of the hill ($E_a$)?

Mathematically, $k$ is proportional to $e^{-E_a}$. Because of the negative sign in the exponent, as $E_a$ increases, the value of $e^{-E_a}$ drops sharply. This is an exponential decay. Physically, it means that reactions with higher activation energies are exponentially slower because fewer molecules have the required energy to cross the barrier. Plot I perfectly illustrates this exponential decay curve. Therefore, Plot I is absolutely correct.

The Temperature Effect

Now, let's shift our focus to the second plot. Here, $k$ is plotted against temperature. But wait, there is a crucial detail: the temperature is given in degrees Celsius (${}^{\circ }\mathrm{C}$), not Kelvin!

We know that the absolute temperature in Kelvin is related to Celsius by the equation:

So, what happens at $0^{\circ }\mathrm{C}$? The absolute temperature is $273.15\text{ K}$. Because the temperature is not absolute zero, the molecules still possess kinetic energy, and the rate constant $k$ will have a finite, positive value. It will not be zero!

As the temperature increases further, the term $-E_a/RT$ becomes less negative, causing the exponential term to grow. Thus, $k$ increases exponentially with temperature. Plot II shows exactly this behavior: it starts with a non-zero y-intercept and curves upwards exponentially. Therefore, Plot II is also correct.

By carefully analyzing the mathematical relationships and the physical meaning behind the Arrhenius equation, we have established that both graphs are accurate representations. Plot I correctly shows the exponential decay with increasing activation energy, and Plot II correctly shows the exponential growth with temperature in Celsius, complete with the non-zero intercept.

Thus, the correct option is (b) Both I and II are correct.