Animated Solution for Chemistry - Chemical Kinetics: Consider an endothermic reaction $X\to Y$ with the activation energies $E_b$ and $E_f$ for the backward and forward reactions respectively. In general

Imagine you are standing at the base of a mountain, and your destination is a valley on the other side. To get there, you can't just teleport; you have to climb all the way to the peak before you can slide down into the valley. This physical journey is the perfect analogy for how chemical reactions occur at the molecular level.

In chemical kinetics, we visualize this journey using an Energy Profile Diagram. The x-axis represents the progress of the reaction (the reaction coordinate), and the y-axis represents the potential energy of the system.

The problem asks us to consider an endothermic reaction, denoted as $X\to Y$. The word "endothermic" is derived from Greek, where "endo" means inside and "therme" means heat. An endothermic reaction absorbs heat from its surroundings.

What does this mean for our energy profile? It means that the product $Y$ has a higher intrinsic potential energy than the reactant $X$. If you look at the graph, the horizontal line representing the energy of $Y$ sits higher than the line for $X$. The difference in their energy levels is the enthalpy of the reaction, denoted by $\Delta H$.

Mathematically, we write this as:

Because the products have more energy, $\Delta H$ is a positive value.

Activation Energy

Now, let's talk about the climb. Molecules of $X$ don't just spontaneously turn into $Y$. They need a spark, an initial push of energy to break old bonds before new ones can form. This peak energy state is called the Transition State or activated complex.

The energy required to climb from the reactant level ($X$) to the transition state is called the Forward Activation Energy, denoted as $E_f$.

But what if the reaction were to run in reverse? What if $Y$ wanted to turn back into $X$? The molecules of $Y$ would also have to climb to that exact same transition state peak. The energy required to climb from the product level ($Y$) to the peak is the Backward Activation Energy, denoted as $E_b$.

If we look at the geometry of our energy profile diagram, a beautiful and simple mathematical relationship emerges. The total height of the mountain from the base (reactant $X$) to the peak is $E_f$.

This total height is composed of two segments: 1. The climb from reactant $X$ to product $Y$, which is $\Delta H$. 2. The remaining climb from product $Y$ to the peak, which is $E_b$.

Therefore, we can write the master equation:

We already established that for an endothermic reaction, $\Delta H$ is a strictly positive number ($\Delta H>0$).

If we substitute a positive value into our master equation, it becomes glaringly obvious that $E_f$ must be larger than $E_b$.

Or, written the other way around:

This makes intuitive sense. If you are starting from a lower valley ($X$) and climbing to a peak, your climb ($E_f$) will always be longer than the climb of someone starting from a higher plateau ($Y$) to reach the same peak ($E_b$).

Thus, for any endothermic reaction, the backward activation energy is always less than the forward activation energy. This perfectly matches option (d). Always remember to visualize the graph; it is the ultimate cheat code for these types of kinetics problems!