Animated Solution for Chemistry - Chemical Kinetics: If the activation energy of a reaction is $80.9{\text{ kJ mol}}^{-1}$, the fraction of molecules at $700\text{ K}$, having enough energy to react to form products is $e^{-x}$. The value of $x$ is ......... (Rounded off to the nearest integer) [Use, $R=8.31{\text{ J K}}^{-1}{\text{ mol}}^{-1}$].

Imagine a microscopic world inside a reaction vessel. Millions of gas molecules are constantly colliding, bouncing off each other in a chaotic dance. However, not every collision leads to a chemical reaction. For a reaction to occur, the colliding molecules must possess a certain minimum amount of kinetic energy, known as the activation energy ($E_a$).

If we plot the kinetic energies of all these molecules, we get the famous Maxwell-Boltzmann distribution curve. Most molecules have an average, moderate energy. Only a tiny fraction of elite molecules sit at the far right tail of the curve, possessing energy greater than $E_a$.

Mathematically, this crucial fraction of molecules is given by the Arrhenius factor:

This elegant expression tells us exactly what portion of the total molecules is capable of reacting at a given absolute temperature $T$. Our goal in this problem is to calculate this fraction and compare it to the given expression $e^{-x}$.

Before we plug in the numbers, we must be extremely careful with our units. This is a classic trap in physical chemistry! The activation energy is given as $80.9{\text{ kJ mol}}^{-1}$, but the universal gas constant $R$ is given in Joules ($8.31{\text{ J K}}^{-1}{\text{ mol}}^{-1}$).

To ensure consistency, we must convert the activation energy into Joules by multiplying by $1000$:

Now, let's substitute our values into the exponent term $\frac{E_a}{RT}$:

First, we multiply the terms in the denominator:

Next, we perform the division:

So, the fraction of molecules is approximately $e^{-13.907}$. The problem states that this fraction is equal to $e^{-x}$. Therefore, by comparing the exponents, we get:

Finally, the question asks us to round off the value of $x$ to the nearest integer. Since $13.907$ is much closer to $14$ than to $13$, we round it up.