Animated Solution for Chemistry - Chemical Kinetics: The rate of a certain biochemical reaction at physiological temperature ($T$) occurs $10^6$ times faster with enzyme than without. The change in the activation energy upon adding enzyme is :

Lowering the Barrier

Imagine you are trying to push a heavy boulder over a steep hill. It takes a massive amount of energy to get it to the top before it can roll down the other side. This hill represents the activation energy ($E_a$) of a chemical reaction. Now, what if someone dug a tunnel through the hill? You wouldn't have to push the boulder as high, making the process much faster and easier. This is exactly what an enzyme (a biological catalyst) does!

In this problem, we are told that an enzyme makes a biochemical reaction $10^6$ times faster. We need to find out exactly how much the activation energy was lowered.

Arrhenius

The relationship between the speed of a reaction (its rate constant, $k$) and its activation energy is beautifully captured by the Arrhenius equation:

To make this equation easier to work with, we often take the base-10 logarithm on both sides. Remember that $\ln x=2.303\log x$. So, the equation becomes:

Let's call this our equation for the uncatalyzed reaction.

When we add the enzyme, the reaction rate skyrockets. The new rate constant is $10^6$ times the original rate constant ($10^{6}k$). The enzyme achieves this by providing a new pathway with a lower activation energy, which we will call $E_a^{\ast }$.

Let's write the Arrhenius equation for this new, super-fast catalyzed reaction:

Taking the base-10 logarithm of this new equation gives us:

Using the properties of logarithms ($\log (ab)=\log a+\log b$), we can split the left side:

Since $\log 10^6$ is simply $6$, we have:

Rearranging to isolate $\log k$:

Now we have two different expressions for $\log k$. Let's set them equal to each other:

The $\log A$ terms cancel out perfectly, leaving us with:

We want to find the change in activation energy, which is the final activation energy minus the initial activation energy ($\Delta E_a=E_a^{\ast }-E_a$). Let's rearrange our equation to group the $E_a$ terms:

Finally, multiplying both sides by $2.303RT$ gives us our answer:

The negative sign is a beautiful confirmation of our physical intuition: the change is negative because the enzyme lowered the activation energy barrier!