Animated Solution for Chemistry - Chemical Kinetics: The rate of a chemical reaction doubles for every $10^{\circ }\text{C}$ rise of temperature. If the temperature is raised by $50^{\circ }\text{C}$, the rate of the reaction increases by about

Imagine a chemical reaction as a bustling dance floor of molecules. When we turn up the heat, we are essentially pumping energy into the room, causing the molecules to dance faster and collide more frequently. The problem states a fascinating rule of thumb: for every $10^{\circ }\text{C}$ rise in temperature, the rate of the reaction doubles. This multiplying factor is known as the temperature coefficient ($\mu$).

In our case, the temperature coefficient $\mu =2$. This means the reaction speed doesn't just grow linearly; it grows exponentially!

To find the new rate after a specific temperature rise, we use a simple yet powerful exponential formula. The ratio of the final rate ($R_2$) to the initial rate ($R_1$) is equal to the temperature coefficient raised to the power of the total temperature change ($\Delta T$) divided by $10$.

Why do we divide by $10$? Because the rate doubles for every $10^{\circ }\text{C}$ interval. Dividing the total temperature rise by $10$ tells us exactly how many of these doubling intervals fit into our temperature change.

Let's plug in our numbers. We know our temperature coefficient $\mu =2$, and our total temperature rise $\Delta T=50^{\circ }\text{C}$. Substituting these values into our equation gives:

First, we simplify the exponent. Fifty divided by ten is simply five. This means the rate will double five times in succession as the temperature climbs by fifty degrees.

Now, we must be careful not to make a silly mistake by multiplying $2$ and $5$. We need to calculate $2$ raised to the power of $5$.

So, the final rate is $32$ times the initial rate.

Just a $50^{\circ }\text{C}$ increase has caused a massive thirty-two-fold jump in the reaction speed! This beautifully illustrates the power of exponential growth in chemical kinetics.