Animated Solution for Chemistry - Chemical Kinetics: Consider the following reactions $A\to P1;B\to P2;C\to P3;D\to P4$, The order of the above reactions are $a,b,c$ and $d$, respectively. The following graph is obtained when $\log [\text{rate}]$ vs $\log [\text{conc.}]$ are plotted :

Imagine you are presented with a graph where multiple lines shoot upwards from a single point on the y-axis. The y-axis represents the logarithm of the reaction rate, $\log [\text{rate}]$, and the x-axis represents the logarithm of the concentration, $\log [\text{conc.}]$. We have four different reactions labeled A, B, C, and D.

Notice how all four lines start from the exact same point on the y-axis but rise with different steepness. Our goal is to determine the order of these reactions based purely on this visual information. To decode this, we need to bridge the gap between the visual graph and the fundamental mathematics of chemical kinetics.

To understand this graph, we must start with the fundamental rate law. For any general reaction of order $n$, the rate is given by the equation:

Here, $k$ is the rate constant, and $n$ is the order of the reaction. However, our graph is plotted in terms of logarithms. Therefore, we need to transform our master equation into a logarithmic format. By taking the logarithm on both sides, we get:

Using the standard properties of logarithms, where $\log (ab)=\log a+\log b$ and $\log (a^b)=b\log a$, we can expand the right side:

Does this new equation look familiar? It perfectly matches the standard equation of a straight line, $y=mx+c$. Let's map the variables:

- $y$ corresponds to $\log (\text{Rate})$ - $x$ corresponds to $\log [\text{conc.}]$ - $c$ (the y-intercept) corresponds to $\log k$ - $m$ (the slope) corresponds to $n$, the order of the reaction.

This mapping is a powerful revelation! The y-intercept $c$ is $\log k$, which explains why all lines start from the exact same point on the y-axis: all four reactions share the same rate constant $k$. Most importantly, the slope $m$ is exactly equal to $n$, the order of the reaction.

So, the secret is out! The slope of the line directly gives us the order of the reaction. Now, let's look at the graph again and compare the steepness of the lines.

Which line is the steepest? Line D has the maximum slope, followed by B, then A, and finally C has the least slope. Mathematically, we can write this as:

Since the slope is equal to the order ($n$), the reaction with the highest slope will have the highest order. Therefore, the order $d$ is greater than $b$, which is greater than $a$, which is greater than $c$:

This perfectly matches option (d). Always remember, in such graphical questions, the slope and the intercept hold the keys to unlocking the physical constants of the reaction!