Arrhenius Equation Derivation
The Arrhenius equation is used for calculating the rate of reaction. It is a crucial part in chemical kinetics. It helps to understand the impact of temperature on the rate of reaction. This equation was first introduced by Svente Arrhenius in 1889.
In the equation, A = Frequency factor
K = Rate constant
R = Gas constant
Ea = Activation energy
T = Kelvin temperature
The collision theory is the foundation for the Arrhenius equation. As per this theory, the reaction is essentially a collision involving two molecules ( of same or different substances) to form the intermediate. This intermediate that is formed is unstable, and itt exists for a short duration of time. The intermediate breaks down thereby giving out two molecules of product. The energy that is used for forming this intermediate is called the activation energy.
If we look at log on both sides of the equation, the equation becomes
Ln is the natural algorithm, and these values can be picked up from a logarithmic table.
For the graphical representation,
When we compare this equation with the straight-line equation, we get
X = \[\frac {1} {T}\]
Y = ln k
M = \[\frac {-Ea} {R}\]
C = ln A
This provides the straight-line graph but has a negative slope.
Plotting the k v/s \[(\frac {1} {T})\].
Impact of Temperature
With the help of the graph, we can conclude that the rate of reactions and temperature are proportional. As temperature increases, the rate of reaction also tends to increase. There is an increase in kinetic energy with temperature. So when the temperature is increased, the number of molecules having kinetic energy higher than activation energy also increases. This leads to a rise in the rate of overall reaction as the activation energy decreases.
For the 10K shift in temperature, the rate is almost doubled.
Let us consider the Arrhenius equation at times T1 and T2 where the rates of reaction are denoted by K1 and K2 respectively.
In K1 = \[\frac {-Ea} {RT_1}\] + In A —--- (1)
In K2 = \[\frac {-Ea} {RT_2}\] + In A —--- (2)
Now we subtract 1 from 2
In K2 - In K1 equals to \[\frac {Ea} {RT_1}\] - \[\frac {Ea} {RT_2}\]
In \[\frac {K_2} {K_1}\] = \[(\frac {Ea} {R})\] \[\frac {1} {T_1}\] - \[\frac {1} {T_2}\]
Converting to log,
Log \[(\frac {Ea} {2.303R})\] \[\frac {T_2-T_1} {T_1T_2}\]
The Arrhenius equation also suggests that uncatalyzed reaction is more impacted by temperature in comparison to the catalyzed reaction.
Real-Life Examples of This Theory:
Milk gets sour faster when it is kept at room temperature instead of being kept in the refrigerator.
Eggs tend to hard boil faster when they are at sea level in comparison to mountains or elevated levels.
The butter tends to become rancid at a faster rate in summer than it does in winter
Cold-blooded animals or species like reptiles and insects become more lethargic during colder days.
Significance of Arrhenius Equation
This equation enables the accounting of factors that have an effect on the rate of reaction and which is not possible to be determined by the rate law.
It helps in finding the impact of energy barrier, frequency, temperature, the orientation of collisions, and presence of catalyst using the equation.
