Arrhenius equation formula activation energy and temperature effect on rate constant
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.
FAQs on Arrhenius Equation and Temperature Dependence of Rate Constant
1. What is the Arrhenius equation for the rate constant?
The Arrhenius equation is k = A e−Ea/(RT), which relates the rate constant to temperature and activation energy.
- k = rate constant
- A = frequency (pre-exponential) factor
- Ea = activation energy (J mol−1)
- R = gas constant (8.314 J mol−1 K−1)
- T = temperature in Kelvin (K)
2. How does temperature affect the rate constant according to the Arrhenius equation?
According to the Arrhenius equation, increasing temperature increases the rate constant exponentially.
- As T increases, the term e−Ea/(RT) becomes larger.
- More molecules gain energy equal to or greater than Ea.
- This results in a faster reaction rate.
3. What is activation energy in the Arrhenius equation?
Activation energy (Ea) is the minimum energy required for reactant molecules to undergo a successful chemical reaction.
- It represents the energy barrier between reactants and products.
- Measured in J mol−1 or kJ mol−1.
- Higher Ea means a slower reaction at the same temperature.
4. What is the frequency factor (A) in the Arrhenius equation?
The frequency factor (A) is a constant that represents the frequency of effective molecular collisions in a reaction.
- It includes collision frequency.
- It accounts for proper molecular orientation.
- It is sometimes called the pre-exponential factor.
5. How do you calculate activation energy using the Arrhenius equation?
Activation energy can be calculated using the logarithmic form ln k = ln A − (Ea/RT) or the two-temperature form ln(k2/k1) = −Ea/R (1/T2 − 1/T1).
- Measure rate constants at two temperatures.
- Substitute values into the two-point equation.
- Solve for Ea.
6. Why does the rate constant increase exponentially with temperature?
The rate constant increases exponentially with temperature because the fraction of molecules exceeding the activation energy rises exponentially.
- Molecular energies follow the Maxwell–Boltzmann distribution.
- Higher temperature shifts more molecules above Ea.
- This dramatically increases successful collisions.
7. What is the linear form of the Arrhenius equation?
The linear form of the Arrhenius equation is ln k = −Ea/R (1/T) + ln A.
- It has the form y = mx + c.
- A plot of ln k vs 1/T gives a straight line.
- Slope = −Ea/R.
- Intercept = ln A.
8. What are the units of the Arrhenius equation variables?
In the Arrhenius equation, T is in Kelvin (K), Ea is in J mol−1, and R is 8.314 J mol−1 K−1.
- The unit of k depends on reaction order.
- The unit of A matches the unit of k.
- Temperature must always be in Kelvin, not °C.
9. How does activation energy affect the rate constant?
A higher activation energy results in a smaller rate constant at the same temperature.
- Because the term e−Ea/(RT) becomes smaller.
- Fewer molecules have sufficient energy to react.
- Reactions with low Ea proceed faster.
10. How does a catalyst affect the Arrhenius equation and rate constant?
A catalyst increases the rate constant by lowering the activation energy in the Arrhenius equation.
- It provides an alternative reaction pathway.
- It decreases Ea.
- The exponential term becomes larger.
