What is Temperature Dependence of the Rate of a Reaction: Factors Affecting Rate of Reaction, Arrhenius Equation, Temperature Coefficient, Applications and Others

Temperature dependence is a crucial factor in determining the rate of a reaction and is a frequently asked concept in competitive exams like JEE MAINS. According to this theory, the rate of a chemical reaction is directly proportional to the temperature i.e. a high temperature can lead to a faster rate of the chemical reaction. This is because increasing the temperature will lead to comparatively more collisions between the particles, which are necessary for speeding up the rate of the reaction.

The arrhenius graph for temperature dependence can be depicted through the following image.

 

This is the formula which is used to depict how change in the temperature will affect the rate of a chemical reaction.

The temperature dependence of the rate constant (  $k$ ) can be given by the Arrhenius equation:

$k=A{e}^{-E_a/RT}$

where:

$k$ : Rate constant.

$A$ : Pre-exponential factor (frequency factor), related to collision frequency and orientation.

$E_a$ : Activation energy (  $\mathrm{J}{\mathrm{m}\mathrm{o}\mathrm{l}}^{-1}$ ), the minimum energy required for a reaction.

$R$ : Gas constant (  $8.314\mathrm{J}{\mathrm{m}\mathrm{o}\mathrm{l}}^{-1}{\mathrm{K}}^{-1}$ ).

$T$ : Absolute temperature (K).

$e$ : Euler’s number

As the temperature T increases, the fraction will also grow, leading to a higher rate constant (k) and a faster speed of the reaction.

Taking the natural logarithm of the Arrhenius equation:

$\mathrm{l}\mathrm{n}k=\mathrm{l}\mathrm{n}A-\frac{E_a}{RT}$

This resembles a straight-line equation (  $y=mx+c$ ), where plotting  $\mathrm{l}\mathrm{n}k$ vs.  $1/T$ yields:

Slope:  $-\frac{E_a}{R}$

Intercept:  $\mathrm{l}\mathrm{n}A$

For two temperatures  $T_1$ and  $T_2$ with rate constants  $k_1$ and  $k_2$ :

$\mathrm{l}\mathrm{n}{k}_1=\mathrm{l}\mathrm{n}A-\frac{E_a}{R{T}_1},\mathrm{}\mathrm{l}\mathrm{n}{k}_2=\mathrm{l}\mathrm{n}A-\frac{E_a}{R{T}_2}$

Subtracting:

$\begin{array}{c}\mathrm{l}\mathrm{n}{k}_2-\mathrm{l}\mathrm{n}{k}_1=\frac{E_a}{R}\left(\frac{1}{T_1}-\frac{1}{T_2}\right)\\ \mathrm{l}\mathrm{n}\frac{k_2}{k_1}=\frac{E_a}{R}\left(\frac{T_2-T_1}{T_1{T}_2}\right)\end{array}$

Converting to base-10 logarithm:

$\mathrm{l}\mathrm{o}\mathrm{g}\frac{k_2}{k_1}=\frac{E_a}{2.303R}\left(\frac{T_2-T_1}{T_1{T}_2}\right)$

This equation is used to calculate  $E_a$ or predict rate constant changes.

The Maxwell-Boltzmann distribution is a technique used to calculate the exact energies of different particles of a gas at a particular temperature. There are different molecules in a gas and each of them will have different speeds. Some will be fast and some will be slow.

Mathematical Representation:

There is a term called distribution function which is used to find particles with the desired speed range. The probability that a particular molecule has an energy E will be:

f(E)=π​2​((kT)3/21​)E1/2e−kTE​

Where:

f(E): Probability density function for kinetic energy

E: Kinetic Energy

k: Boltzmann constant (1.38 × 10⁻²³ J/K)

T: Absolute temperature (K)

The collision theory is a principle used in chemistry to understand the mechanism behind the occurrence of chemical reactions. For a particular reaction to take place, the particles need to collide with each other, and the higher energies these particles have while colliding with each other, higher will be the rate of the reaction.

Collision theory is mathematically represented by:

k=pZe−EaRTk = pZ e^{-\frac{E_a}{RT}}k=pZe−RTEa​​

Where,

p: steric factor (orientation probability, 0 < p ≤ 1)

Z: collision frequency

e−Ea/RTe^: fraction of collisions with sufficient energy

Some key factors necessary for a collision to occur are as follows:

1. Collision Frequency:

These are the number of particles which collide with each other in a particular time frame. This frequency depends on

• Temperature: High temperature will lead to higher kinetic energy of the particles, causing collisions with more frequency.
• Surface Area: More surface area will ensure that more number of particles will interact with each other to cause a collision.
• Concentration of Reactants: The volume of reactants also majorly determines the intensity of the collisions.

2. Activation Energy

This is referred to the minimum energy threshold required for a successful collision to take place. Particles will only collide with each other if their kinetic energy is more than the Ea​ (activation energy).

3. Orientation

According to this factor, the molecules must be in proper alignment with each other to produce a high frequency collision. This is determined by a sterlic factor p, which states that:

Rate∝p⋅Z⋅e−RTEa​​

Click to Read: NCERT Chapter 3 Chemistry: Chemical Kinetics

Here are some of the practical uses of this concept of temperature dependence in our daily lives:

• Heating and Cooking
• Preservation of Food Items
• Drug and Chemical Manufacturing
• Biomedical Sector
• Environmental Science
• Fuel Combustion
• Food Industry
• Pharmaceuticals
• Biological Enzymes
• Electrical Appliances (such as a thermometer)

The notes given below are designed with the goal of helping candidates in their preparation for high level competitive exams: