 # Temperature Dependence of the Rate of a Reaction While cooking, if we keep the gas on low temperature, the food cooks slowly. But when we increase the temperature to its maximum, the food cooks quickly. Therefore, temperature increases the rate of a reaction. This dependence of rate on temperature can be explained by Arrhenius equation. Let’s learn about and deduce this equation.

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### Suggested Videos        ## Temperature and Rate – the Relationship

By now, we know that temperature influences the rate of a reaction. As the temperature increases, the rate of a reaction increases. For example, the time taken to melt a metal will be much higher at a lower temperature but it will decrease as soon as we increase the temperature. It has been found that the rate constant is nearly doubled for a chemical reaction with a rise in temperature by 10°. We can explain the dependence of the rate of a chemical reaction on temperature by Arrhenius equation.

## Arrhenius Equation

The equation was first proposed by Dutch chemist, J.H. Van’t Hoff but Swedish chemist, Arrhenius provided its physical justification and interpretation. The Arrhenius equation is based on the Collision theory. It is not an equation that is born out of pure math that we can derive. It is an empirical equation that fits experimental data in most of the situations. The Arrhenius equation looks like this,

k = A e -Ea/RT……………(I) (Source: chemguide.co.uk)

where A is the Arrhenius factor or the frequency factor. It is also known as the pre-exponential factor. This constant is specific to a particular reaction. R is the gas constant and Eis the activation energy which we measure in joules/mole.

According to the Arrhenius equation, a reaction can only take place when a molecule of one substance collides with the molecule of another to form an unstable intermediate. This intermediate exists for a very short time and then breaks up to form two molecules of the product. The energy required to form this intermediate is known as activation energy (Ea). (Source: en.wikipedia.org)

In a graph of potential energy vs reaction coordinate, the reaction coordinate represents the profile of energy change when reactants change into products.  Some of the energy releases when the complex decomposes to form products. Therefore, the final enthalpy of the reactions depends only on the nature of the reactants and products.

Obviously, all the molecules do not have the same energy. The distribution of kinetic energy can be described by plotting the fraction of molecules with given kinetic energy vs kinetic energy.  The peak of the curve in the graph corresponds to the most probable kinetic energy. When the temperature increases, the maximum of the curve moves to the higher energy value. Therefore, the curve broadens. (Source: wps.prenhall.com)

Increasing the temperature increases the fraction of molecules, which collide with energies greater than the activation energy Ea.

### Temperature dependence of Rate of Reaction in Arrhenius Equation

In Arrhenius equation, the factor e -Ea/RT  corresponds to the fraction of molecules colliding with activation energies more than Ea. Taking natural logarithms of both sides of the equation I, we get,

ln k = -Ea/RT + ln A …………..(II)

Therefore, from the Arrhenius equation, we can find that increasing the temperature or decreasing the activation energy will result in an increase in the rate of the reaction and an exponential increase in the rate constant. In a graph of activation energy vs rate of reaction, slope = -Ea/R and intercept = ln A.

At temperature T1, equation II will be

ln k1 = -Ea/RT1 + ln A …………………(III)

At temperature T, equation II will be

ln k2 = -Ea/RT2 + ln A …………………(IV) (k1 and k2 are the rate constants at temperature T1 and T2)

Subtracting equation III from equation IV, we get

ln k2 – ln k1 = Ea/RT1 – Ea/RT2

∴ ln kk= (E/R)[1/T1 – 1/T2]

∴ log kk= (E/2.303R)[(T– T1)/T1T2]

## A Solved Question for You

Q: Why does the rate of a reaction increase when the temperature increases?

Solution: When the temperature increases, the fraction of molecules that have kinetic energies more than the activation energy of the reaction increases. Therefore, the total activation energy of the reaction decreases. Thus, the rate of the reaction increases.

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