Why does a small increase in temperature have a large effect on rate and difference between enthalpy change of reaction and energy of activation of reaction?
The Arrhenius equation explains the exponential dependence (i.e., high dependence) of the rate constant (reaction rate) of a chemical reaction on the absolute temperature.
Rate constant = Constant x e(-Ea/RT).
Ea is the energy of activation. A similar equation (exponential dependence) is established for the enthalpy of activation. The energy (or enthalpy) of activation is an energy barrier necessary to go through the transition state which is higher in energy than the reactants. After the reaction, the products return to an energy state lower than the enthalpy of activation.
The enthalpy of reaction is the difference between the enthalpies of the products and of the reactants, so it is different (smaller in absolute value) from the enthalpy of activation. The attached figures may help (taken in "15.6: The Effect of Temperature on Reaction Rate - Chemistry LibreTexts").
Case (a) is for an exothermic reaction (negative energy or enthalpy of reaction), case (b) for an endothermic reaction, with positive energy and enthalpy of reaction.
Increasing the temperature increases reaction rates because of the disproportionately large increase in the number of high energy collisions. It is only these collisions which result in a reaction. If the temperature is increased: the reactant particles move more quickly they have more energy the particles collide successfully more often. The effect of temperature on a rate constant always increases the rate constant. This is because the temperature increases the number of collisions which increases the chance of a collision having enough energy to overcome the activation energy barrier. Temperature affects the rate of a reaction because increasing the temperature will increase the kinetic energy of the reactant particles. This results in more successful collisions between reactant particles and a faster rate of reaction. By increasing temperature, reactants have more kinetic energy and move faster, making them collide more often. By increasing number of collisions, number of reactions is increased and so reaction rate increases. Enthalpy change measures the difference in initial and final states while activation energy is the energy required to start a reaction. Activation enthalpy is the energy required to start the reaction while enthalpy of reaction is equal to the amount of heat released or absorbed during the reaction. The minimum amount of energy required by a molecule i.e., reacting species to undergo a chemical reaction is known as energy of activation for the reaction whereas enthalpy is the heat energy absorbed or released from the system during a chemical reaction performed at a constant pressure. For both exothermic as well as endothermic reactions it not a necessary condition for the activation energy to be equal to the Enthalpy change as activation energy is always greater than the Enthalpy change. As the temperature increases, the molecules move faster and therefore collide more frequently. The molecules also carry more kinetic energy. Thus, the proportion of collisions that can overcome the activation energy for the reaction increases with temperature.
It is not true that "The effect of temperature on a rate constant always increases the rate constant." In the gas phase and in special conditions, inverse effects of T may be observed (negative activation energy). As far as I can remember...
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Activation enthalpy is the energy required to start the reaction while enthalpy of reaction is equal to the amount of heat released or absorbed during the reaction. Enthalpy change measures the difference in initial and final states (the net energy released or added) while activation energy is the energy required to start a reaction. The minimum amount of energy required by a molecule i.e., reacting species to undergo a chemical reaction is known as energy of activation for the reaction whereas enthalpy is the heat energy absorbed or released from the system during a chemical reaction performed at a constant pressure. Endothermic reactions have greater activation energy of the forward reaction than the reverse reaction because the products of this reaction have more energy than the reactants. We know endothermic reactions require energy/heat... this energy is required to overcome the greater energy barrier.Thus, Ea>ΔH. Q. For an endothermic reaction, energy of activation is Ea and enthalpy of reaction is ΔH (both of these in kJ/mol). In exothermic reaction, change in enthalpy is negative while in endothermic reaction, change in enthalpy is positive. Activation energy has no link with the endothermic and exothermic part of any reaction as it relates with only rate of any chemical reaction. Increasing the temperature increases reaction rates because of the disproportionately large increase in the number of high energy collisions. It is only these collisions which result in a reaction. The effect of temperature on a rate constant always increases the rate constant. This is because the temperature increases the number of collisions which increases the chance of a collision having enough energy to overcome the activation energy barrier. As the temperature increases, the particles are given more energy and can move faster. As a result, there is likely to be a greater number of collisions and a faster rate of reaction. Therefore, the rate of reaction increases as the temperature increases. When you lower the temperature, the molecules are slower and collide less. That temperature drop lowers the rate of the reaction.