The rate of a certain reaction doubles for every \(10^{\circ} \mathrm{C}\) rise in temperature. (a) How much faster does the reaction proceed at \(45^{\circ} \mathrm{C}\) than at \(25^{\circ} \mathrm{C}\) ? (b) How much faster does the reaction proceed at \(95^{\circ} \mathrm{C}\) than at \(25^{\circ} \mathrm{C}\) ?

The Arrhenius equation is a formula that provides deep insight into the effects of temperature on the rates of chemical reactions. It is expressed as:
\[ k = A e^{-\frac{E_a}{RT}} \]
where:\

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• \(k\) is the rate constant of the reaction,\
• \(A\) is the frequency factor, representing the number of times particles collide with the correct orientation per unit time,\
• \(E_a\) is the activation energy of the reaction,\
• \(R\) is the universal gas constant, and\
• \(T\) is the absolute temperature (in Kelvin).\

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The equation demonstrates that an increase in temperature leads to an exponential increase in the reaction rate, as seen in our exercise example where the rate doubles for every 10°C rise. This exponential relationship is because higher temperatures increase the number of particles that have enough energy to overcome the activation energy barrier. When explaining how a reaction speeds up as the temperature rises, we often simplify this concept to the rule of thumb presented in the textbook exercise, but the Arrhenius equation gives us a quantitative way to calculate the effect of temperature changes on reaction rates.

Chemical kinetics is the branch of chemistry that is concerned with the rates of chemical reactions and the mechanisms by which they occur.
Understanding kinetics allows chemists to determine how different conditions, such as temperature, concentration, and the presence of a catalyst affect the speed of a reaction. It relies on measurements of reaction rates, which tell us how quickly reactants are transformed into products.
In our textbook exercise, the rate of a reaction doubling with each 10°C temperature rise is an application of kinetics. It's important to note that not all reactions will have such a simply defined rate increase with temperature, but kinetics gives us the tools to predict and explain these rate changes. The rate at which reactions occur can be quite complex, involving multiple steps, reactant collisions, and activation energies, which are all described more deeply by the principles of kinetics.

Activation energy, or \(E_a\), is a term in chemistry that represents the minimum amount of energy needed for reactants to transform into products during a chemical reaction.
It's like the initial ‘push’ needed to start a chemical process. In essence, for reactants to successfully react and form products, their molecules must collide with sufficient energy to overcome the activation energy barrier.
This concept is crucial in the context of our exercise because it helps explain why increasing the temperature results in a faster reaction rate. Increasing temperature boosts the energy of molecules involved in the reaction, increasing the likelihood that collisions will have enough energy to overcome the activation energy barrier. However, it’s not just about reaching this threshold - the distribution of molecular energies at any given temperature can also affect how many reactants have sufficient energy, which ties back to the Arrhenius equation and the temperature dependence of the reaction rate.