Question

A standard "rule of thumb" for thermally activated reactions is that the reaction rate doubles for every \(10 \mathrm{K}\) increase in temperature. Is this statement true independent of the activation energy (assuming that the activation energy is positive and independent of temperature \() ?\)

Short answer

In summary, the "rule of thumb" that the reaction rate doubles for every 10 K increase in temperature may not hold true for all cases. While it could be true for specific activation energy values, there can be different reactions with other activation energies where this statement does not apply. The Arrhenius equation provides a general framework for understanding how reaction rate constants vary with temperature, but the statement's validity ultimately depends on the specific activation energy of a given reaction.

Step-by-step solution

Introduction to Arrhenius Equation

The Arrhenius equation provides a quantitative basis for the temperature dependence of reaction rates. The Arrhenius equation can be given as: \[k = A \cdot e^{-\frac{E_a}{RT}}\] Where: - \(k\) is the reaction rate constant - \(A\) is the pre-exponential factor or frequency factor - \(E_a\) is the activation energy of the reaction - \(R\) is the gas constant - \(T\) is the temperature in kelvins

Investigate the 10 K temperature increase effect on reaction rate constant

Let's consider the initial reaction rate constant at a given temperature T, as \(k_1\): \[k_1 = A \cdot e^{-\frac{E_a}{RT}}\] Now, we will calculate the reaction rate constant, \(k_2\), at the increased temperature (T + 10 K): \[k_2 = A \cdot e^{-\frac{E_a}{R(T+10)}}\]

Calculating the ratio of the reaction rate constants

To investigate whether the reaction rate doubles for every 10 K increase in temperature, calculate the ratio of the reaction rate constants \(k_2/k_1\): \[\frac{k_2}{k_1} = \frac{A \cdot e^{-\frac{E_a}{R(T+10)}}}{A \cdot e^{-\frac{E_a}{RT}}}\] Simplify the expression by canceling out the pre-exponential factor (A): \[\frac{k_2}{k_1} = \frac{e^{-\frac{E_a}{R(T+10)}}}{e^{-\frac{E_a}{RT}}}\] Now, we will use the properties of exponents to simplify the expression further: \[\frac{k_2}{k_1} = e^{\frac{E_a}{RT} - \frac{E_a}{R(T+10)}}\]

Evaluating the ratio under the given condition

Finally, we want to determine under which conditions the ratio of the reaction rate constants, \(\frac{k_2}{k_1}\), is equal to 2. We can write this as: \[e^{\frac{E_a}{RT} - \frac{E_a}{R(T+10)}} = 2\] To evaluate the activation energy dependency, we will need to solve this equation for \(E_a\). For this exercise, the temperature dependence of activation energy is assumed to be negligible. Therefore, we can conclude that the "rule of thumb" statement about the doubling of the reaction rate for every 10 K increase in temperature could be true under a specific activation energy value that satisfies the equation above. However, there could be different reactions with other activation energies in which this statement may not hold true.

Key concepts

Reaction Rate Dependence on Temperature

Understanding the relationship between reaction rate and temperature is crucial for chemists and it's encapsulated in the Arrhenius equation. The crux of this relationship is that as the temperature of a system increases, the molecules within it have more energy. This increase in molecular energy translates into a higher probability that molecular collisions will have sufficient energy to overcome the activation energy barrier, leading to an increased rate of reaction.

Reaction rates generally increase with temperature because molecules move faster and collide more frequently. These collisions need to be energetic enough to result in a reaction, which is where the activation energy comes in. Simply put, an increase in temperature can increase reaction rates by providing more energy to the molecules involved, thus increasing the likelihood of successful collisions. The rule of thumb that the rate doubles with each 10 K increase in temperature is an approximation based on typical reaction conditions, but the exact rate increase would depend on the specific activation energy of the reaction in question.

It's important to note that this temperature effect is not linear and follows an exponential pattern as indicated by the Arrhenius equation. Therefore, while the 'rule of thumb' is a useful guideline, the actual change in rate can vary and it is not universally applicable to all reactions.

Activation Energy

The activation energy, denoted as \(E_a\), is a key term in understanding how chemical reactions occur. It is defined as the minimum energy that must be input to a chemical system with potential reactants to result in a chemical reaction.

Activation energy can be thought of as a barrier. For a reaction to proceed, reactants must overcome this barrier. The higher the activation energy, the slower the reaction rate because fewer molecules will have sufficient energy to surpass the barrier at a given temperature. Conversely, a lower activation energy means that more molecules can potentially react, leading to a faster reaction rate.

According to the Arrhenius equation, reaction rates increase exponentially with the reciprocal of the temperature and inversely with the activation energy. Hence, even a small decrease in activation energy can greatly increase the reaction rate. This principle underlies the use of catalysts in chemical reactions, which work by providing an alternative reaction pathway with a lower activation energy.

Thermally Activated Reactions

Thermally activated reactions are chemical processes that require the absorption of heat to proceed. In essence, when substances are heated, the kinetic energy of their molecules increases. As the kinetic energy reaches a certain threshold—the activation energy—the molecules can undergo chemical transformation.

These reactions are often visualized on an energy profile diagram showing the energy of reactants and products, with the activation energy being the peak that must be overcome. It's important to note that not every collision between reactant molecules will result in reaction, even at high temperatures. A certain orientation or alignment might be required, further complicating the process.

With increased temperature, the fraction of molecules with energy equal to or greater than the activation energy also increases, described by the Maxwell-Boltzmann distribution. This distribution is the basis for the reaction rate dependence on temperature, and it explains why the 'rule of thumb' about reaction rates has its limitations—while temperature increases can speed up reaction rates, there is no simple, universal factor for this increase because it depends on the specific activation energy of the particular reaction.