Question

How do reaction rates typically depend on temperature? What part of the rate law is temperature dependent?

Short answer

Reaction rates typically increase exponentially with increasing temperature, as described by the Arrhenius equation. The temperature dependence of the reaction rate is embodied in the rate constant \(k\) of the rate law.

Step-by-step solution

Understanding the General Temperature Dependence

The dependence of reaction rates on temperature is generally described by the Arrhenius equation, which states that the rate of a chemical reaction increases exponentially with an increase in temperature. The equation is given by \( k = A e^{-\frac{E_a}{RT}} \), where \(k\) is the rate constant, \(A\) is the frequency factor, \(E_a\) is the activation energy, \(R\) is the universal gas constant, and \(T\) is the temperature in kelvins.

Identifying the Temperature Dependent Part of the Rate Law

In the rate law \(Rate = k[A]^n[B]^m\) where \(k\) is the rate constant and the concentrations of reactants are given by \( [A] \) and \( [B] \), the part that is temperature dependent is the rate constant \(k\). The concentrations of reactants \( [A] \) and \( [B] \) are not temperature dependent within the context of the rate law.

Key concepts

Arrhenius Equation

The Arrhenius equation is pivotal to understanding how reaction rates increase as temperatures rise. Imagine a scenario where you're trying to bake a cake: a higher oven temperature typically decreases the baking time because heat accelerates the chemical reactions in the cake mixture. Similarly, in a chemical reaction, as the temperature increases, the reaction rate often increases exponentially. This relationship is described mathematically by the Arrhenius equation: \[\begin{equation}k = A e^{-\frac{E_a}{RT}}\end{equation}\]where:

\t
• \t\t\(k\) is the rate constant representing the reaction rate at a given temperature.
\t
• \t\t\(A\) is the pre-exponential factor, also known as the frequency factor, which relates to the frequency of collisions with the proper orientation.
\t
• \t\t\(E_a\) is the activation energy required for the reaction to proceed.
\t
• \t\t\(R\) is the universal gas constant (8.314 J/(mol·K)).
\t
• \t\t\(T\) is the temperature in Kelvin.

Increasing the temperature causes the exponential term (\[\begin{equation}e^{-\frac{E_a}{RT}}\end{equation}\]) to increase, thus increasing the value of \(k\) and the overall rate of the reaction. This equation underscores the exquisite sensitivity of reaction rates to temperature changes.

Rate Constant

The rate constant, denoted as \(k\), is a fundamental component in the rate law equation (\[\begin{equation}Rate = k[A]^n[B]^m\end{equation}\]) and serves as the bridge between the chemical kinetics and the thermodynamic properties of a reaction. It's akin to setting the speed on a cruise control in a vehicle: the rate constant dictates how fast or slow the reaction proceeds under certain conditions.Significantly, the rate constant is directly influenced by temperature, as described by the Arrhenius equation. It isn't something static but varies with ambient conditions, particularly temperature. Understanding what affects the rate constant can allow chemists to manipulate reaction speeds for desirable outcomes in industries such as pharmaceuticals, where controlling reaction rates can be critical for the creation of drugs.However, the rate constant doesn't depend on reactant concentrations, and it is specific to each chemical reaction. It encapsulates many factors, including how frequently reacting molecules collide, the energy and orientation of those collisions, and how many collisions lead to a successful reaction. By studying changes in the rate constant, chemists can infer a great deal about the mechanism and dynamics of a reaction.

Activation Energy

Activation energy (\(E_a\)) can be thought of as the 'toll fee' that reacting particles must pay to convert reactants into products; it's the minimum amount of energy required to initiate a chemical reaction. A common real-world analogy might be the energy needed to push a boulder over a hill before it can roll down the other side.The lower the activation energy, the easier it is for a reaction to occur, and vice versa. For instance, striking a match involves providing a little bit of energy through friction to overcome the activation energy, allowing the chemicals in the match head to react and produce a flame.In the context of the Arrhenius equation, the activation energy is what determines the sensitivity of the rate constant to temperature changes. A high activation energy means that a small increase in temperature will result in a substantial increase in the rate constant, hence a faster reaction rate. Conversely, a lower activation energy implies that temperature changes will have a less dramatic effect on the reaction rate.Understanding activation energy is essential not only for predicting how reactions will behave at different temperatures but also for the design of chemical processes where specific temperatures are used to control the speed and outcome of reactions. It's a critical factor in fields such as material science, energy production, and biochemistry.