Temperature Dependence and Reaction Theories

Temperature Dependence Of The Rate Of A Reaction

General Observation: It is generally observed that the rate of a chemical reaction increases significantly with an increase in temperature. A common rule of thumb is that for many reactions, the rate approximately doubles for every 10°C (or 10 K) rise in temperature. However, this is just an approximation and the actual temperature dependence is more complex.

Arrhenius Equation: The relationship between the rate constant ($k$) and temperature ($T$) is quantitatively described by the Arrhenius equation:

$$k = A e^{-E_a / RT}$$

Where:

$k$ is the rate constant.
$A$ is the pre-exponential factor or the frequency factor. It represents the frequency of collisions with the correct orientation. It is related to the frequency of collisions and the probability that these collisions have the correct orientation.
$e$ is the base of the natural logarithm.
$E_a$ is the activation energy of the reaction. This is the minimum amount of energy that reacting molecules must possess for a collision to be effective and lead to a chemical reaction. It's an energy barrier that must be overcome.
$R$ is the ideal gas constant (8.314 J K$^{-1}$ mol$^{-1}$).
$T$ is the absolute temperature (in Kelvin).

Explanation of the Arrhenius Equation:

Activation Energy ($E_a$): The exponential term $e^{-E_a / RT}$ represents the fraction of molecules that have kinetic energy equal to or greater than the activation energy at a given temperature $T$. As temperature increases, the value of $-E_a/RT$ becomes less negative (closer to zero), and thus $e^{-E_a / RT}$ increases, leading to a higher rate constant $k$. This means a larger fraction of molecules possess the minimum energy required for a reaction.
Pre-exponential Factor ($A$): This factor accounts for the frequency of collisions and the probability of those collisions having the correct orientation. While it can have some temperature dependence, it is often treated as constant over moderate temperature ranges compared to the exponential term.

Graphical Method to Determine $E_a$ and $A$:

We can linearize the Arrhenius equation by taking the natural logarithm of both sides:

$$\ln k = \ln(A e^{-E_a / RT})$$ $$\ln k = \ln A + \ln(e^{-E_a / RT})$$ $$\ln k = \ln A - \frac{E_a}{RT}$$

This equation is in the form of a straight line, $y = mx + c$, where:

$y = \ln k$
$x = \frac{1}{T}$
$m = -\frac{E_a}{R}$ (the slope)
$c = \ln A$ (the y-intercept)

By plotting $\ln k$ against $\frac{1}{T}$ for various temperatures, we can obtain a straight line. From the slope of this line, the activation energy ($E_a$) can be calculated ($E_a = - \text{slope} \times R$). The y-intercept can be used to determine the pre-exponential factor ($A$).

Example: The rate constant for a particular reaction is $1.0 \times 10^{-2} \text{ s}^{-1}$ at 298 K and $2.0 \times 10^{-2} \text{ s}^{-1}$ at 313 K. Calculate the activation energy ($E_a$) for this reaction.

Example 1. The rate constant for a particular reaction is $1.0 \times 10^{-2} \text{ s}^{-1}$ at 298 K and $2.0 \times 10^{-2} \text{ s}^{-1}$ at 313 K. Calculate the activation energy ($E_a$) for this reaction.

Answer:

We use the two-point form of the Arrhenius equation, derived from $\ln k = \ln A - \frac{E_a}{RT}$:

$$\ln\left(\frac{k_2}{k_1}\right) = \frac{E_a}{R}\left(\frac{1}{T_1} - \frac{1}{T_2}\right)$$

Given:

$k_1 = 1.0 \times 10^{-2} \text{ s}^{-1}$
$T_1 = 298$ K
$k_2 = 2.0 \times 10^{-2} \text{ s}^{-1}$
$T_2 = 313$ K
$R = 8.314 \text{ J K}^{-1} \text{ mol}^{-1}$

Calculate the ratio of rate constants:

$$\frac{k_2}{k_1} = \frac{2.0 \times 10^{-2}}{1.0 \times 10^{-2}} = 2$$

Substitute the values into the equation:

$$\ln(2) = \frac{E_a}{8.314 \text{ J K}^{-1} \text{ mol}^{-1}}\left(\frac{1}{298 \text{ K}} - \frac{1}{313 \text{ K}}\right)$$

$0.693 = \frac{E_a}{8.314 \text{ J K}^{-1} \text{ mol}^{-1}}\left(\frac{313 - 298}{298 \times 313}\right)$

$0.693 = \frac{E_a}{8.314 \text{ J K}^{-1} \text{ mol}^{-1}}\left(\frac{15}{93274}\right)$

$0.693 = \frac{E_a}{8.314 \text{ J K}^{-1} \text{ mol}^{-1}} \times 1.608 \times 10^{-4} \text{ K}^{-1}$

Now, solve for $E_a$:

$$E_a = \frac{0.693 \times 8.314 \text{ J K}^{-1} \text{ mol}^{-1}}{1.608 \times 10^{-4} \text{ K}^{-1}}$$ $$E_a \approx \frac{5.762 \text{ J mol}^{-1}}{1.608 \times 10^{-4}}$$ $$E_a \approx 35833 \text{ J mol}^{-1}$$

Convert to kJ/mol:

$$E_a \approx 35.83 \text{ kJ mol}^{-1}$$

The activation energy for this reaction is approximately 35.83 kJ/mol.

Collision Theory Of Chemical Reactions

Collision Theory: Collision theory is a model that explains how chemical reactions occur and why reaction rates differ. It is based on the idea that for a reaction to take place, reactant molecules must collide with each other. However, not all collisions result in a reaction.

Key Postulates of Collision Theory:

Collision Requirement: Reactant particles must collide in order for a chemical reaction to occur.
Energy Requirement (Activation Energy): The colliding molecules must possess a minimum amount of kinetic energy, called the activation energy ($E_a$), for the collision to be effective. This energy is required to overcome the energy barrier and break existing bonds, allowing new bonds to form.
Orientation Requirement: The colliding molecules must have the proper spatial orientation or configuration for the reactive atoms or bonds to come into contact.

Factors Influencing Collision Theory:

Concentration: Higher concentrations of reactants lead to more frequent collisions between reactant molecules. This increases the rate of effective collisions, and thus the reaction rate.
Temperature: An increase in temperature increases the average kinetic energy of the molecules. This leads to:
- Increased Collision Frequency: Molecules move faster, leading to more collisions per unit time.
- Increased Fraction of Effective Collisions: A larger proportion of molecules will possess energy equal to or greater than the activation energy ($E_a$). This latter effect is more significant in increasing the reaction rate. This is captured by the $e^{-E_a/RT}$ term in the Arrhenius equation.
Physical State and Surface Area: For reactions involving solids, the reaction occurs at the surface. Increasing the surface area of a solid reactant (e.g., by grinding it into a powder) increases the number of reactant particles exposed for collision, thereby increasing the reaction rate.
Catalysts: Catalysts are substances that increase the rate of a chemical reaction without being consumed in the process. They do this by providing an alternative reaction pathway with a lower activation energy ($E_a$), or by increasing the frequency of effective collisions.

The Pre-exponential Factor (A):

The pre-exponential factor ($A$) in the Arrhenius equation can be thought of as the product of two factors:

$$A = P \times Z$$

Where:

$Z$ is the collision frequency. It represents the total number of collisions per unit volume per unit time between reactant molecules. Collision frequency is influenced by factors like concentration, temperature, and the size and mass of the molecules.
$P$ is the probability factor or the steric factor. It accounts for the requirement of proper orientation during collisions. If $P=1$, it implies that all collisions are effective, which is rare. For most reactions, $P$ is significantly less than 1, indicating that only a small fraction of collisions have the correct orientation.

Limitations of Collision Theory:

Collision theory assumes that molecules behave like hard spheres that collide. It doesn't fully account for the detailed mechanisms of bond breaking and formation, or the role of intermediate species.
It is most successful for simple bimolecular reactions. For unimolecular reactions and reactions involving complex mechanisms, other theories like Transition State Theory are often more appropriate.

Example: Explain why powdered sugar dissolves faster in water than granulated sugar.

Example 1. Explain why powdered sugar dissolves faster in water than granulated sugar.

Answer:

The process of dissolving can be considered a type of reaction at the molecular level, where sugar molecules interact with water molecules. According to collision theory, the rate of interaction is influenced by the frequency of collisions between sugar molecules and water molecules. Powdered sugar has a much larger surface area compared to granulated sugar because it is broken down into many smaller particles. A larger surface area means that more sugar molecules are exposed to the water at any given time. This increased exposure leads to a higher frequency of collisions between sugar molecules and water molecules, resulting in a faster rate of dissolution.