Lecture 11.3 Reaction Rates and temperature: Calculations

Understanding Reaction Rates and Temperature

Overview of Lecture Goals

• The lecture focuses on the relationship between temperature and reaction rates, emphasizing calculations related to this relationship.

• Key objectives include understanding how to calculate the effect of temperature on reaction rates using the Arrhenius equation and explaining its connection to Collision Theory.

Collision Theory Fundamentals

• For a reaction to occur, reactant molecules must collide with proper orientation and sufficient energy. This is crucial for successful reactions.

• Activation energy (EA) is defined as the minimum potential energy barrier that must be overcome for reactants to convert into products. It represents the threshold energy required for a reaction to proceed.

Graphical Representation of Collision Energy

• A plot illustrates collision energy on the x-axis versus the fraction of collisions on the y-axis, showing how many molecules have enough energy for a successful reaction.

• At low temperatures, only a few molecules possess sufficient energy (above EA), limiting successful reactions; increasing temperature raises this number significantly without altering EA itself.

Impact of Temperature on Reaction Rates

• Increasing temperature does not decrease activation energy but increases the number of molecules with adequate energy, leading to more frequent collisions and higher reaction rates. Thus, higher temperatures enhance reaction probabilities due to increased molecular movement and collision frequency.

Calculating Activation Energy

Example Problem Setup

• An example problem involves determining activation energy (EA) and collision frequency from experimental data at 700 Kelvin using graphical methods. The data points can be plotted for analysis.

Graphical Method Explanation

• By plotting data points on graph paper, one can derive slope values which help in calculating EA through linear relationships represented by the Arrhenius equation: ln K = -E_a/R cdot 1/T + C . Here, K is rate constant, R is gas constant, and T is temperature in Kelvin.

Slope Calculation Details

• The slope derived from plotting should equal -EA/R; thus rearranging allows calculation of EA when given specific slope values from experimental data points plotted against their respective temperatures.

Final Calculation Insights

• Using numerical values such as R = 8.314 J/(mol·K) alongside calculated slopes enables determination of significant activation energies necessary for understanding chemical kinetics effectively; in this case yielding an EA value around 1819186 Joules or approximately 1.82 MJ/mol after adjustments are made for signs during calculations.

Activation Energy and Rate Constants Calculation

Understanding Activation Energy

• The activation energy is expressed in joules per mole, which can be converted to kilojoules by dividing by a factor of 1000. This results in an activation energy of 181 kJ/mol.

Calculating Frequency Factor

• To find the frequency factor (A), the natural logarithm (Ln) is used: A = e^(26.662), leading to a value of approximately 3.79 x 10^11 for first-order reactions, with units being inverse seconds.

Graphical Method for Activation Energy

• The graphical method involves using the Arrhenius equation to calculate activation energy from two rate constants at different temperatures (K1 and K2). The formula includes terms for temperature (T1 and T2) and requires taking the natural logarithm of both sides.

Example Calculation Steps

• For K1 = 0.011 and K2 = 0.035 at temperatures T1 = 700K and T2 = 730K:

• Calculate Ln(0.011) - Ln(0.035) to find a value of approximately -1.1157.

• Rearranging gives: EA/R = -(-1.1157), where R is the gas constant (8.314 J/(mol·K)).

Finalizing Activation Energy Calculation

• After performing calculations involving multiplication by R and division by small numbers, an estimated activation energy comes out to be around 158 kJ/mol after conversion from joules per mole to kilojoules per mole through appropriate factors.

Fast Order Reaction Example

• In another example involving a fast order reaction with K1 = 0.034 at T1 = 298K:

• Using EA provided as 50.2 kJ, convert this into joules for calculation purposes.

• Apply the Arrhenius equation again to solve for K2 at T2 = 350K, resulting in a calculated value of approximately 0.69 after applying anti-logarithmic transformations on both sides of the equation during calculations involving Ln(K).

Activation Energy Calculation in Chemical Reactions

Understanding the Basics of Activation Energy

• The calculation of activation energy (EA) requires using a calculator and the Arrhenius equation, specifically the natural logarithm form.

• In Example 3, given temperature T1 at 45°C with rate constant K1 as 0.8, and T2 at 135°C with K2 as 2.4, we aim to find the activation energy for this reaction.

Applying the Arrhenius Equation

• The formula involves calculating ln(K1) - ln(K2) = -EA/R left( 1/T1 - 1/T2 right) , where R is the gas constant (8.314 J/mol·K).

• Converting temperatures: T1 becomes 318 K (273 + 45), and T2 becomes 408 K (273 + 135). This leads to calculations involving ln(0.8) - ln(2.4) = -EA/8.314(1/318 - 1/408) .

Rearranging for Activation Energy

• To isolate EA, multiply both sides by -1 to make it positive; then rearrange to solve for EA.

• After calculations, EA is found to be approximately 1316143 J/mol or about 13 kJ/mol when converted.

Further Example on Reaction Rate Increase

• A new problem states that if a reaction's rate increases by a factor of ten when temperature rises from 25°C to 50°C, we need to determine EA again.

• Here, K1 is assumed as 1 and K2 as 10 due to their ratio being equal to ten; thus K_ratio = K_1/K_2 = 10/1 =10.

Final Calculations for Activation Energy

• Convert temperatures: T1 is now at 298 K (25 + 273), and T2 at 323 K (50 + 273).

• Using ln(K_ratio) = ln(10), which equals approximately 2.30, we set up our equation again with these values.

• Correcting earlier mistakes in ratios leads us back through similar calculations yielding an updated value for EA.

Conclusion on Results

• After performing all necessary multiplications and divisions based on previous steps, the final calculated activation energy comes out around 73 kJ/mol after conversion from J/mol by dividing by a factor of one thousand.