Chapter 4: Problem 87
The rate constant, the activation energy and the Arrhenius parameter of a chemical reaction at \(25^{\circ} \mathrm{C}\) are \(3.0 \times 10^{-4} \mathrm{~s}^{-1}, 104.4 \mathrm{~kJ} \mathrm{~mol}^{-1}\) and \(6 \times 10^{14} \mathrm{~s}^{-1}\) respectively:
Short Answer
Expert verified
By inserting the values into the Arrhenius equation and simplifying, we verify that the given rate constant, activation energy, and Arrhenius parameter are consistent with the equation.
Step by step solution
01
Understand the Arrhenius Equation
The Arrhenius Equation is given by the formula: \( k = A e^{\frac{-E_a}{RT}} \), where \( k \) is the rate constant, \( A \) is the Arrhenius parameter (frequency factor), \( E_a \) is the activation energy, \( R \) is the gas constant, and \( T \) is the temperature in Kelvin.
02
Convert Temperature to Kelvin
First, convert the temperature from Celsius to Kelvin by adding 273.15 to the Celsius temperature. \( T(\text{K}) = 25 + 273.15 = 298.15 \text{K} \).
03
Convert Activation Energy to J/mol
Convert the activation energy from kJ/mol to J/mol by multiplying by 1000. \( E_a = 104.4 \text{ kJ/mol} \times 1000 = 104400 \text{ J/mol} \).
04
Prepare the Constants
Use the gas constant \( R \) in J/(mol K), which is approximately \( R = 8.314 \text{ J/(mol K)} \).
05
Plug Values into the Arrhenius Equation
Insert the known values into the Arrhenius equation: \( 3.0 \times 10^{-4} \text{s}^{-1} = 6 \times 10^{14} \text{s}^{-1} \times e^{\frac{-104400 \text{ J/mol}}{(8.314 \text{ J/(mol K)}) \times 298.15 \text{K}}} \).
06
Solve for the Exponent
Isolate the exponential term by dividing both sides of the equation by \( A \): \( e^{\frac{-E_a}{RT}} = \frac{k}{A} \). Calculate the exponent: \( e^{\frac{-104400}{(8.314) \times 298.15}} = \frac{3.0 \times 10^{-4}}{6 \times 10^{14}} \).
07
Calculate the Exponential Value
Calculate the value of the exponent and the corresponding exponential to check the calculation: \( e^{\frac{-104400}{(8.314) \times 298.15}} = e^{\frac{-104400}{2473.9171}} = e^{-42.207} \). Then, calculate the right side of the equation: \( \frac{3.0 \times 10^{-4}}{6 \times 10^{14}} = 5 \times 10^{-19} \).
08
Verify the Identity
Verify that the left-hand side (calculated exponential) is approximately equal to the right-hand side. This would ensure that the given rate constant, activation energy, and Arrhenius parameter satisfy the Arrhenius equation.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding the Chemical Reaction Rate
The rate at which a chemical reaction occurs can be thought of as the speed of the reaction. It tells us how quickly reactants are converted into products. This rate can be affected by various factors, including the presence of a catalyst, the concentration of the reactants, and the temperature at which the reaction takes place.
For instance, increasing the temperature generally increases the reaction rate, as the particles of the reacting substances have more energy to collide with each other more frequently and with greater force. Mathematically, the chemical reaction rate is often expressed in terms of the change in concentration of a reactant or product per unit time. Simple reactions might follow the equation rate = \( k[Reactant]^{order} \), where \( k \) is the rate constant and the order is the power to which the concentration of the reactant is raised.
Understanding the factors affecting reaction rates is crucial in various fields, from industrial synthesis to biochemistry, as it allows chemists to control and optimize reactions efficiently.
For instance, increasing the temperature generally increases the reaction rate, as the particles of the reacting substances have more energy to collide with each other more frequently and with greater force. Mathematically, the chemical reaction rate is often expressed in terms of the change in concentration of a reactant or product per unit time. Simple reactions might follow the equation rate = \( k[Reactant]^{order} \), where \( k \) is the rate constant and the order is the power to which the concentration of the reactant is raised.
Understanding the factors affecting reaction rates is crucial in various fields, from industrial synthesis to biochemistry, as it allows chemists to control and optimize reactions efficiently.
Activation Energy: The Energy Threshold for Chemical Reactions
Activation energy (\( E_a \) ) is a term that describes the minimum amount of energy required for a chemical reaction to proceed. It's like the 'entrance fee' for reactants to transform into products. When molecules collide, they must do so with enough energy to exceed the activation energy before a reaction can occur.
This concept is key in understanding why certain reactions occur spontaneously while others require additional energy, such as heat or light, to get started. The activation energy can also explain why certain reactions that release energy (exothermic reactions) might still be slow because the reactant molecules must overcome an initial energy barrier.
To visualize this, think of a ball needing to get over a hill (the activation energy) before it can roll down (the reaction proceeding). Catalysts are substances that lower the activation energy, making it easier for the reaction to occur without changing the overall energy balance.
This concept is key in understanding why certain reactions occur spontaneously while others require additional energy, such as heat or light, to get started. The activation energy can also explain why certain reactions that release energy (exothermic reactions) might still be slow because the reactant molecules must overcome an initial energy barrier.
To visualize this, think of a ball needing to get over a hill (the activation energy) before it can roll down (the reaction proceeding). Catalysts are substances that lower the activation energy, making it easier for the reaction to occur without changing the overall energy balance.
Rate Constant: The Proportionality Factor in Chemical Kinetics
In chemical kinetics, the rate constant (\( k \) ) is a proportionality factor that connects the reaction rate to the concentrations of reactants. It is a unique value for every reaction at a given temperature, providing insights into the reaction's speed under specific conditions.
The value of the rate constant is influenced by the nature of the reactants, the presence of a catalyst, and other environmental conditions like temperature or solvent. For example, as temperature increases, the rate constant typically increases as well, due to the increased energy and frequency of collisions between reactant molecules.
Knowing the rate constant allows us to predict how quickly a reaction will reach completion, which is particularly important in designing chemical processes and manufacturing, where time and efficiency are of the essence.
The value of the rate constant is influenced by the nature of the reactants, the presence of a catalyst, and other environmental conditions like temperature or solvent. For example, as temperature increases, the rate constant typically increases as well, due to the increased energy and frequency of collisions between reactant molecules.
Knowing the rate constant allows us to predict how quickly a reaction will reach completion, which is particularly important in designing chemical processes and manufacturing, where time and efficiency are of the essence.
Temperature Conversion: Celsius to Kelvin for Chemical Reactions
Temperature plays a pivotal role in the study of chemical kinetics, and it's important to express it accurately. In scientific contexts, especially in chemistry, temperature is usually measured in Kelvin (K). The Kelvin scale is an absolute temperature scale, starting at absolute zero, the theoretical point where particles have no kinetic energy.
To convert Celsius (°C), which is more common in everyday use, to Kelvin, one must add 273.15 to the Celsius value. This conversion is crucial for calculations involving the Arrhenius Equation or any temperature-dependent constant in chemistry because these constants are often defined using the Kelvin scale.
The relationship between temperature and reaction rate can be explored through the Arrhenius Equation, which is significantly affected by even slight changes in temperature. Therefore, accurate temperature conversion ensures that predictions and measurements involving reaction rates and energies are correct.
To convert Celsius (°C), which is more common in everyday use, to Kelvin, one must add 273.15 to the Celsius value. This conversion is crucial for calculations involving the Arrhenius Equation or any temperature-dependent constant in chemistry because these constants are often defined using the Kelvin scale.
The relationship between temperature and reaction rate can be explored through the Arrhenius Equation, which is significantly affected by even slight changes in temperature. Therefore, accurate temperature conversion ensures that predictions and measurements involving reaction rates and energies are correct.