Chapter 15: Problem 4
Fill in all of the details in the derivation of Equation 15.32 for the bimolecular reaction rate, given in Section 15.2 .4.
Short Answer
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The derivation of the bimolecular reaction rate is complex, involving integration over all speeds of a Maxwell-Boltzmann distribution. The final result should resemble the Arrhenius formula adjusting the pre-exponential factor based on details of the bimolecular process.
Step by step solution
01
Starting Point- Arrhenius Equation
The Arrhenius equation is a mathematical model that describes the temperature dependence of reaction rates. It has the form: \(k = A \exp\left(\frac{-E_a}{k_bT}\right)\), where \(k\) is the rate constant, \(A\) is the pre-exponential factor, \(E_a\) is the activation energy, \(k_b\) is the Boltzmann constant and \(T\) is the absolute temperature.
02
Integration Over All Energies
In a bimolecular reaction, there are particles with different energies. Therefore, for the entire system, we must integrate over all possible molecular speeds. The Maxwell-Boltzmann distribution allows us to determine the fraction of molecules that have each possible kinetic energy. Therefore, our equation evolves into an integral: \(k = 2\pi N^2 \int \sigma(u) u \exp\left(\frac{-E_a}{k_bT}\right)f(u) du\) where \(u\) is the relative speed of two colliding molecules, \(\sigma(u)\) is the cross-section of the collision, \(N\) is the number of molecules and \(f(u)\) represents the Maxwell-Boltzmann distribution.
03
Solving the Integral
This integral is not trivial to solve, and it often requires special mathematical techniques or approximations. It would depend on the specific function forms for \(\sigma(u)\) and \(f(u)\). If, for example, the cross-section were a constant independent of speed and the collision happened only if the kinetic energy exceeds a certain value (the activation energy), the integral could be solved exactly. The final result would then be \(k = A_1 \exp\left(\frac{-E_a}{k_bT}\right)\), where \(A_1\) contains the pre-exponential factor \(A\) as well as constants from the solution of the integral.
04
Interpretation of the Derived Equation
The derived equation should look similar to the original Arrhenius equation, but with possible adjustments to the pre-exponential factor depending on the specific circumstances of the bimolecular reaction. The exponential dependence on \(-E_a/k_bT\) illustrates how the reaction rate increases with temperature and decreases with higher activation energy, which matches known chemical behaviour.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Arrhenius Equation
The **Arrhenius Equation** is a vital concept when studying the rates of chemical reactions. Named after the Swedish scientist Svante Arrhenius, this equation provides a clear mathematical model for understanding how temperature affects the speed of a chemical reaction. The equation is expressed as:
The term \(E_a\) is the activation energy, crucial for determining the minimum energy required for a reaction to occur. The Boltzmann constant \(k_b\) connects macroscopic and microscopic physical quantities, while \(T\) is the absolute temperature.
From this equation, you see that even small temperature changes can noticeably affect the rate of a reaction because of the exponential nature of the expression. A higher temperature can significantly increase the rate constant \(k\), making reactions happen faster.
- \[ k = A \exp\left(\frac{-E_a}{k_bT}\right) \]
The term \(E_a\) is the activation energy, crucial for determining the minimum energy required for a reaction to occur. The Boltzmann constant \(k_b\) connects macroscopic and microscopic physical quantities, while \(T\) is the absolute temperature.
From this equation, you see that even small temperature changes can noticeably affect the rate of a reaction because of the exponential nature of the expression. A higher temperature can significantly increase the rate constant \(k\), making reactions happen faster.
Maxwell-Boltzmann Distribution
The **Maxwell-Boltzmann Distribution** is an essential concept in the field of statistical mechanics and chemistry, as it describes the distribution of speeds among molecules in a gas. This is particularly important for understanding bimolecular reactions, where such speed distributions impact how often molecules collide and react.In essence, the Maxwell-Boltzmann distribution tells us that not all molecules in a gas move at the same speed. Instead, there is a spread of speeds, with some molecules moving slowly and others more rapidly. The distribution can be visualized as a curve, with the most probable speed corresponding to the peak of this curve.
This distribution is mathematically represented by a function \(f(u)\), which quantifies the likelihood that a particle in the gas is traveling at a particular speed \(u\). Integrating over this distribution assists in calculating how many molecules exceed the activation energy and consequently partake in the reaction. The relationship between speed, collision frequency, and energy is essential for predicting reaction rates in gases.
Ultimately, the Maxwell-Boltzmann distribution provides a fundamental underpinning for interpreting reaction rates and dynamics based on molecular speed distributions.
This distribution is mathematically represented by a function \(f(u)\), which quantifies the likelihood that a particle in the gas is traveling at a particular speed \(u\). Integrating over this distribution assists in calculating how many molecules exceed the activation energy and consequently partake in the reaction. The relationship between speed, collision frequency, and energy is essential for predicting reaction rates in gases.
Ultimately, the Maxwell-Boltzmann distribution provides a fundamental underpinning for interpreting reaction rates and dynamics based on molecular speed distributions.
Rate Constant
The **Rate Constant**, symbolized as \(k\), is a key component of kinetics in chemical reactions. It serves to quantify the speed at which a reaction proceeds under specific conditions. The rate constant is heavily influenced by various factors including temperature, pressure, and the presence of catalysts.In the context of the Arrhenius equation, the rate constant is expressed as:
Furthermore, the value of the rate constant can guide scientists in optimizing conditions to achieve efficient and faster reactions, especially in industrial processes where reaction speed is a critical factor.
- \[ k = A \exp\left(\frac{-E_a}{k_bT}\right) \]
Furthermore, the value of the rate constant can guide scientists in optimizing conditions to achieve efficient and faster reactions, especially in industrial processes where reaction speed is a critical factor.