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Chapter 13: Problem 63

What is the rule of thumb for how reaction rate changes as the temperature changes?

Short Answer

Expert verified
The rule of thumb for how reaction rate changes as the temperature changes is that the reaction rate approximately doubles for every 10掳C increase in temperature. This can be explained using the Arrhenius equation, which relates the reaction rate constant (k) to the temperature (T): \( k = A e^{-\frac{E_a}{RT}} \). A higher temperature increases the kinetic energy of reactants, leading to more successful collisions and a higher reaction rate. Keep in mind that this rule is a generalization and can vary for different reactions.

Step by step solution

01

Understand the factors that affect reaction rate

Reaction rates are influenced by several factors, such as concentration of reactants, temperature, presence of a catalyst, and surface area of reactants. In this exercise, we will focus on how temperature affects the reaction rate.
02

Know the Arrhenius equation

The Arrhenius equation is a mathematical equation that illustrates the dependency of reaction rate (`k`) on temperature (`T`). The equation is given by: \( k = A e^{-\frac{E_a}{RT}} \) Where: - \(k\) is the reaction rate constant, - \(A\) is the pre-exponential factor (also called frequency factor), which represents the frequency of collisions between reactants, - \(e\) is the base of the natural logarithm (~2.718), - \(E_a\) is the activation energy of the reaction (in J/mol), - \(R\) is the gas constant (8.314 J/(mol K)), - \(T\) is the temperature in Kelvin (K).
03

Understand the effect of temperature on the reaction rate

According to the Arrhenius equation, an increase in temperature leads to an increase in the reaction rate constant (`k`). A higher temperature means that the reactants have more kinetic energy, which increases the likelihood of successful collisions and the rate of the reaction. The rule of thumb for the effect of temperature on reaction rate is that the rate approximately doubles for every 10掳C increase in temperature. It's important to note that this rule is a generalization and can vary for different reactions.
04

Apply the rule of thumb to a specific temperature change

Suppose the reaction rate is `k1` at temperature `T1` and `k2` at temperature `T2`. If the temperature rises by 10掳C, we can apply the rule of thumb to estimate how the reaction rate will change: 1. Convert the 10掳C temperature increase to Kelvin: \( \Delta T = 10 掳C * \frac{1 K}{1 掳C} = 10 K \) 2. Calculate the new temperature: \( T2 = T1 + \Delta T \) 3. Estimate the new reaction rate: \( k2 鈮� 2 * k1 \) Keep in mind that this rule of thumb is an approximation and the actual change in reaction rate may differ. In summary, the rule of thumb for how reaction rate changes as the temperature changes is that the reaction rate approximately doubles for every 10掳C increase in temperature. This can be explained using the Arrhenius equation, which shows the dependency of the reaction rate on temperature.

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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 vital for understanding how temperature influences the reaction rate. It provides a mathematical description that links the reaction rate constant, denoted as \(k\), with the absolute temperature \(T\). This relationship is expressed as follows:
\[k = A e^{-\frac{E_a}{RT}}\]
Here:
  • \(A\), the pre-exponential factor, showcases the frequency of effective collisions.
  • \(E_a\), representing activation energy, is a barrier that reactants must overcome to react.
  • The equation uses the natural exponent \(e\) and the universal gas constant \(R = 8.314 \text{\ J/(mol K)}\).
By manipulating variables in this equation, one can discern how changing temperatures alter reaction speed. As temperature rises, the value of \(e^{-\frac{E_a}{RT}}\) decreases, increasing \(k\). This means more collisions become energetic enough to surpass \(E_a\), propelling the reaction forward.
Activation Energy
Activation energy, symbolized as \(E_a\), is crucial for chemical reactions to proceed. It forms the energy threshold that reactants need to gain to transition into products. Imagine \(E_a\) as a hill that reactants must go over during the reaction.
Reactants gain energy mainly through collisions. The more energy they receive and surpass \(E_a\), the faster the reaction occurs. Activation energy is an intrinsic property of the reaction, and its magnitude can dramatically affect the reaction rate.
  • High \(E_a\): Reactions are slower at the same temperature as only a few collisions have enough energy.
  • Low \(E_a\): Reactions proceed faster due to more collisions exceeding this energy.
Temperature notably impacts this, as a slight temperature increase causes a significant rise in reactant energy, increasing the number of collisions that successfully surpass \(E_a\).
Factors Affecting Reaction Rate
Multiple factors influence how fast reactions occur. Here, we focus on the varying elements that can either slow down or speed up a reaction:
  • Concentration of Reactants: Higher concentrations lead to more frequent collisions, potentially increasing reaction speed.
  • Temperature: As mentioned, an increase in temperature typically accelerates the reaction rate by raising kinetic energy.
  • Presence of Catalysts: Catalysts reduce \(E_a\) without being consumed, effectively speeding up the reaction.
  • Surface Area: Finer division of solid reactants increases exposure, facilitating more collisions.
While temperature is a notable factor, these other components also play significant roles in shaping the pace of a reaction. A balanced combination of these factors can lead to desired reaction rates in practical applications.
Temperature Dependence on Reaction Rate
Temperature profoundly affects how quickly reactions happen. The general "rule of thumb" suggests reaction rates roughly double with every 10掳C rise. This rule is a solid starting point but not universally exact.
The underlying concept can be understood using the Arrhenius equation. As temperature increases:
  • Reactant molecules gain kinetic energy
  • There are more collisions and more have adequate energy to overcome activation energy \(E_a\)
Consequently, the reaction progresses faster. However, this rule certainly varies across different reactions due to distinct \(E_a\) and reaction mechanisms. This variance highlights the importance of context-specific understanding of reaction dynamics when controlling or predicting outcomes in chemistry.

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Most popular questions from this chapter

What is one benefit of understanding a reaction's mechanism?

The rate law for the reaction \(2 \mathrm{NO}+\mathrm{Br}_{2} \rightarrow 2 \mathrm{NOBr}\) is Rate \(=k[\mathrm{NO}]^{2}\left[\mathrm{Br}_{2}\right] .\) How will the rate change when: (a) [NO] is doubled? (b) \(\left[\mathrm{Br}_{2}\right]\) is tripled? (c) [NO] is tripled? (d) \(\left[\mathrm{Br}_{2}\right]\) is quadrupled? (e) [NO] is doubled and \(\left[\mathrm{Br}_{2}\right]\) is tripled?

Given the general form of the rate law, Rate \(=k[\operatorname{Reactant} 1]^{x}[\text { Reactant } 2]^{y}\) answer the following questions: (a) Which part of the rate law reflects the inherent factors of the reaction? (b) What is the general name for the exponents \(x\) and \(y ?\) (c) How do we calculate the overall order of a chemical reaction? (d) Suppose the reaction is second-order with respect to reactant 1 and first- order with respect to reactant \(2 .\) What are the values of \(x\) and \(y\), and what is the overall order of a reaction with only these two reactants? (e) Suppose reactant 1 does not appear in the rate law. What does this say about the value of its order? What is the meaning of the value of its order?

Which of the following are substitution reactions: (a) \(\mathrm{CH}_{3} \mathrm{Br}+\mathrm{I}^{-} \rightarrow \mathrm{CH}_{3} \mathrm{I}+\mathrm{Br}^{-}\) (b) \(\mathrm{H}_{2}+\mathrm{Br}_{2} \rightarrow 2 \mathrm{HBr}\) (c) \(\mathrm{CH}_{2}=\mathrm{CH}_{2}+\mathrm{H}_{2} \rightarrow \mathrm{CH}_{3}-\mathrm{CH}_{3}\) (d) \(\mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{Cl}+\mathrm{OH}^{-} \rightarrow \mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{OH}+\mathrm{Cl}^{-}\)

Explain why the value of \(k\) gets larger as the temperature of a reaction mixture is increased.
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