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Chapter 14: Problem 51

(a) What factors determine whether a collision between two molecules will lead to a chemical reaction? (b) Does the rate constant for a reaction generally increase or decrease with an increase in reaction temperature? (c) Which factor is most sensitive to changes in temperature-the frequency of collisions, the orientation factor, or the fraction of molecules with energy greater than the activation energy?

Short Answer

Expert verified
(a) For a collision between two molecules to lead to a chemical reaction, they must possess enough energy (activation energy) and be properly oriented during the collision. (b) The rate constant generally increases with an increase in reaction temperature, as described by the Arrhenius equation. (c) The fraction of molecules with energy greater than the activation energy is most sensitive to changes in temperature, as it directly impacts the probability of effective collisions and reaction rate.

Step by step solution

01

Answer (a): Factors determining a chemical reaction during molecular collision

For a collision between two molecules to lead to a chemical reaction, the molecules must satisfy certain conditions: 1. They should possess enough energy during the collision. This minimum energy required for the reaction is called activation energy, represented by \(E_a\). The activation energy is needed to break old bonds and form new bonds. 2. The molecules should be properly oriented. During the collision, the molecules must be aligned properly so that the atoms involved in the bond-breaking and bond-forming process can interact effectively. In summary, a collision between two molecules will lead to a chemical reaction if they both have sufficient energy (greater than or equal to the activation energy) and are properly oriented during the collision.
02

Answer (b): Effect of temperature on the rate constant

The rate constant generally increases with an increase in reaction temperature. According to the Arrhenius equation, the rate constant (\(k\)) of a reaction is dependent on the temperature (\(T\)): \(k = A \times e^{-\frac{E_a}{RT}}\) where: - \(A\) is the pre-exponential factor - \(E_a\) is the activation energy - \(R\) is the gas constant - \(T\) is the temperature (in Kelvin) As the temperature increases, the exponential term, \(e^{-\frac{E_a}{RT}}\), becomes larger, which in turn increases the value of the rate constant \(k\). This increase in the rate constant leads to a faster reaction rate.
03

Answer (c): Temperature sensitivity of collision frequency, orientation factor, and fraction of molecules with energy greater than activation energy

Out of the factors mentioned, the fraction of molecules with energy greater than the activation energy is most sensitive to changes in temperature. 1. Collision frequency is influenced by the concentration of reactants and the temperature. However, it is relatively less sensitive to temperature changes compared to the activation energy factor. 2. The orientation factor may be influenced by temperature, but its effect is less considerable than the effects of the other factors. The fraction of molecules with energy greater than the activation energy is determined by the Boltzmann distribution. As temperature increases, a larger fraction of molecules will have energy greater than or equal to the activation energy, which leads to a higher probability of effective collisions and a faster reaction rate. This sensitivity to temperature changes explains why the activation energy factor is the most significant factor among the three.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Activation Energy
Activation energy is a key concept in chemical reaction kinetics. It's the minimum energy that molecules need when they collide to undergo a chemical reaction. Think of it as an energy barrier that molecules must overcome for a reaction to occur. Without enough energy to surpass this barrier, the molecules will simply bounce off each other without reacting.

Here are some important points to keep in mind about activation energy:
  • The higher the activation energy, the slower the reaction, because fewer molecules will have enough energy to get over the barrier.
  • The activation energy is often represented as \(E_a\) in equations.
  • Catalysts can lower the activation energy, allowing more molecules to participate in the reaction even at lower energies.
Understanding activation energy helps us predict how conditions like temperature or the presence of a catalyst can affect the reaction rate.
Arrhenius Equation
The Arrhenius equation provides a quantitative basis for understanding how temperature affects reaction rates. This equation relates the rate constant (\(k\)) of a reaction to the temperature (\(T\)), giving us insight into the kinetics of chemical reactions.

The equation is mathematically expressed as:
\[k = A \times e^{-\frac{E_a}{RT}}\]
In this formula:
  • \(k\) stands for the rate constant, which changes with temperature.
  • \(A\) is the pre-exponential factor, reflecting the frequency of collisions with the correct orientation.
  • \(E_a\) stands for activation energy.
  • \(R\) is the gas constant (8.314 J/mol路K).
  • \(T\) is the temperature in Kelvin.
The Arrhenius equation shows that as the temperature increases, the exponential factor becomes less negative, leading to a larger rate constant. This means reactions tend to go faster at higher temperatures.
Temperature Effect on Reaction Rate
Temperature plays a critical role in influencing the rate of chemical reactions. According to the Arrhenius equation, as temperature increases, so does the rate of most chemical reactions. But why does this happen?

Here's a closer look at how temperature affects reaction rates:
  • Higher temperatures provide more energy to molecules, increasing their speed and the frequency of collisions.
  • More energy means more molecules have the necessary activation energy to react, as depicted by the Boltzmann distribution.
  • The increase in collisions and in the number of molecules with sufficient energy results in a faster reaction rate.
However, not all components of the reaction are equally sensitive to temperature changes. The factor most impacted by temperature changes is the fraction of molecules with energy greater than the activation energy, as their numbers increase significantly with temperature rise, leading directly to faster reactions. Understanding this temperature dependence allows chemists to control the speed of reactions by adjusting the temperature accordingly.

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Most popular questions from this chapter

The isomerization of methyl isonitrile \(\left(\mathrm{CH}_{3} \mathrm{NC}\right)\) to acetonitrile \(\left(\mathrm{CH}_{3} \mathrm{CN}\right)\) was studied in the gas phase at \(215^{\circ} \mathrm{C}\), and the following data were obtained: $$ \begin{array}{rc} \hline \text { Time (s) } & {\left[\mathrm{CH}_{3} \mathrm{NC}\right](M)} \\ \hline 0 & 0.0165 \\ 2000 & 0.0110 \\ 5000 & 0.00591 \\ 8000 & 0.00314 \\ 12,000 & 0.00137 \\ 15,000 & 0.00074 \\ \hline \end{array} $$ (a) Calculate the average rate of reaction, in \(M / s\), for the time interval between each measurement. (b) Calculate the average rate of reaction over the entire time of the data from \(t=0\) to \(t=15,000 \mathrm{~s} .(\mathbf{c})\) Which is greater, the average rate between \(t=2000\) and \(t=12,000 \mathrm{~s}\), or between \(t=8000\) and \(t=15,000 \mathrm{~s} ?(\mathbf{d})\) Graph \(\left[\mathrm{CH}_{3} \mathrm{NC}\right]\) versus time and determine the instantaneous rates in \(M / \mathrm{s}\) at \(t=5000 \mathrm{~s}\) and \(t=8000 \mathrm{~s}\).

For each of the following gas-phase reactions, write the rate expression in terms of the appearance of each product and disappearance of each reactant: (a) \(\mathrm{O}_{3}(g)+\mathrm{H}_{2} \mathrm{O}(g) \longrightarrow 2 \mathrm{O}_{2}(g)+\mathrm{H}_{2}(g)\) (b) \(4 \mathrm{NH}_{3}(g)+5 \mathrm{O}_{2}(g) \longrightarrow 4 \mathrm{NO}(g)+6 \mathrm{H}_{2} \mathrm{O}(g)\) (c) \(2 \mathrm{C}_{2} \mathrm{H}_{2}(g)+5 \mathrm{O}_{2}(g) \longrightarrow 4 \mathrm{CO}_{2}(g)+2 \mathrm{H}_{2} \mathrm{O}(g)\) (d) \(\mathrm{C}_{3} \mathrm{H}_{7} \mathrm{NH}_{2}(g) \longrightarrow \mathrm{C}_{3} \mathrm{H}_{6}(g)+\mathrm{NH}_{3}(g)\)

As described in Exercise 14.41 , the decomposition of sulfuryl chloride \(\left(\mathrm{SO}_{2} \mathrm{Cl}_{2}\right)\) is a first-order process. The rate constant for the decomposition at \(660 \mathrm{~K}\) is \(4.5 \times 10^{-2} \mathrm{~s}^{-1}\). (a) If we begin with an initial \(\mathrm{SO}_{2} \mathrm{Cl}_{2}\) pressure of \(60 \mathrm{kPa}\), what is the partial pressure of this substance after 60 s? (b) At what time will the partial pressure of \(\mathrm{SO}_{2} \mathrm{Cl}_{2}\) decline to one-tenth its initial value?

Consider a hypothetical reaction between \(\mathrm{A}, \mathrm{B},\) and \(\mathrm{C}\) that is zero order in A, second order in B, and first order in C. (a) Write the rate law for the reaction. (b) How does the rate change when [A] is tripled and the other reactant concentrations are held constant? (c) How does the rate change when [B] is doubled and the other reactant concentrations are held constant? (d) How does the rate change when [C] is tripled and the other reactant concentrations are held constant? (e) By what factor does the rate change when the concentrations of all three reactants are doubled? (f) By what factor does the rate change when the concentrations of all three reactants are cut in half?

Urea \(\left(\mathrm{NH}_{2} \mathrm{CONH}_{2}\right)\) is the end product in protein metabolism in animals. The decomposition of urea in \(0.1 \mathrm{M} \mathrm{HCl}\) occurs according to the reaction $$ \mathrm{NH}_{2} \mathrm{CONH}_{2}(a q)+\mathrm{H}^{+}(a q)+2 \mathrm{H}_{2} \mathrm{O}(l) \longrightarrow 2 \mathrm{NH}_{4}^{+}(a q)+\mathrm{HCO}_{3}^{-}(a q) $$ The reaction is first order in urea and first order overall. When \(\left[\mathrm{NH}_{2} \mathrm{CONH}_{2}\right]=0.200 \mathrm{M},\) the rate at \(61.05^{\circ} \mathrm{C}\) $$ \text { is } 8.56 \times 10^{-5} \mathrm{M} / \mathrm{s} $$ (a) What is the rate constant, \(k\) ? (b) What is the concentration of urea in this solution after \(4.00 \times 10^{3} \mathrm{~s}\) if the starting concentration is \(0.500 \mathrm{M}\) ? (c) What is the half-life for this reaction at \(61.05^{\circ} \mathrm{C}\) ?
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