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Chapter 12: Problem 4

How does temperature affect \(k\), the rate constant? Explain.

Short Answer

Expert verified
The rate constant (k) and temperature (T) in a chemical reaction are related through the Arrhenius equation: \(k = Ae^{-Ea/RT}\), where A is the pre-exponential factor, Ea is the activation energy, R is the universal gas constant, and T is the temperature in Kelvin. As temperature increases, the rate constant (k) generally increases, signifying an increased reaction rate. This is because a higher temperature allows a larger fraction of molecules to have enough energy to overcome the activation energy barrier. Moreover, reactions with higher activation energies are more sensitive to changes in temperature, as even a small change in temperature can significantly impact the rate constant (k).

Step by step solution

01

Understand the Rate Constant

The rate constant (k) is a proportionality constant that appears in the rate law of a chemical reaction. It reflects the likelihood of a successful collision between reactant molecules that leads to the formation of products. The magnitude of the rate constant is influenced by various factors such as temperature, concentration, and the activation energy of the reaction.
02

Introduce the Arrhenius Equation

The Arrhenius equation is a mathematical formula that describes the relationship between the rate constant (k) and temperature (T) in a chemical reaction. The equation is: \[k = Ae^{-Ea/RT}\] where: - k is the rate constant - A is the pre-exponential factor or the frequency factor, which reflects the likelihood of a successful molecular collision - Ea is the activation energy required for the reaction to occur (in joules per mole, J/mol) - R is the universal gas constant (8.314 J/mol·K) - T is the temperature in Kelvin
03

Explain the Temperature Dependence

As temperature increases, the fraction of molecules with enough energy to overcome the activation energy barrier also increases. According to the Arrhenius equation, a higher temperature results in a larger value of the rate constant, k. This means that the reaction rate generally increases as the temperature rises. Moreover, the exponential term in the Arrhenius equation, \(e^{-Ea/RT}\), indicates that even a small change in temperature can significantly impact the rate constant.
04

Refer to the Activation Energy

Activation energy (Ea) is the minimum amount of energy required for a reaction to occur. A higher activation energy means that a larger fraction of the reactant molecules need to gain energy to successfully react. The higher the activation energy, the more sensitive the rate constant is to changes in temperature. In summary, the relationship between the rate constant (k) and temperature (T) can be described by the Arrhenius equation. As temperature increases, so does the rate constant, leading to an increased reaction rate. The effect of temperature on the rate constant is more pronounced for reactions with higher activation energies.

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Most popular questions from this chapter

Consider the following initial rate data for the decomposition of compound \(\mathrm{AB}\) to give \(\mathrm{A}\) and \(\mathrm{B}\) : Determine the half-life for the decomposition reaction initially having \(1.00 M \mathrm{AB}\) present.

Would the slope of \(a \ln (k)\) versus \(1 / T\) plot (with temperature in kelvin) for a catalyzed reaction be more or less negative than the slope of the \(\ln (k)\) versus \(1 / T\) plot for the uncatalyzed reaction? Explain. Assume both rate laws are first-order overall.

The rate law for the reaction $$ 2 \mathrm{NOBr}(g) \longrightarrow 2 \mathrm{NO}(g)+\mathrm{Br}_{2}(g) $$ at some temperature is $$ \text { Rate }=-\frac{\Delta[\mathrm{NOBr}]}{\Delta t}=k[\mathrm{NOBr}]^{2} $$ a. If the half-life for this reaction is \(2.00 \mathrm{~s}\) when \([\mathrm{NOBr}]_{0}=\) \(0.900 M\), calculate the value of \(k\) for this reaction. b. How much time is required for the concentration of \(\mathrm{NOBr}\) to decrease to \(0.100 \mathrm{M}\) ?

A certain first-order reaction is \(45.0 \%\) complete in \(65 \mathrm{~s}\). What are the values of the rate constant and the half-life for this process?

A certain reaction has the following general form: $$ \mathrm{aA} \longrightarrow \mathrm{bB} $$ At a particular temperature and \([\mathrm{A}]_{0}=2.80 \times 10^{-3} M\), concentration versus time data were collected for this reaction, and a plot of \(1 /[\mathrm{A}]\) versus time resulted in a straight line with a slope value of \(+3.60 \times 10^{-2} \mathrm{~L} / \mathrm{mol} \cdot \mathrm{s}\) a. Determine the rate law, the integrated rate law, and the value of the rate constant for this reaction. b. Calculate the half-life for this reaction. c. How much time is required for the concentration of \(\mathrm{A}\) to decrease to \(7.00 \times 10^{-4} M ?\)
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