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Chapter 13: Problem 39

Some reactions are described as parallel in that the reactant simultaneously forms different products with different rate constants. An example is and $$ \begin{array}{l} \mathrm{A} \stackrel{k_{1}}{k_{2}} \mathrm{~B} \\ \mathrm{~A} \stackrel{\longrightarrow}{\longrightarrow} \mathrm{C} \end{array} $$ The activation energies are \(45.3 \mathrm{~kJ} / \mathrm{mol}\) for \(k_{1}\) and \(69.8 \mathrm{~kJ} / \mathrm{mol}\) for \(k_{2}\). If the rate constants are equal at \(320 \mathrm{~K},\) at what temperature will \(k_{1} / k_{2}=2.00 ?\)

Short Answer

Expert verified
We first find the relationship between the pre-exponential factors for each reaction, then substitute it into the formula representing the ratio of rate constants, \(k_1 / k_2\). Then we solve the equation to find the required temperature.

Step by step solution

01

Understand The Arrhenius Equation

The Arrhenius equation is: \(k = A \exp{ \left(- \frac{E}{RT}\right)}\) where:- k is the rate constant- E is the activation energy- R is the gas constant- T is the temperatureHere, the pre-exponential factor A can be determined using the condition given, where the rate constants are equal at 320 K.
02

Find the pre-exponential factor for each reaction

From the Arrhenius equation and given that \(k_1 = k_2\) at \(T = 320K\), we can write:\(A_{1} \exp{ \left(- \frac{E_{1}}{R \cdot 320}\right)} = A_{2} \exp{ \left(- \frac{E_{2}}{R \cdot 320}\right)}\)Solving will give us a relationship between \(A_1\) and \(A_2\).
03

Determine the temperature for \(k_1 / k_2=2\)

We know that:\(k_1 / k_2 = A_{1} / A_{2} \cdot \exp{ \left( \frac{E_{2} - E_{1}}{RT} \right)}\)Given that \(k_1 / k_2 = 2\), we can substitute \(A_{1} / A_{2}\) from step 2: \( 2 = \exp{ \left( \frac{E_{2} - E_{1}}{RT} \right)}\)Solving the equation will yield the temperature.

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Most popular questions from this chapter

The rate constant of a first-order reaction is \(4.60 \times\) \(10^{-4} \mathrm{~s}^{-1}\) at \(350^{\circ} \mathrm{C} .\) If the activation energy is \(104 \mathrm{~kJ} /\) mol, calculate the temperature at which its rate constant is \(8.80 \times 10^{-4} \mathrm{~s}^{-1}\)

Is the rate constant \((k)\) of a reaction more sensitive to changes in temperature if \(E_{\mathrm{a}}\) is small or large?

Specify which of the following species cannot be isolated in a reaction: activated complex, product, intermediate.

Consider the first-order reaction $$ \mathrm{CH}_{3} \mathrm{NC}(g) \longrightarrow \mathrm{CH}_{3} \mathrm{CN}(g) $$ Given that the frequency factor and activation energy for the reaction are \(3.98 \times 10^{13} \mathrm{~s}^{-1}\) and \(161 \mathrm{~kJ} / \mathrm{mol}\), respectively, calculate the rate constant at \(600^{\circ} \mathrm{C}\).

To carry out metabolism, oxygen is taken up by hemoglobin (Hb) to form oxyhemoglobin (HbO \(_{2}\) ) according to the simplified equation $$ \mathrm{Hb}(a q)+\mathrm{O}_{2}(a q) \stackrel{k}{\longrightarrow} \mathrm{HbO}_{2}(a q) $$ where the second-order rate constant is \(2.1 \times 10^{6} / M \cdot \mathrm{s}\) at \(37^{\circ} \mathrm{C}\). (The reaction is first order in \(\mathrm{Hb}\) and \(\mathrm{O}_{2}\).) For an average adult, the concentrations of \(\mathrm{Hb}\) and \(\mathrm{O}_{2}\) in the blood at the lungs are \(8.0 \times 10^{-6} \mathrm{M}\) and \(1.5 \times 10^{-6} M,\) respectively. (a) Calculate the rate of formation of \(\mathrm{HbO}_{2}\). (b) Calculate the rate of consumption of \(\mathrm{O}_{2}\). (c) The rate of formation of \(\mathrm{HbO}_{2}\) increases to \(1.4 \times 10^{-4} M / \mathrm{s}\) during exercise to meet the demand of increased metabolism rate. Assuming the \(\mathrm{Hb}\) concentration to remain the same, what must be the oxygen concentration to sustain this rate of \(\mathrm{HbO}_{2}\) formation?
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