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Chapter 13: Problem 132

A gas mixture containing \(\mathrm{CH}_{3}\) fragments, \(\mathrm{C}_{2} \mathrm{H}_{6}\), molecules, and an inert gas (He) was prepared at \(600 \mathrm{~K}\) with a total pressure of 5.42 atm. The elementary reaction $$ \mathrm{CH}_{3}+\mathrm{C}_{2} \mathrm{H}_{6} \longrightarrow \mathrm{CH}_{4}+\mathrm{C}_{2} \mathrm{H}_{5} $$ has a second-order rate constant of \(3.0 \times 10^{4} / M \cdot\) s. Given that the mole fractions of \(\mathrm{CH}_{3}\) and \(\mathrm{C}_{2} \mathrm{H}_{6}\) are 0.00093 and \(0.00077,\) respectively, calculate the initial rate of the reaction at this temperature.

Short Answer

Expert verified
After calculating the initial concentrations of CH3 and C2H6 from their partial pressures and then using the rate law for a second order reaction, one can find the initial rate of the reaction.

Step by step solution

01

Calculate the initial concentration of CH3 and C2H6

The initial concentration of a component in the gas phase can be calculated using the ideal gas law. We know that P = nRT/V, where P is the pressure, n is the number of moles, R is the ideal gas constant, and T is the temperature. Therefore, the number of moles, n, is given by n = PV/RT. Since the volumes for all gasses in the mixture are the same, we can ignore the volume and directly calculate the mole amount from the pressures of the components. In this case, the total pressure is 5.42 atm, and the mole fractions of CH3 and C2H6 are 0.00093 and 0.00077, respectively. Therefore, the partial pressures of CH3 and C2H6 are 0.00093*5.42 atm and 0.00077*5.42 atm, respectively.
02

Use the rate law for a second order reaction

In this case, the rate of reaction is given by r = k*[CH3]*[C2H6], where k is the rate constant, and * indicates multiplication. We substitute the values from Step 1 into this equation to get the initial rate of reaction.
03

Convert units if necessary

If the units of the rate constant, k, and the pressure (which is proportional to concentration for gases), are not consistent, it may be necessary to convert units. Here, the pressure is given in atm, while the rate constant is given in /M*s, so we need to convert atm to M (moles/liter).

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Most popular questions from this chapter

The following expression shows the dependence of the half-life of a reaction \(\left(t_{\frac{1}{2}}\right)\) on the initial reactant concentration [A] \(_{0}:\) $$ t_{\frac{1}{2}} \propto \frac{1}{[\mathrm{~A}]_{0}^{n-1}} $$ where \(n\) is the order of the reaction. Verify this dependence for zero-, first-, and second-order reactions.

An instructor performed a lecture demonstration of the thermite reaction (see Example 6.10 ). He mixed aluminum with iron(III) oxide in a metal bucket placed on a block of ice. After the extremely exothermic reaction started, there was an enormous bang, which was not characteristic of thermite reactions. Give a plausible chemical explanation for the unexpected sound effect. The bucket was open to air.

The rate constants for the first-order decomposition of an organic compound in solution are measured at several temperatures: $$ \begin{array}{cccccc} k\left(\mathrm{~s}^{-1}\right) & 0.00492 & 0.0216 & 0.0950 & 0.326 & 1.15 \\ T(\mathrm{~K}) & 278 & 288 & 298 & 308 & 318 \end{array} $$ Determine graphically the activation energy and frequency factor for the reaction.

Determine the molecularity and write the rate law for each of the following elementary steps: (a) \(\mathrm{X} \longrightarrow\) products (b) \(\mathrm{X}+\mathrm{Y} \longrightarrow\) products (c) \(\mathrm{X}+\mathrm{Y}+\mathrm{Z} \longrightarrow\) products (d) \(\mathrm{X}+\mathrm{X} \longrightarrow\) products (e) \(\mathrm{X}+2 \mathrm{Y} \longrightarrow\) products

What are the units for the rate constants of zeroorder, first-order, and second-order reactions?
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