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Chapter 13: Problem 112

The first-order rate constant for the decomposition of dimethyl ether $$ \left(\mathrm{CH}_{3}\right)_{2} \mathrm{O}(g) \longrightarrow \mathrm{CH}_{4}(g)+\mathrm{H}_{2}(g)+\mathrm{CO}(g) $$ is \(3.2 \times 10^{-4} \mathrm{~s}^{-1}\) at \(450^{\circ} \mathrm{C} .\) The reaction is carried out in a constant-volume flask. Initially only dimethyl ether is present and the pressure is 0.350 atm. What is the pressure of the system after 8.0 min? Assume ideal behavior.

Short Answer

Expert verified
The final pressure is the initial pressure minus the calculated pressure decrease. Substitute all known values into the equation to obtain the final pressure at the end of 8 minutes.

Step by step solution

01

Calculate the remaining concentration of dimethyl ether

The expression for first-order reactions is \( k = ln [A]_0/[A]_{t} / t \), where k is the rate constant, [A]_{0} is the initial concentration, [A]_{t} is the concentration at time t and t is the time elapsed. From the problem, we know k= \(3.2 \times 10^{-4} \mathrm{~s}^{-1}\), t = 8 min = 480 s, and \( [A]_{0}\) = 0.350 atm / 0.0821 L·atm/K·mol / (450 + 273) K, as we assume that initially only dimethyl ether is present. \nWe solve for \([A]_t\), yielding \([A]_{t} = [A]_0 \times e^{(-k \times t)}\)
02

Calculate the decrease in concentration of dimethyl ether

The decrease in concentration of dimethyl ether is the initial concentration \([A]_0\) minus the concentration at time t \([A]_{t}\). So, the decrease in concentration = \([A]_0 - [A]_{t}\).
03

Convert the decrease in concentration into a decrease in pressure

We know from the ideal gas law that P = nRT/V, or equivalently n = PV/RT. Therefore, decreases in n (the concentration) result in corresponding decreases in P (pressure). Thus, decrease in pressure is equal to the decrease in concentration times RT/V, which simplifies to the decrease in pressure = (decrease in concentration) x (RT). The value of R is 0.0821 L·atm/K·mol (as per ideal gas law conditions) and T is the temperature in Kelvins. Then, we subtract this pressure decrease from the initial pressure to find the final pressure.

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Most popular questions from this chapter

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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

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