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Rate of Reaction

The thermal d...

Question

The thermal decomposition of dimethyl ether as measured by finding the increase in pressure of the reaction $(C H_{3})_{2} O (g) \to C H_{4} (g) + H_{2} (g) + C O (g)$ at $500^{\circ} C$ is as follows:

Time (sec) 390 1195 3155 $\infty$
Pressure increase (mm Hg) 96 250 467 619
The initial pressure of either was $312$ mm Hg. The rate constant of reaction is:

A

0.000428 s e c^{- 1}

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B

0.000544 s e c^{- 1}

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C

0.000567 s e c^{- 1}

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D

0.000586 s e c^{- 1}

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Solution

The correct option is D $0.000428 s e c^{- 1}$
The unit of rate constant indicates it is the first-order reaction.
The initial pressure of ether is $a = 312 m m H g$ .
After 390 s, the pressure of ether is $a - x$ .
Total pressure is $a - x + x + x + x = a + 2 x$ .
$312 + 2 x = 312 + 96$
$x = 48$
Thus, the pressure of ether is $a - x = 312 - 48 = 264$ .
The expression for the rate constant of first order reaction is

$k = \frac{2.303}{t} l o g \frac{a}{a - x} = \frac{2.303}{390} l o g \frac{312}{264} = 0.000428 s e c^{- 1}$ .

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