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Question: Consider a first order gas phase decomposition reaction given below: \[A_{(g)}\overset{\quad\quad}{...

Consider a first order gas phase decomposition reaction given below:

A(g)→B(g)+C(g)A_{(g)}\overset{\quad\quad}{\rightarrow}B_{(g)} + C_{(g)}

The initial pressure of the system before decomposition of A was pip_{i}. After lapse of time ‘t’ total pressure of the system increased by x units and became ′pt′'p_{t}'.

The rate constant k for the reaction is given as______

A

k=2.303tlog⁡pipi−xk = \frac{2.303}{t}\log\frac{p_{i}}{p_{i} - x}

B

k=2.303tlog⁡pi2pi−ptk = \frac{2.303}{t}\log\frac{p_{i}}{2p_{i} - p_{t}}

C

k=2.303tlog⁡pi2pi+ptk = \frac{2.303}{t}\log\frac{p_{i}}{2p_{i} + p_{t}}

D

k=2.303tlog⁡pipi+xk = \frac{2.303}{t}\log\frac{p_{i}}{p_{i} + x}

Answer

k=2.303tlog⁡pi2pi−ptk = \frac{2.303}{t}\log\frac{p_{i}}{2p_{i} - p_{t}}

Explanation

Solution

If x atm be the decreases in pressure of A at time t and one mole each of B and C is being formed, the increase in pressure of B and C will also be x atm each.

A(g)A_{(g)} →\rightarrow B(g)B_{(g)} ++ C(g)C_{(g)}

At t = 0 piatmp_{i}atm 0 atm 0 atm

At time t (pi−x)atm(p_{i} - x)atm x atm x atm

Where pip_{i} is the initial pressure at time t=0

pt=(pi−x)+x+x=pi+xp_{t} = (p_{i} - x) + x + x = p_{i} + x

x=(pt−pi)x = (p_{t} - p_{i})

Where, PA=pi−x=pi−(pt−pi)=2pi−ptP_{A} = p_{i} - x = p_{i} - (p_{t} - p_{i}) = 2p_{i} - p_{t}

k=(2.303t)(log⁡pipA)=2.303tlog⁡pi(2pi−pt)k = \left( \frac{2.303}{t} \right)\left( \log\frac{p_{i}}{p_{A}} \right) = \frac{2.303}{t}\log\frac{p_{i}}{(2p_{i} - p_{t})}