A gas phase reaction may be represented as:

$A2P$

The reaction is known to be second order with respect to component A and is to be carried out isothermally in a tubular plug flow reactor $($PFR$)$ at a constant pressure of $200$kPa and temperature of $450K.$ Component A is fed to the PFR at $250mol{s}^{-1},$ but molecules of A are first mixed with an equal number of moles of an inert gas. The conditions in the PFR are such that the pressure in the reactor is essentially constant, and the fractional conversion achieved is $0\mathrm{.}82.$

Noting that for a PFR

$={\displaystyle \underset{=0}{\overset{-}{}}}\frac{1}{}d$

where $$ is the residence time $($s$)$ for the reactor, $$ is the fractional conversion and $$ is the rate of reaction $)$

$($a$)$ Show that the volume $V(m^3)$ of the reactor required for the reaction may be written as

$V=(\frac{RT}{P}{)}^2\frac{F}{k}{\displaystyle \underset{=0}{\overset{-}{}}}\frac{(2+{)}^2}{(1-{)}^2}d$

where:

$R$ is the gas constant

$T$ is the absolute temperature $(K)$

$P$ is the pressure $(Pa)$

$F$ is the molar flow rate of reactant $(mol{s}^{-1})$

$k$ is the reaction rate constant $(m^{3}mo{l}^{-1}{s}^{-1})$

$($b$)$ Given that the rate constant for this reaction at the stated conditions is $15\mathrm{.}325m^{3}mo{l}^{-1}{s}^{-1},$ calculate the volume of the reactor.

$($c$)$ If the quantity of the inert gas was increased, how would this affect the performance of the reactor.

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