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Problem $-1($Desulfurization unit of a $NG$ reforming process $-$ mole balance, extent of the reaction at equilibrium and equilibrium conversion calculation$)$

Consider the following gas$-$phase reaction between methyl mercaptan $(C{H}_3-S-H)$ and hydrogen $($in a desulfurization unit of a $NG$ reforming process$)$ to form methane and hydrogen sulfide $(H_2(S)).$ takes place in a fixed bed reactor.

$($desulfurization reaction$)$

$C{H}_3-S-H+H_{2}harrC{H}_4+H_{2}S$

$($A$)$

$($B$)($C$)$

At equilibrium, the mole fractions of the four reactive species are related by the reaction equilibrium constant $($assumed value$).$

$K_y=\frac{y_C{y}_D}{y_A{y}_B}=4\mathrm{.}87$

$($i$)$ Draw and label a flow diagram of the desulfurization unit $($only$)$

$($ii$)$ Suppose the feed to the reactor consists of $n_{A0}$;$n_{B0}$;$n_{C0}$ and $n_{D0},$ moles of $A,B,C,$ and $D,$ respectively. Write expressions for the moles of each species in the final product, $n_A,n_B,n_C,$ and $n_D$ in terms of incoming moles and the extent $()$ be of the reaction.

$($iii$)$ Then use these expressions and the given equilibrium constant to derive an equation for ${}_e,$ the equilibrium extent of reaction, in terms of $n_{A0}$;$n_{B0}$;$n_{C0}$ and $n_{D0}.$

$($iv$)$ If the feed to the reactor contains equimolar quantities of mercaptan $(C{H}_3-S-H)$ and hydrogen, calculate the equilibrium fractional conversion.

$($v$)($iv$)$ It is desired to produce $70$mol of methane starting with $75$mol of mercaptan $(C{H}_3-S-H).$ If the reaction proceeds to equilibrium, how much hydrogen must be fed? What is the composition of the final product?