Question

Derive the relation between Kp and Kc.

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Class ⇒ 11.

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Answer 1

hope it help more in future so

Answer 2

${\pink{\maltese}}\:\:{\underline{\underline{\textbf{\textsf{\red{To\:Derivate\:}}}}}}:$

${\quad{\hookrightarrow \: \: {\underline{\rm{We \: have \: to \: derive \: the \: relation \: between \:{\red{\bf{K_p}}}\rm\: and \: {\red{\bf{K_c}.}}}}}}}$

${\pink{\maltese}}\:\:{\underline{\underline{\textbf{\textsf{\red{Required\:Solution\:}}}}}}:$

${\pink{\clubsuit }\: \: {\underline{\sf{Meaning \: of \:{\red{\bf{K_p}}}\sf \:and \: {\red{\bf{K_c}}}}}}} :$

$\hookrightarrow \: \: \rm{They \: are \: equilibrium \: constant \: and \: dependent \: on \qquad}\\ \qquad \rm{partial \: pressures \: exerted \: by \: the \: gaseous \: components.}$

${\pink{\clubsuit }\: \: {\underline{\sf{Definition \: of \:{\red{\bf{K_p}}}\sf \:and \: {\red{\bf{K_c}}}}}}} :$

$\hookrightarrow \: \: {\red{\bf{K_p}}} \: \to \: \rm{It \:is \:defined \: by \:{\textbf{\textsf{molar\: concentrations.}}}}$

$\hookrightarrow \: \: {\red{\bf{K_c}}} \: \to \: \rm{It \:is \:defined \: by \:the\:{\textbf{\textsf{partial\: pressures}}} \: of\:the\:gases\:inside.}$

${\pink{\clubsuit }\: \: {\underline{\sf{For\:a\:general\:reaction\:}}}} :$

$\hookrightarrow \: \: {\red{\sf{aA \: + \: bB \: \leftrightharpoons \: cC \: + \: dD}}}$

${\pink{\clubsuit }\: \: {\underline{\sf{Representation \: of \:{\red{\bf{K_p}}}}}}} :$

${\hookrightarrow \: \: {\red{\bf{K_p}}} \: \to \: \rm{ \dfrac{[P {}^{c}_C] [P {}^{d}_D]}{[P {}^{a}_A][P {}^{b}_B]}} ---({\boldsymbol{Equation\:1)}}}.$

${\pink{\clubsuit }\: \: {\underline{\sf{Representation \: of \:{\red{\bf{K_c}}}}}}} :$

${\hookrightarrow \: \: {\red{\bf{K_c}}} \: \to \: \rm{ \dfrac{[C]^c[D]^d}{[A]^a[B]^b}} ---({\boldsymbol{Equation\:2)}}}.$

${\pink{\clubsuit }\: \: {\underline{\sf{Ideal\:Gases\:Equation\:}}}} :$

${\hookrightarrow \: \: {\sf{pV = nRT}}}.$

${\hookrightarrow \: \: {\sf{p = \dfrac{n}{V}RT}}}.$

${\hookrightarrow \: \: { \red{\sf{P = CRT}}}. \: \: \lbrack \rm\because \: Concentration \:is \: in \: { \bf{mol \: L^{-1} }}\: or \:{ \bf{ mol \: dm^{-3}}}\rbrack}$

${\pink{\clubsuit }\: \: {\underline{\sf{Partial\: pressure\:of\:the\:{\red{\bf{reacants}}}\:and\:{\red{\bf{products}}}}}}} :$

${\hookrightarrow \: \: {\sf{P {}^{a}_A = [A]^a \cdot[RT]^a}}}$

${\hookrightarrow \: \: {\sf{P {}^{b}_B = [B]^b \cdot[RT]^b}}}$

${\hookrightarrow \: \: {\sf{P {}^{c}_C = [C]^c \cdot[RT]^c}}}$

${\hookrightarrow \: \: {\sf{P {}^{d}_D = [D]^d \cdot[RT]^d}}}$

${\pink{\clubsuit }\: \: {\underline{\sf{\bf{\red{Calculations}}}}}} :$

${\hookrightarrow \: \: {\red{\bf{K_p}}} \: = \: \sf{ \dfrac{[P {}^{c}_C] [P {}^{d}_D]}{[P {}^{a}_A][P {}^{b}_B]}} }.$

${\bullet \: \: \rm {Substituting\:the\:Partial\: pressure\:of\:the\:{\red{\bf{reacants}}}\:and\:{\red{\bf{products}}}}:}$

${\hookrightarrow \: \: {\red{\bf{K_p}}} \: = \: \sf{ \dfrac{ [C]^c \cdot[RT]^c[D]^d \cdot[RT]^d}{[A]^a \cdot[RT]^a[B]^b \cdot[RT]^b}} }.$

${\hookrightarrow \: \: {\red{\bf{K_p}}} \: = \: \sf{ \bigg(\dfrac{[C]^c[D]^d}{[A]^a[B]^b} \bigg)(RT)^{(c + d) - (a + b)} } }.$

${\bullet \: \: \rm {Here, \: \bf \Delta n \: \rm is \: the \:\bf difference \rm\: between \: {\sf the \: sum \: of \: number \: of \: moles \: of \: products } \: and \:{ \sf the \: sum \: of \: number \: of \:moles \: of \: reactants. }}}$

${ \hookrightarrow \: \: \bf\Delta n = \sf(the \: sum \: of \: number \: of \: moles \: of \: products) - (the \: sum \: of \: number \: of \:moles \: of \: reactants).}$

${ \hookrightarrow \: \:{\bf{\red{\Delta n = \sf(c+d) - (a+b).}}}}$

${\bullet \: \: \rm {\therefore, \: Substituting\:\bf \Delta n \: \rm and \: \bf Equation \:2:}}$

${\hookrightarrow \: \: {\bf{\purple{K_p \: = \: \sf{K_c(RT)^{\Delta n_{(g)}}}}}}} \: \: \bigstar.$

${\pink{\clubsuit }\: \: {\underline{\sf{\bf{\red{Here}}}}}} :$

${ \hookrightarrow \: \:{\bf{P \to \sf Partial \: Pressure \: \rm(in \: bar).}}}$

${ \hookrightarrow \: \:{\bf{N \to \sf Amount \: of \: gas \: \rm (in mol).}}}$

${ \hookrightarrow \: \:{\bf{V \to \sf Volume \: \rm (in \: c^3).}}}$

${ \hookrightarrow \: \:{\bf{T \to \sf Kelvin \: Temperature.}}}$

${\hookrightarrow \: \: \bf{R \to \sf Gas \: Constant \: \rm(0.831 \: litre \: bar \: K^{-1} \: mol^{-1}).}}$

${\hookrightarrow \: \: \bf{C \to \sf Concentration \: \rm (in \: mol \: L^{-1} \: or \: dm^{-3}).}}$