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Prove alpha=sqrt((K(p)/(P+K(p)))) for ...

Prove `alpha=sqrt((K_(p)/(P+K_(p))))` for
`PCl_(5) hArr PCl_(3)+Cl_(2)`
where `alpha` is the degree of dissociation at temperature when equilibrium constant is `K_(p)`.

Text Solution

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`{:(,PCl_(5),hArr,PCl_(3),+,Cl_(2)),("Initial moles",1,,0,,0),("moles at equilibrium",1-alpha,,alpha,,alpha):}`
where `alpha` is the degree of dissociation. Total moles `=1+alpha`
`:. P'_(Cl_(2))=[alpha/(1+alpha)]xxP, P'_(Cl_(3))=[alpha/(1+alpha)]xxP`
`P'_(Cl_(5))=[(1-alpha)/(1+alpha)]P`
`:. K_(p)=(P'_(Cl_(3))xxP'_(Cl_(2)))/(P'_(Cl_(5)))=([(alphaxxP)/(1+alpha)][(alphaxxP)/(1+alpha)])/([(1-alpha)/(1+alpha)]P)`
`K_(P)=(alpha^(2)P)/(1-alpha^(2)) :. alpha=sqrt((K_(p)/(P+K_(p))))`
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