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A certain ideal solution of two liquids ...

A certain ideal solution of two liquids `A` and `B` has mole fraction of `0.3` and `0.5` for the vapour phase and liquid phase, respectively. What would be the mole fraction of `B` in the vapour phase, when the mole fraction of `A` in the liquid is `0.25`?

Text Solution

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`chi_(A)("vap") = 0.3, chi_(B)("liq")=0.5,chi_(A)("liq")=0.5`
`p_(A)=chi_(A)p_(A)@ = 0.5 p_(A)@, p_(B) = chi_(B)p_(B)@ =0.5 p_(B)@`
`p_("total") = p_(A) + p_(B) = 0.5 p_(A)@ + 0.5 p_(B)@ = 0.5 (p_(A)@ + p_(B)@)`
`chi_(A)(vap) = (p_(A))/(p_("total"))`
`0.3 = (0.5 p_(A)@)/(0.5 (p_(A)@ + p_(B)@)) = 0.3 = (p_(A)@)/(p_(A)@ + p_(B)@)`
On rewriting it,
`1/0.3 = (p_(A)@ + p_(B)@) / (p_(A)@) = 1+ (p_(B)@)/(p_(A)@)`
`rArr (p_(B)@)/(p_(A)@) = (1)/(0.3)-1 = (0.7)/(0.3) = 7/3`
`:. (p_(A)@)/(p_(B)@) = (0.3)/(0.7) rArr p_(A)@ = 3/7 p_(B)@`
`chi_(A)(liq)= 0.25, chi_(B)(liq) = 0.75`
`p_(A) = 0.25 p_(A)@ = 0.25 xx 3/7 p_(B)@`
`p_(B)=0.75 p_(B)@`
`:. P_("total")=p_(A) + p_(B) = 0.25 xx 3/7 p_(B)@ + 0.75 p_(B)@`
`= 0.75 xx 8/7 p_(B)@`
`chi_(B)(vap) = (p_(B))/(p_("total")) = (0.75p_(B)@)/(0.75xx(8)/(7)p_(B)@)=(7)/(8)`
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