The composition of vapour over a binary ideal solution is determined by the composition of the liquid. If `x_(A)` and `y_(A)` are the mole fractions of A in the liquid and vapour, respectively find the value of `x_(A)` for which `(y_(A)-x_(A))` has maximum. What is the value of the pressure at this composition?

Explanation

Since
`y_(A) = (x_(A)P_(A)^(@))/(P_(B)^(@)+(P_(A)^(@) -P_(B)^(@))x_(A))`
Substracting `x_(A)` from both the sides, we get
`y_(A) = x_(A) = (x_(A)P_(A)^(@))/(P_(B)^(@)+(P_(A)^(@)-P_(B)^(@))x_(A)) -x_(A)`
Differentiating this with respect to `x_(A)`, we get
`(d(y_(A)-x_(A)))/(dx_(A)) =(P_(A)^(@))/(P_(A)^(@)+(P_(A)^(@)-P_(B)^(@))x_(A))- (x_(A)P_(A)^(@)(P_(A)^(@)-P_(B)^(@)))/({P_(B)^(@)+(P_(A)^(@)-P_(B)^(@))x_(A)}^(2)) -1`
The value of `x_(A)` at which `y_(A) -x_(A)` has a maximum value can be obtained by setting the above differential equal to zero. Thus, we have
`(P_(A)^(@))/(P_(A)^(@)+(P_(A)^(@)-P_(B)^(@))x_(A)) -(x_(A)P_(A)^(@)(P_(A)^(@)-P_(B)^(@)))/({P_(B)^(@)+(P_(A)^(@)-P_(B)^(@))x_(A)}^(2)) -1 =0`
Solving for `x_(A)`, we get `x_(A) = (sqrt(P_(A)^(@)P_(B)^(@))-P_(B)^(@))/(P_(A)^(@)-P_(B)^(@))`
The value of `P` at this composition is
`P = x_(A) P_(A)^(@) +x_(B) P_(B)^(@)`
or `P = P_(B)^(@) +(P_(A)^(@) -P_(B)^(@)) x_(A)`
or `P = P_(B)^(@) +(P_(A)^(@) -P_(B)^(@)) ((sqrt(P_(A)^(@)P_(B)^(@))-P_(B)^(@))/(P_(A)^(@)-P_(B)^(@)))`
or `P = sqrt(P_(A)^(@)P_(B)^(@))`