InterviewSolution
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InterviewSolution
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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? |
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Answer» 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)^(@))` |
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