d
Total pressure, \(\mathrm{P}_{\mathrm{T}}=\mathrm{P'}_{\mathrm{A}}+\mathrm{P'}_{\mathrm{B}}\)

We know that, \(\mathrm{P}_{\mathrm{A}}^{\prime}=\mathrm{P}_{\mathrm{A}} \mathrm{x}_{\mathrm{A}}\)

\(\mathrm{P}_{\mathrm{B}}^{\prime}=\mathrm{P}_{\mathrm{B}} \mathrm{x}_{\mathrm{B}}\)

substituting the values of \(\mathrm{P}_{\mathrm{A}}^{\prime}\) and \(\mathrm{P}_{\mathrm{B}}^{\prime}\) in eq. (i)

\(\mathrm{P}_{\mathrm{T}}=\mathrm{P}_{\mathrm{A}} \mathrm{x}_{\mathrm{A}}+\mathrm{P}_{\mathrm{B}} \mathrm{x}_{\mathrm{B}}\)

\(\left[\mathrm{x}_{\mathrm{A}}+\mathrm{x}_{\mathrm{B}}=1\right.\)

\(\left.x_{A}=1-x_{B} \text { or } x_{B}=1-x_{A}\right]\)

\(=\mathrm{P}_{\mathrm{A}} \mathrm{x}_{\mathrm{A}}+\mathrm{P}_{\mathrm{B}}\left(1-\mathrm{x}_{\mathrm{A}}\right)\)

\(\left.=P_{A} x_{A}+P_{B} \cdot P_{B} x_{A}\right)\)

\(\mathrm{P}_{\mathrm{T}}=\mathrm{P}_{\mathrm{B}}+\mathrm{x}_{\mathrm{A}}\left(\mathrm{P}_{\mathrm{A}}-\mathrm{P}_{\mathrm{B}}\right)\)