Question

Assume that you have an ideal solution of two volitile liquids B and T (benzene and toluene do form a rather ideal solution).

a) Determine the formula for the total vapor pressure P above the solution in terms of the mole fraction of B in the vapor, x^V_B. Your formula will only contain x^V_Band the vapor pressures of the two pure liquids, P*_Band P*_T .

b) According to your formula what is the value of P when i) x^V_B= 0 and ii) x^V_B= 1?

Answer 1

Sol:-

(a).For liquid state :-

Let Mole fraction of B (benzene) in liquid state = xLB and

Mole fraction of T ( toluene) in liquid state = xLT

Now apply Raoult's Law i.e. Partial pressure of a component is equal to the product of its mole fraction and vapour pressure in its pure state , we have

P_B = xLB.P*_B and P_T = xLT.P*_T .................(1)

here P_B and P_T are the partial pressure of B and T respectively.

P*_B and P*_T are the pure vapour pressure of B and T respectively

xLB and xLT are the mole fractions of B and T in the liquid state respectively

Now

From Dalton law of partial pressure i.e. Total pressure will be the sum of the partial pressure of indvidual components, we have the total pressure (P) is :

P = P_B + P_T

P = xLB.P*_B + xLT.P*_T ................(2)

For vapour phase :-

Let xVB and xVT are the mole fractions of B and T in vapour phase respectively, P*_B and P*_T are the pure vapour pressure of B and T respectively and P is the total vapur pressure.

From the application of Dalton law of partial pressure i.e. Pressure of the gas in gaseous mixture is equal to the product of its mole fraction and total pressure, we have

P_B = xVB.P

xVB = P_B/P ...................(3)

Put value of P_B in equation (1), we have

xVB. P = xLB. P*_B ....................(4)

and P_T = xVT.P

xVT = P_T/P ....................(5)

Put value of P_T in equation (1), we have

xVT. P = xLB. P*_B ....................(6)

Substitute the values of equation (3) and (5) in equation (1), we have

P = xLB. P*_B +xLT. P*_T

P = xLB .P*_B + (1 - xLB) .P*_T ( because, xLB + xLT = 1)

P = xLB .P*_B + P*_T - xLB. P*_T

xLB = P - P*_T / P*_B - P*_T .....................(7)

Substitute the value of xLB in equation (4), we have

xVB.P = (P -P*_T) P*_B / P*_B - P*_T

on cross-multiplication

xVB.P (P*_B - P*_T ) = (P - P*_T) P*_B

xVB.P (P*_B -P*_T ) - P.P*_B = - P*_T.P*_B

P[(P*_B -P*_T ).xVB - P*_B] = - P*_T.P*_B

P[(P*_T -P*_B ).xVB + P*_B] = P*_T.P*_B

P =  P*_T.P*_B / P*_B + ( P*_T -P*_B ).xVB .......................(Required equation)

Hence this is the total vapour pressure in term of xVB.

(b).

(i) Put xVB = 0 in the required equation, Total vapur pressure becomes

P = P*_T.P*_B / P*_B + (P*_T -P*_B )(0)

P = P*_T.P*_B / P*_B

(ii) Put xVB = 1 in the required equation, Total vapur pressure becomes

P = P*_T.P*_B / P*_B + ( P*_T -P*_B )(1)

P = P*_T.P*_B / P*_B + P*_T - P*_B

P =  P*_T.P*_B / P*_T

P = P*_B

i.e.

Total pressure equal is to the vapour pressure of B(benzene) in its pure state.