Q.P$_1$ and P$_2$ are the vapour pressures of pure liquid components, 1 and 2 respectively of an ideal binary solution. If x$_1$ represents the mole fraction of component 1, the total pressure of the solution formed by 1 and 2 will be

(i) Raoult's Law: each component's vapour pressure in a solution is proportional to its mole fraction. (ii) The van der Waals volume term (V - nb) corrects for the real, finite size of gas molecules, which reduces the free space actually available for their motion.

Answering part (a), since (a)(i) and (a)(ii) are given as the primary alternative (the OR alternative (b), deriving the van't Hoff equation, is not required unless (a) is unanswerable):

(a)(i) Raoult's Law: For a solution formed by mixing two (or more) volatile, miscible liquids, Raoult's Law states that the partial vapour pressure of each component in the solution is directly proportional to its mole fraction in the liquid solution, at a given temperature:

p_A = p_A(degree) . x_A and p_B = p_B(degree) . x_B

where p_A(degree) and p_B(degree) are the vapour pressures of the PURE components A and B, and x_A, x_B are their respective mole fractions in solution. For an ideal solution obeying Raoult's Law throughout the whole composition range, the total vapour pressure of the solution is simply the sum: P(total) = p_A + p_B = p_A(degree)x_A + p_B(degree)x_B.

(a)(ii) Volume correction in the van der Waals equation: The ideal gas law (PV = nRT) assumes gas molecules are point particles with zero volume, free to move throughout the entire container volume V. In reality, gas molecules occupy a small but finite volume of their own, so the space genuinely AVAILABLE for their free movement is somewhat LESS than the total measured volume V of the container.

Van der Waals corrected this by replacing V with (V - nb) in the equation of state, where:

• n = number of moles of gas …