How Do Mole Fractions Determine Vapor Pressure in a Binary Solution?

[brake4country]

Homework Statement

Benzene and toluene combine to form an ideal solution. At 80 C, vapor pressure of pure benzene is 800 mmHg and the vapor pressure of pure toluene is 300 mmHg. If the vapor pressure of the solution is 400 mmHg, what are the mole fractions of benzene and toluene?
(A) 60% benzene and 40% toluene
(B) 50% benzene and 50% toluene
(C) 40% benzene and 60% touene
(D) 20% benzene and 80% toluene

Homework Equations

P_v=x_aP_a + x_bP_b

The Attempt at a Solution

I could easily back solve by trying each one out. D is correct. (0.2)(800) + (0.8)(300) = 400 but I really wanted to try this problem using the equation above. Instead of backsolving, is there an accurate way to put this formula to use? Thanks in advance.

 

[Borek]

Hint: x_a + x_b = ?

 

[brake4country]

but we don't know the moles of either benzene or toluene. Is it safe to assume 100 g of each converted to moles?

 

[Borek]

You don't need moles nor masses. Deal with mole fractions only.

What is the definition of a mole fraction in a mixture? What do the mole fractions of a mixture always add up to?