The vapor pressure of pure cyclohexane (C6H12) and pure toluene (C7H8) at 20°C are 66.9 and 21.1 torr, respectively. The total pressure above a mixture of these two liquids is 50.0 torr. Calculate the mole fraction of cyclohexane in the liquid mixture?

#chi_("Cy"(l)) = 0.6310#
#chi_("Tol"(l)) = 0.3690#
#chi_("Cy"(g)) = 0.8443#
#chi_("Tol"(g)) = 0.1557#

The total pressure for ideal solutions is the simple sum:

#P_("tot") = P_("Cy") + P_"Tol"#

where #P_A# is the partial pressure of component #A#.

When forming an ideal solution (which cyclohexane and toluene do!), it follows Raoult's law, i.e.

#P_A = chi_(A(l))P_A^"*"#

where #"*"# indicates the pure component and #chi_(A(l))# is the mol fraction in the solution phase.

Thus,

#P_"tot" = chi_("Cy"(l))P_"Cy"^"*" + chi_("Tol"(l))P_"Tol"^"*"#

#= chi_("Cy"(l))P_"Cy"^"*" + (1-chi_("Cy"(l)))P_"Tol"^"*"#

#= chi_("Cy"(l))(P_"Cy"^"*" - P_"Tol"^"*") + P_"Tol"^"*"#

As a result, upon using the total vapor pressure, we can easily get the mol fraction of cyclohexane in the solution phase:

#color(blue)(chi_("Cy"(l))) = (P_"tot" - P_"Tol"^"*")/(P_"Cy"^"*" - P_"Tol"^"*")#

#= ("50.0 torr" - "21.1 torr")/("66.9 torr" - "21.1 torr")#

#= color(blue)(0.6310)#

Note that we have no way of knowing the mol fraction in the vapor phase simply by knowing the mol fraction in the solution phase --- they're not related.

Instead, we know that the partial pressure of one component above a solution is based on the mol fraction in the vapor phase:

#P_"Tol" = chi_("Tol"(g))P_"tot"#

Therefore, the mol fraction of toluene in the vapor phase is:

#color(blue)(chi_("Tol"(g))) = P_"Tol"/P_"tot"#

#= (chi_("Tol"(l))P_"Tol"^"*")/P_"tot"#

#= ((1 - chi_("Cy"(l)))P_"Tol"^"*")/P_"tot"#

#= ((1 - 0.6310)("21.1 torr"))/("50.0 torr")#

#= color(blue)(0.1557)#