Demonstrate mathematically that if you are using an ideal solution model for a liquid phase, the liquid will always be stable (no miscibility gap).

Question

What will be an ideal response?

Answer 1

Applying the stability criteria requires us to quantify.?Gmix. ?Gmixcan be expressed with the equation below:
(??G_mix)/RT=?G^E/RT+x_1 ln?(x_1 )+x_2 ln?(x_2 )
The general term for the Gibbs excess energy is expressed as:
?G^E/RT=x_1 ln?(?_1 )+x_2 ln?(?_2 )
For an ideal solution, ?_i=1 therefore:
?G^E/RT=x_1 ln?(1)+x_2 ln?(1)
?G^E/RT=0

The first equation then simplifies to:
(??G_mix)/RT=x_1 ln?(x_1 )+x_2 ln?(x_2 )

For a mixture to be stable:
((?^2 ??G_mix)/(?x_1^2 ))_(T,P)>0
We can use the expression for ??G_mix to check if the solution is stable.
??G_mix=RTx_1 ln?(x_1 )+RT(1-x_1 ) ln?(1-x_1 )
((???G_mix)/(?x_1 ))_(T,P)=RT?ln?(x_1 )-RT?ln?(1-x_1)
((?^2 ??G_mix)/(?x_1^2 ))_(T,P)=RT(1/x_1 +1/((1-x_1 ) ))

There is no situation where the above expression is NOT greater than 0, so there is no miscibility gap. If desired this can be shown more explicitly with further algebra:
RT(1/x_1 +1/((1-x_1 ) ))>0
1/x_1 +1/((1-x_1 ) )>0
((1-x_1 ))/((x_1)(1-x_1 ) )+((x_1 ))/(x_1 )(1-x_1 ) >0
((1-x_1 )+x_1)/((x_1)(1-x_1 ) )>0
1/((x_1)(1-x_1 ) )>0
1>0
Therefore for the ideal solution model, there is no miscibility gap.