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