Find `"K"_("b")`, the ionization constant of tartaric acid if a 0.100 molal aqueous solution of tartaric acid freezes at `-0.205^(@)"C"`. Assume that only the first ionization is of importance and that `0.1"m"=0.1"M". " K"_("f")=1.86 "kg mol"^(-1)"K".`

Explanation

Assuming that the tartaric acid be monobasic as AH. It ionizes as
`{:(, AH hArr, A+, H^(+)),("Initially conc", C,0,0),("Con.After dissociation",C(1-alpha),Calpha,Calpha):}`
Here `i=(C(1-alpha)+C alpha+C alpha)/(C)` where `alpha` = degree of dissociation
`=(C(1+alpha))/(C)=1+alpha`
Molala concentration =0.1
`DeltaT_(1)=K_(T)xxC_(m)xx1`
`0.205=1.86xx0.1xx(1+alpha), :. alpha=0.1`
`K_(s)=([A^(-)][H^(+)])/([AH])=(C alphaxx C alpha)/(C(1-alpha))=(alpha^(2)C)/(1-alpha)`
`K_(a)=((0.1)^(2)xx0.1)/(1-0.1)=1.11xx10^(-3)`
`:.K_(s)=1.11xx10^(-3)`