Find `"K"_("a")`, 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 a monobasic as AH. It ionizes as
`{:(,AH,hArr,A^(Theta),+,H^(+)),("Initially conc.",C,,0,,0),("Conc. 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`
Molal concentration =0.1
`Delta"T"_("f")="K"_("f")xx"C"_("m")xx"i "implies0.205=1.86xx0.1xx(1+alpha)thereforealpha=0.1`
`"K"_("a")=(["A"^(-1)]["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)`
`therefore"K"_("b")=1.11xx10^(-3)`