A dilute solution contains m mol of solute A in 1 kg of a solvent with molal elevation constant `K_(b)`. The solute dimerises in solution as `2A hArr A_(2)`. Show that equilibrium constant for the dimer formation is
`K =(K_(b)(K_(b)m-DeltaT))/((2DeltaT_(b)-K_(b)m)^(2))` where `DeltaT_(b)` is the elevation of the boiling point for the given solution. Assume molarity = molarity

Explanation

`{:(2A,hArr,A_(2),),(m,,0,"initially"),((m-malpha),,(malpha)/(2),"after dimerization"),(m(1-alpha),,(malpha)/(2),):}`
Where `alpha` is the degree of dimerization and m is molality which is also molarity (given). Hence due to dimerization final molality `=m`
`m(1-alpha) +(malpha)/(2) = m (1-(alpha)/(2))`
`i= 1 +(n-1)alpha`
`= 1+ ((1)/(2)-1)alpha`
`= (1-(alpha)/(2))`
`DeltaT_(b)= K_(b) xx m xx i= K_(b) xx m(1-(alpha)/(2))`
`:. alpha =(2(K_(b)m-DeltaT_(b)))/(K_(b)m)`
equilibrium constant `K` for the dimer formation is
`K =([A_(2)])/([A]^(2)) = ((malpha)/(2))/(m^(2)(1-alpha)^(2)) =(alpha)/(2m(1-alpha)^(2))`
`K = 2 [(K_(b)m-DeltaT_(s))/(K_(b)m)]//2m[1-(2(K_(b)m-DeltaT_(b)))/(K_(b)m)]^(2)`
`K = (K_(b)(K_(b)m-DeltaT_(b)))/((K_(b)m-2K_(b)m+2DeltaT_(b))^(2))`
`K = (K_(b)(K_(b)m-DeltaT_(b)))/((2DeltaT_(b)-K_(b)m)^(2))` Proved