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
`2AhArrA_(2)`
Select the correct exprression of equilibrium constant `'K'` for the dimer formation. (assume molarity `=` molality)

Options

A. `K=(K_(b)(K_(b)m-DeltaT_(b)))/((K_(b)m+2DeltaT_(b))^(2))`
B. `K=(alpha)/(2m(1-alpha)^(2))`
C. `K=(K_(b)(K_(b)m-DeltaT_(b)))/((2DeltaT_(b)-K_(b)m)^(2))`
D. `K=(2(K_(b)m-DeltaT_(b)))/(K_(b)m)`

Answer

B, C

Explanation

`2A hArr A_(2)`
moles at `t=0, m, 0`
moles at `t_(eq), m-malpha, (malpha)/2`
Total moles `=m-(m alpha)/2=m(1-(alpha)/2)`
`DeltaT_(b)=iK_(b)m=K_(b)m(1-(alpha)/2)`
`1-(alpha)/2=(DeltaT_(b))/(K_(b)m)`
`alpha=(2(K_(b)m-DeltaT_(b)))/(K_(b)m)`
Equilibrium constant `K` for the dime formation is
`K=([A_(2)])/([A]^(2))=(malpha)/(2[m(1-alpha)]^(2))`
`K=(alpha)/(2m(1-alpha)^(2))`
`K=(2(K_(b)m-DeltaT_(b)))/(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))`