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Question

Calculate the equilibrium concentration of B+ ions in a solution obtained by mixing equal volume of 0.02 M AOH and 0.2 M BOH solution.

Given that,
Kb(AOH)=1×108
Kb(BOH)=2×109

A
1.16×105 M
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B
6×108 M
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C
4.55×103 M
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D
3×108 M
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Solution

The correct option is A 1.16×105 M
Denoting a for AOH and b for BOH.
Let the equal volumes of solution a and b as V, i.e. Va=Vb=V
So, C1=CaVaVa+Vb=0.02×VV+VC1=0.01 M
Similarly, C2=Cb×VbVa+Vb=0.2×VV+VC2=0.1 M
Kb1=1×108 M
Kb2=2×109 M

For AOH :

AOH(aq)OH(aq)+ A+(aq)

at t=0 C1 0 0

at t=teq C1C1α1 (C1α1+C2α2) C1α1

For BOH :

BOH(aq)OH(aq)+ B+(aq)

at t=0 C1 0 0

at t=teq C2C2α2 (C1α1+C2α2) C2α2


Dissociation constant for AOH Kb1 :
Kb1=(C1α1+C2α2)(C1α1)C1(1α1)
Dissociation constant for BOH Kb2 :
Kb2=(C1α1+C2α2)(C2α2)C2(1α2)
Since α1 and α2 are very small in comparison to unity for weak bases. So 1α11 and 1α21.
C1Kb1+C2Kb2 = (C1α1+C2α2)2
[OH] = C1α1+C2α2 = C1Kb1+C2Kb2
[OH] = C1Kb1+C2Kb2[OH]=(0.01×1×108)+(0.1×2×109)
[OH]=3×1010[OH]=1.73×105 M
From the equation of Kb2 and using approximation 1α21
Kb2=(C1α1+C2α2)(C2α2)C2
At equilibrium, [B+]=C2α2 and [OH]=(C1α1+C2α2)Kb2×C2[OH]=C2α2=[B+][B+]=2×109×0.11.73×105=1.16×105 M

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