`{:(I step,H_(3)PO_(4) hArr H^(+)+H_(2)PO_(4)^(-),K_(a_(1))=7.5xx10^(-3)),(II step,H_(2)PO_(4)hArr H^(+)+HPO_(4)^(2-),K_(a_(2))=6.2xx10^(-8)),(III step,HPO_(4)^(2-)hArrH^(+)+PO_(4)^(3-),K_(a_(3))=3.6xx10^(-13)):}`
For `I` step :
`underset(0.1(1-alpha))underset(0.1)(H_(3)PO_(4))hArr underset(0.1alpha)underset(0)(H^(+))+underset(0.1alpha)underset(0)(H_(2)PO_(4)^(-))`
`K_(a_(1))=([H^(+)][H_(2)PO_(4)^(-)])/([H_(3)PO_(4))]rArr(0.1alpha^(2))/((1-alpha))=7.5xx10^(-3)....(1)`
Excepting `alpha ltlt1,alpha^(2)=7.5xx10^(-2):.alpha=0.274` (not negligible)
so solve quadratic equation (1) & get=`alpha=0.24`
`:.[H^(+)]=0.24MrArr [OH^(-)]=(10^(-14))/(0.024)=4.17xx10^(-13)M` ("insignificant")
&`[H_(2)PO_(4)^(-)]=0.024M`
Also `[H_(3)PO_(4)^(-)]=0.1-0.024=0.076M`
The value of `K_(a_(2))` and `K_(a_(3))`.Also dissociations of `II` and `III` steps occur in present of `H^(+)` furnished in `I` step and thus ,dissocations of `II` and `III` steps are further suppressed due to common ion effect.
For `II` step `underset(0.024-y)underset(0.024)(H_(2)PO_(4)^(-))hArr underset(0.024-y)underset(0.024)H^(+)+underset(y)underset(0)(HPO_(4)^(2))`
The dissocation of `H_(2)PO_(4)^(-)` occurs in presence of `[H^(+)]` furnished in step `I`.
Thus `K_(a_(2))=([H^(+)][HPO_(4)^(-)])/([H_(2)PO_(4)^(-)])`
`6.2xx10^(-8)=((0.024+y)y)/((0.24-y))`
`:.y` is small `:.0.024-y~~0.024` and neflecting `y^(2)`
`6.2xx10^(-8)=(0.024y)/(0.024)`
`y=6.2xx10^(-8)M`
or `[HPO_(4)^(2-)]=K_(a_(2))=6.2xx10^(-8)M` (insignificant)
for `III` step `underset((6.2xx10^(-5)-x))(HPO_(4)^(2-))hArr underset((0.024+x))(H^(+))+underset(x)(PO_(4)^(3-))`
`K_(3)=([H^(+)][PO_(4)^(3-)])/([HPO_(4)^(2-)])=((0.24+x)x)/((6.2xx10^(-8)-x))`
because `x ` is small :.Again neglecting `x^(2)` and assuming `6.2xx10^(-6)-x=6.2xx10^(-8)`
`:.3.6xx10^(-13)=(0.024x)/(6.2xx10^(-8))`
`:.x=(3.6xx10^(-13)xx6.2xx10^(-8))/(0.024)=9.3xx10^(-19)M`
or `[PO_(4)^(3-)]=9.3xx10^(-19)M` (insufficant)
Note: For weak polyprotic acid solution having no other electrolyte the anion concentration produced in step of dissocation is always equal to `K_(a_(2))` if concentration is reasonable.
