Ka vs Kb Question

[icewave667]

Can someone help me with this question? I thought the answer was C, Ka>Kb because 5.7 is slightly acidic. However, the answer is actually A (Kb greater than Ka). How is this possible if the pH is 5.7? There was an alternative explanation that pOH is 8.3 so that is why Kb is greater than Ka but I have a hard time digesting this explanation because you would normally calculate Ka / pH I think...not pOH. Any ideas? Thanks!

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[sniderwes]

Ka+Kb=14 for conjugate ions, since Ka is 5.7 then Kb is bigger.

 

[icewave667]

The pH is 5.7, not the Ka. I'm still confused. Thanks for your response though

 

[sniderwes]

Sorry, I went in totally the wrong direction with my answer.

I think this should be right: so you add 10 moles of something (concentration becomes 10M). So the amount of hydrogen that dissociates from the compound is x, and there's an equation that says x^2/(F-x)=Ka. Should look familiar.
So pH is 5.7 which means x has to be 10^-5.7 (definition of pH). so plug this for x and solve for Ka, which is around 10^-13. So Kb is 10^-1 which is a whole lot bigger

 

[Czarcasm]

From a quantitative perspective:

Ka = [H3O+][A-]/[HA]
Ka = [2x10^-6 M][2x10^-6 M] / [10M]* (the difference in 10M-x is negligible so we can ignore).
Ka = 3.98 x 10^-13 M

Since pKa + pKb = 14, we can calculate Kb:
Ka x Kb = 1x10^-14, therefore Kb: 0.025 M

Kb is greater than Ka. Choice A.

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Conceptually:

A weak to moderate acid, as shown in this example has a mildly stable conjugate base. As acids get stronger, the conjugate base becomes more stable. For strong acid's it's more intuitive in that we can say a strong acid has a considerably lower pKa and therefore a higher pKb, because their inversely related (which makes sense because the conjugate base is very stable). I'll clarify this in a bit.

However, for weak acid's and their conjugate bases, it's less intuitive as these values can go either way and so generally the best way to approach these types of questions is by analyzing the effect of the concentrations:

Here we know: Ka: [H3O][A-]/[HA]. Assuming HA is constant (which usually is the case because the difference is negligble), as the acid becomes stronger, H3O+ dissociates more; But because the concentration is below 1 (that is 10 to some - power: ie. 10^-7), you are multiplying by a fraction (or dividing a whole number); Therefore, the more the weak acid dissociates, the smaller Ka becomes (take another look at that equation). And because Kb is inversely related to Ka, the larger the Kb.

In the more intuitive scenario, where we have a strong acid that dissociates, say 1M of HCl, well, this would dissociate completely to produce 1M [H3O+]. Bare in mind there is no fraction to consider here because we are not dividing; the acid is consumed entirely (the essense of a strong acid). Therefore the whole denominator term at the bottom can be neglected and you can simply relate: Ka = [H3O+][A-]. In this case, regardless if you have some fraction or a whole number, as long as the amount that dissociate increases, Ka will increase. There is a direct relationship here.

If this is at all confusing, save yourself the misery and do the quick and simple calculation!
Hope this makes sense.