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Well, I found that lead cation $\ce{Pb^2+}$ is present in both group 1 and group 2 of cation analysis. But why?

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Simply because lead(II) fits both analytical groups:

One reason for putting lead in both groups is that the chloride is somewhat of a borderline case in terms of being insoluble enough to appear in Group I. Group II represents a more complicated analysis but is more sensitive for lead.

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    $\begingroup$ Lead (II) chloride has enough solubility so it could slip through the cracks in Group I. Group II where the sulfide is precipitated is more sensitive. $\endgroup$ – Oscar Lanzi Jan 29 '19 at 15:46
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    $\begingroup$ @OscarLanzi You are right, probably it would be worth adding that there are no sharp margins between these 6 groups and various sources add or remove cations from one group to another arbitrarily. $\endgroup$ – andselisk Jan 29 '19 at 15:49

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