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There is an extract in my book which i am not able to get. It says: Place strip of metallic zinc in aqueous solution of copper nitrate for about one hour. You may notice that the strip becomes coated with reddish metallic copper and the blue colour of the solution disappears. Formation of Zn2+ ions among the products can easily be judged when the blue colour of the solution due to Cu2+ has disappeared. If H2S gas is passed through the colorless solution containing Zn2+ ions, appearance of white zinc sulphide, ZnS can be seen on making solution alkaline with ammonia.

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The final solution contains $Zn^{2+}$ ions, which reacts with $H_2S$ to produce white $ZnS$. But you may not avoid that maybe some of the initial $Cu^{2+}$ ions still remains in the solution. And these rare $Cu^{2+}$ ions do react with $H_2S$ to produce Copper sulphide $CuS$ which is dark black. $$\ce{Cu^{2+} + H_2S -> CuS + 2 H^+}$$So the obtained final precipitate is probably a mixture $ZnS$ + some $CuS$ ; it is probably not perfectly white, but grey or even black. To be able to see the white color of $ZnS$, you add some drops of ammonia $NH_3$ in the dark mixture containing $ZnS$. This will dissolve the copper sulphide, producing a solution containing $[Cu(NH_3)_4]^{2+}$. $$\ce{CuS + 4 NH_3 -> [Cu(NH_3)_4]^{2+} + S^{2-}}$$So the black component of the precipitate will disappear, and you will be able to see the white color of insoluble $ZnS$.

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You are seeing the difference between a metal that can be precipitated by hydrogen sulfide in acidic solution and one that requires the sulfide bearing solution to be alkaline or at least not as acidic. In my student days (1970s-1980s) they were called Group II and Group III metals in qualitative analysis, although that may have changed.

Both copper and zinc react as follows:

$\ce{M^{2+} + H2S <=> MS(s) + 2H^+}$

With copper the sulfide has such a low dissociation constant that the equilibrium constant for the precipitation is driven up to very high values, and the copper sulfide precipitates more or less completely even in in strongly acidic solutions.

In the second stage you have just the solution, having decanted it from the solid copper. For zinc instead of copper the equilibrium constant of the above reaction is not as high and you need to consume the hydrogen ions to drive the precipitation if zinc sulfide forward. Hence make the solution alkaline with ammonia. (A strong base might set off redissolution of the zinc sulfide by a different reaction.)

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