Among the following cations, the number of cations which will give  characteristic precipitate in their identification tests with $$K_4[Fe(CN)_6]$$  is  $$\underline{\hspace{2cm}}$$. $$Cu^{2+},\; Fe^{3+},\; Ba^{2+},\; Ca^{2+},\; NH_4^{+},\; Mg^{2+},\; Zn^{2+}$$

We need to determine how many of the given cations give a characteristic precipitate with potassium ferrocyanide $$K_4[Fe(CN)_6]$$.

Key Concept: Potassium ferrocyanide $$K_4[Fe(CN)_6]$$ is used as a group reagent and gives characteristic coloured precipitates with certain metal ions by forming insoluble double salts or complex compounds.

Testing each cation:

1. $$Cu^{2+}$$: Gives a chocolate brown (reddish-brown) precipitate of copper(II) ferrocyanide: $$Cu_2[Fe(CN)_6]$$. This is a characteristic test for $$Cu^{2+}$$. ✓

2. $$Fe^{3+}$$: Gives a deep Prussian blue precipitate of iron(III) ferrocyanide: $$Fe_4[Fe(CN)_6]_3$$. This is a very well-known confirmatory test for $$Fe^{3+}$$. ✓

3. $$Ba^{2+}$$: Does not give a characteristic precipitate with $$K_4[Fe(CN)_6]$$. Barium is typically identified using sulphate or chromate reagents. ✗

4. $$Ca^{2+}$$: Does not give a characteristic precipitate. Calcium is identified by flame test (brick red) or oxalate precipitation. ✗

5. $$NH_4^+$$: Does not give a precipitate with $$K_4[Fe(CN)_6]$$. $$NH_4^+$$ is detected by Nessler's reagent or by heating with NaOH. ✗

6. $$Mg^{2+}$$: Does not give a characteristic precipitate with $$K_4[Fe(CN)_6]$$. ✗

7. $$Zn^{2+}$$: Gives a white (bluish-white) precipitate of zinc ferrocyanide: $$Zn_2[Fe(CN)_6]$$. This is a standard test for $$Zn^{2+}$$. ✓

Count: $$Cu^{2+}$$, $$Fe^{3+}$$, and $$Zn^{2+}$$ give characteristic precipitates. That is 3 cations.

The answer is 3.