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• the blues of Cu(H₂O)₆²⁺
• the green of CuCl₂(H₂O)₄
• to the brown of CuCl₄^-2

are shown by  Copper(II) compounds.

This contrasts with the pale yellows and whites of Copper(I) compounds....
Copper is a transition element. These colors reflect the d⁹ electron configuration of Cu(II) and the d¹⁰ configuration of Cu(I).

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This synthesis starts with copper metal which is converted to the nitrate by reaction with concentrated Nitric Acid.

This equation is not balanced
Cu + HNO₃--> Cu(NO₃)₂ + H₂O +NO₂
Please balance this redox reaction.
If worse gets to worse, see your Lab Manual.
Why is Nitric acid rather than Hydrochloric acid used in this step?

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This copper(II) nitrate needs to be converted to copper(II) chloride which is then reduced to the desired copper(I) chloride.
This is done by first precipitating the copper(II) as copper carbonate and then dissolving in an excess of HCl.

Cu(NO₃)₂ + Na₂CO₃ ------> CuCO₃ + 2 NaNO₃

CuCO₃ + 2 HCl -----> CuCl₂ + CO₂ + H₂O

The green Copper(II) Chloride is converted to the dark brown tetrachlorocuprate(II) ion with excess Hydrochloric Acid and reduced with copper metal to a pale yellow copper(I) chloro complex.

CuCl₄^-2 + Cu -----> 2 CuCl₃^-2

The color of the solution shows the degree of reaction

The above transitions happen in the course of 15 to 30 minutes.

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Finally the product is obtained by diluting the copper(I) chloro complexes with ice and water.