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What happened when?
Copper wire is dipped in silver nitrate solution

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Hint: The standard reduction potential is a measure of the ease of an element to lose its electrons. These values are obtained with reference to the hydrogen electrode. Higher the value of reduction potential higher is the ability of a metal to undergo the reduction. This results in the replacement of existing ions and forming a new composition of ions.

Complete step by step answer:
Copper ion $\text{ C}{{\text{u}}^{\text{2+}}}\text{ }$ accepts two-electron and reduced to copper $\text{ Cu }$ metal
The reduction potential of copper is given as follows,
$\text{ C}{{\text{u}}^{\text{2+}}}\text{ + 2}{{\text{e}}^{-}}\text{ }\to \text{ Cu }{{\text{E}}^{\text{0}}}\text{ = +0}\text{.34 V }$
The silver $\text{ Ag }$ also undergoes the reduction reaction. silver ion $\text{ A}{{\text{g}}^{\text{+}}}\text{ }$ accepts the one electrons and reduces to elements. the reduction potential of the silver is,
$\text{ A}{{\text{g}}^{\text{+}}}\text{ + }{{\text{e}}^{-}}\text{ }\to \text{ Ag }{{\text{E}}^{\text{0}}}\text{ = +0}\text{.80 V }$
From the above reduction values, it is clear that the reduction potential of the silver is more than that of the copper. Thus, copper will replace the silver ions.
When the copper wire is dipped in the silver nitrate solution, the silver ions which have more reduction value easily undergo the reduction, and copper which has low reduction potential with respect to silver undergoes the oxidation.
The silver is reduced from $\text{ A}{{\text{g}}^{\text{+}}}\text{ }$to the $\text{ Ag }$while the copper $\text{ Cu }$loses its two electrons and oxidize to$\text{ C}{{\text{u}}^{\text{2+}}}\text{ }$.
The reaction reduction of silver and oxidation of copper is as shown below,
$\text{ Cu + AgN}{{\text{O}}_{\text{3}}}\text{ }\to \text{ Cu(N}{{\text{O}}_{\text{3}}}{{\text{)}}_{\text{2}}}\text{ + A}{{\text{g}}^{\text{+}}}\text{ }$
As the reaction proceeds, Silver metal is precipitated as a white solid .The copper ion forms a compound with nitrate ion .The formed Copper nitrate is an inorganic compound and it is a blue crystalline solid .Thus the solution gradually turns into blue in colour.

Therefore, this is the reason when the copper wire is dipped in the silver nitrate solution, the solution turns blue.

Note: The tendency of a metal to lose electrons can be expressed in two ways. As a standard reduction potential $\text{ }{{\text{E}}^{\text{0}}}\text{ (SRP ) }$ or as the standard oxidation potential $\text{ }{{\text{E}}^{\text{0}}}\text{ (SOP ) }$ . Higher positive value indicates the elements easily reduced however high negative value indicates the element is readily oxidized. The SOP and SRP are related as,
$\text{ }{{\text{E}}^{\text{0}}}\text{ (SRP ) = }-{{\text{E}}^{\text{0}}}\text{ (SOP ) }$