Why might the copper and silver nitrate react this way?
The copper is coated immediately with a black substance. After some time has passed, the silver nitrate solution transforms from colorless to blue. A silvery substance then collects on the copper wire.
When you react solid copper with silver nitrate a redox reaction occurs as shown below:
Cu + 2AgNO3 --> 2Ag + Cu(NO3)2
The solid copper reacts with the silver nitrate solution (in water) to produce solid silver and copper nitrate. This is called a redox reaction. The copper gives up 2 electrons (oxidized) and transfers them to 2 silver ions to produce solid silver (reduction). The nitrate ions are merely spectator ions; they play no role in the net ionic equation. The black that immediately forms on the copper is probably copper nitrate that then eventually dissolves in the water thus giving the bluish color due to the presence of the Cu2+ ion. The silvery substance that collects on the copper wire is the solid silver forming.