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Reaction
No reaction will take place in a copper sulfate half cell by itself. Imagine, however, that you connected a wire from the copper electrode to a zinc electrode in another half cell, then added a "salt bridge" which allowed movement of charges between the two electrolytes. You would now get a current between the two half cells because a chemical reaction is taking place. This reaction involves oxidation (loss of electrons) of the zinc and reduction (gain of electrons) of the copper, although the two halves of the reaction are taking place in different cells.
Potentials
Suppose you measured the voltage between your copper and zinc half cells using a voltmeter. Assuming the solute concentration in both is one molar and the copper and zinc are pure, you would get a value close to +1.10 volts. If you put the copper sulfate half cell together with a half cell containing a silver electrode and a silver chloride electrolyte, however, you would get a voltage of roughly +0.12 (again, assuming one molar concentration in both electrolytes). Clearly, the voltage depends on the two half cells you choose.
Standard Potentials
To provide a standard of comparison, chemists use so-called "standard potentials" measured with reference to a "standard hydrogen electrode," or SHE. There's nothing particularly special about a SHE, per se; it is just a convenient benchmark (any other half cell could have served as a benchmark instead). Compared to a SHE, a copper sulfate half cell has a voltage of +0.34 volts, so this is its standard potential. To get the standard potential, E°, of any other half cell combined with the copper sulfate half cell, you use their standard potentials. The standard potential of a zinc half cell, for example, is -0.76, so if you add 0.76 and 0.34, you get 1.10. The standard potential for a silver chloride half cell, by contrast, is 0.22, so the standard potential for the combination of the two half cells is 0.12.
Nernst Equation
Standard potentials are measured at standard atmospheric pressure using electrolytes with a one molar concentration. If the concentration of copper sulfate is not one molar, however, the voltage will be different -- and this is exactly what will happen as a battery runs down. To calculate the voltage given other concentrations, you can use the Nernst equation. At room temperature, this famous equation is as follows: E = E° - (0.05917 / n) log Q, where n is the number of electrons that change hands during the reaction and Q is the reaction quotient. For any reaction wW + xX --> yY + zZ, you can approximate the reaction quotient as ( [Z]^z [Y]^y ) / ( [X]^x [W]^w ), where a number in brackets denotes the concentration. Note, however, that if a reactant or product is a pure solid or a pure liquid, it does not appear in the reaction quotient; you just leave it out completely.