Answer:
  These are the main reasons:

• The potentials of both metals affect each other when both are connected with lead wire.

• It is more difficult for hydrogen ion to be reduced on the Zn plate than on the Cu plate (i.e. activation energy is higher on the Zn plate).

  The details for these reasons are explained next. Although some parts may be too difficult for high school students, please read it as it may cause you to be more interested in electrochemistry.

1. In the case that only Zn is dipped in water solution
  Zn metal has a large ionization tendency (low ionization energy). So Zn ion is dissolved in the water solution and electrons remain on the Zn metal when it is dipped in water solution (see Eq 1)

  Zn ↔ Zn²⁺ + 2e^-  (1)

  This reaction is continued until electrons accumulate at the electrode and the potential becomes –0.76 V. The decrease of the potential stops when the concentration of Zn²⁺ ions around the Zn metal increases and becomes concentrated enough to be reduced by Zn metal. The value of –0.76 V is enough potential to reduce the hydrogen ion around Zn and generate hydrogen gas. Therefore, Zn metal is dissolved and hydrogen gas is generated when Zn metal is dipped in water solution containing  hydrogen ion.

2. In the case where Zn and Cu (or platinum, Pt, if possible) are connected with lead wire and these are dipped in water solution
  In this case, the electric potential of Cu metal also becomes –0.76 V since Zn metal and Cu metal are connected with lead wire. Suppose that Cu is the favourable metal for reduction of hydrogen ion, then reaction (2) occurs on the surface of Cu metal:

  H⁺ + e^- ↔ 1/2 H₂  (E₀ = 0 V)　　（２）

  If this reaction occurs smoothly, hydrogen ions around Cu metal are reduced and changed to hydrogen gas as long as the electric potential of Cu metal is negative. This reaction consumes electrons in Cu metal, which gradually makes the potential of Cu metal more positive. Since Zn and Cu are connected with lead wire, the electric potential of Zn metal becomes no longer – 0.76 V but close to 0 V.
  Because the electric potential of Zn metal is forced to be around 0 V (more than -0.76 V which is the original equilibrium potential), more Zn metal dissolves into the solution. On the other hand, the reduction reaction rate of hydrogen ion on the surface of Zn becomes slow.
  Therefore, when Zn and Cu are connected with lead wire or small resistance such as a fairy lamp, the hydrogen gas generated on the surface of Cu and Zn dissolves into the solution on the surface of Zn because both potentials of Zn and Cu mutually influence one another. On the other hand, when Zn and Cu are connected by a large resistance such as a big motor, the potentials of Zn and Cu no longer influence one another. So nothing happens on the Cu surface, and hydrogen gas is generated on the Zn surface. Thus, we can say “the power of a voltaic cell is small”. (Above photo: Experiment of voltaic cell, from School of Physics, UNSW, with permission)

3. The relationship between the reduction of hydrogen ion and the surface of an electrode
  Actually reaction (2), the reduction of hydrogen ion, occurs only slightly at the potential expected from the equilibrium potential, if a special electrode is not employed. Platinum (Pt) electrode has a high activity (low activation energy) for this reaction. On the other hand, this reaction hardly occurs at all on the mercury (Hg) electrode. So if Zn is covered with a thin layer of Hg, the reduction of hydrogen ion will hardly occur even in sulfuric acid solution. This is the reason why the surface of the Zn negative electrode in a battery is covered with mercury. In a battery without Hg, H⁺ in solution is reduced and Zn is dissolved naturally, that is, the life of the battery is shortened.
  Therefore, the important properties of materials for anodes are:

• materials which do not dissolve. (materials whose ionization tendency is low.)

• materials whose activity for reduction of hydrogen ion is high.

  Pt has both properties so that it is a very suitable material for anodes. However, Pt is an expensive metal so Cu is often employed instead.

Summary
In a voltaic battery, hydrogen gas is generated at the Cu electrode, because the activation energy for the reduction of hydrogen ion is lower at Cu than at Zn, furthermore, the potentials of both Cu and Zn influence one another. Hydrogen gas is generated on the surface of Zn only if Zn is dipped into sulfuric acid solution.


Acknowledgement
We would like to thank an anonymous professor for his careful teaching and kind support for this answer.
 
 

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