No, the degree of ionization of a pure metal electrode does not increase with dilution when it is dipped in water and the volume of water surrounding it is increased.
When a pure metal electrode is dipped in water, it does not undergo significant ionization in the same way as an acid. Instead, the metal electrode acts as an electron donor, releasing electrons into the solution. This process is known as oxidation, and the metal electrode is oxidized to form metal cations in the solution.
The concentration of metal cations at the electrode-water interface is determined by the solubility of the metal and the equilibrium established between the metal and its ions. Once this equilibrium is reached, the concentration of metal cations remains constant and does not increase with dilution.
Increasing the volume of water surrounding the metal electrode would dilute the concentration of metal cations, but it would not lead to an increased degree of ionization. The process at the metal electrode-water interface is governed by the solubility equilibrium, not by the dilution of the solution.
In summary, the degree of ionization of a pure metal electrode, when dipped in water, is determined by the solubility of the metal and does not increase with dilution.