Statement I: When acidified zinc sulphate solution is electrolysed between zinc electrodes, it is zinc that is deposited at the cathode and evolution of hydrogen gas does not take place.

Statement II: The electrode potential of zinc becomes more negative than hydrogen as the overvoltage for the hydrogen evolution on zinc is quite large.

Statement I is false but Statement II is correct

The correct answer is option 4. Statement I is false but Statement II is correct.

Let us break down why Statement I is false and Statement II is correct:

Statement I: When acidified zinc sulphate solution is electrolysed between zinc electrodes, it is zinc that is deposited at the cathode and evolution of hydrogen gas does not take place.

"When acidified zinc sulphate solution is electrolyzed between zinc electrodes, it is zinc that is deposited at the cathode and evolution of hydrogen gas does not take place."

Electrolysis of Zinc Sulphate Solution:

In a typical setup with zinc electrodes and an acidified zinc sulfate solution, the reactions at the electrodes depend on the conditions. 

At the anode, zinc undergoes oxidation:

\(\text{Zn (s)} \rightarrow \text{Zn}^{2+} + 2\text{e}^-\)

At the cathode, two scenarios can occur If the solution is not strongly acidified, zinc can be deposited:

\(\text{Zn}^{2+} + 2\text{e}^- \rightarrow \text{Zn (s)}\)

If the solution is strongly acidified, hydrogen evolution can become more favorable due to the lower overvoltage for hydrogen evolution compared to zinc deposition. The reaction would be:

\(2\text{H}^+ + 2\text{e}^- \rightarrow \text{H}_2 \text{(g)}\)

In acidified conditions, the hydrogen evolution reaction might dominate, leading to the production of hydrogen gas rather than zinc deposition.

Thus, Statement I is false because under strongly acidic conditions, hydrogen gas can evolve at the cathode instead of zinc being deposited.

Statement II: The electrode potential of zinc becomes more negative than hydrogen as the overvoltage for the hydrogen evolution on zinc is quite large.

"The electrode potential of zinc becomes more negative than hydrogen as the overvoltage for the hydrogen evolution on zinc is quite large."

Electrode Potentials: The standard electrode potential of zinc (\(\text{Zn}^{2+}/\text{Zn}\)) is -0.76 V, which is more negative compared to the standard electrode potential of hydrogen (\(\text{2H}^+/ \text{H}_2\)) which is 0.00 V.

Overvoltage for Hydrogen Evolution: Overvoltage is the extra voltage required to drive a reaction at a rate faster than its theoretical prediction. For hydrogen evolution on zinc, the overvoltage is high, making the hydrogen evolution less favorable compared to zinc deposition under some conditions.

High overvoltage implies that the actual conditions needed to evolve hydrogen are more extreme, thus zinc deposition might be preferred in less acidic or neutral solutions.

Therefore, Statement II correctly explains that due to the large overvoltage for hydrogen evolution on zinc, hydrogen gas is less likely to evolve, which can lead to zinc being deposited instead.

Conclusion:

Statement I is false because under strongly acidic conditions, hydrogen evolution is more likely than zinc deposition. Statement II is correct because it accurately describes why hydrogen evolution is less likely on zinc due to the high overvoltage.

Thus, the correct answer is: Statement I is false but Statement II is correct.