Electrolysis Problem -- Half reactions and gasses produced

[i_love_science]

Homework Statement

Water is electrolyzed by passing a current of 0.500 amps through 1.00 L of 1.00 M Na2SO4 solution for 120 minutes. One gas is collected at the anode and a different gas is collected at the cathode.
a) Write the balanced equation for the cathode half-reaction.
b) Give the formula for the gas produced at the anode.

Relevant Equations

half reactions

The solution for a) is 2H2O+2e- → H2 +2OH-, and the solution for b) is O2, because the half reaction at the anode is
2H2O→O2 +4H+ + 4e-.
I don't understand why the electrolysis does not involve Na2SO4 being reduced or oxidized, and only considers water. Thanks.

 

[BvU]

Na2SO4 being reduced or oxidized

What would be the result at anode ?
And at cathode ?

 

[i_love_science]

What would be the result at anode ?
And at cathode ?

For the cathode, SO3 gas might be produced, though I'm really not sure. I don't think Na2SO4 will be oxidized.
Do you imply that since Na2SO4 is not likely to be reduced or oxidized, and only water is reduced or oxidized?

 

[BvU]

Dissolved Na₂SO₄ is fully dissociated.

Ionic mobility of H⁺ is 36.23 10^-4 cm²/(s.V) , Na⁺ has 5.19
Na has an ionization energy of 496 kJ/mol, H has 1312, so who wins ?

Ionic mobility of OH^- is 20.64 10^-4 cm²/(s.V) , SO₄^2- has 8.29
SO₄^2- ions are rather big and slow
(this guy, at 3'50" mentions a pecking order: halide anions, OH^-, all other anions)

If I sound a bit careful, it's because I'm not entirely sure of the actual processes and the decisive properties.

@Borek ?

(numbers are from Peter Atkins: Physical Chemistry)
$\$

 

[Borek]

Imagine somehow metallic Na is produced on the electrode - what would happen?