yeah I am unsure how to approach the calculation. I know the gas in the water will cause the density to decrease which will mean the weight of water displaced is less than the weight of the vessel which will cause it to sink.
I know it is a play on archemedies but unsure what to do.
The point at which the vessel sinks is when it reaches zero bouyancy. This depends on the weight of fluid displaced by the weight of the vessel. Say the vessel displaces 1/2 of it's total volume normally in water. If the gas cut were to reduce the effective density of the water/gas to 1/2 of the original density the vessel would displace its entire volume of the water/gas mix. Therefor causing it to likely sink.