Minimum gas pressure to measure its volume with an eudiometer

[Irl495]

Hello,

As you can see in the attached hand drawing+calculation, my question concerns the typical method of evolved gas volume measurement by water displacement in an inverted graduated cylinder or "eudiometer". Once a bubble (cavity to be precise) of gas leaves the end of the tube through which it enters the inverted water column, a net buoyancy force easily carries it to the closed space at the top of the column, thereby displacing (pushing down) the water column by an amount Δh and increasing the pressure there by the amount ρ_wgΔh, where ρ_w is the water density. The open bath into which the bottom open end of the cylinder is submerged is exposed to atmospheric pressure p_atm.

Now, suppose the gas input tube terminates at a location in the water column exactly level with the free bath surface so that the water pressure there is p_w,A=p_atm (see attachment if unclear).

My question: I'm looking for confirmation or correction in my thinking about the minimum gas input pressure required for bubbles to actually detach from the input tube so that they may rise to fill the space at the top of the column. According to "Laplace's law", the equilibrium pressure p_i inside a single-walled "bubble" such as the gas cavities in this example, must be larger than the equilibrium pressure in the surrounding fluid, p_o, by the amount 2γ/R, where γ is the interfacial tension and R is the bubble radius. In the present system, I can therefore estimate a minimum required pressure of gas flow into the base of the water column (call it p_g,A), below which gas bubbles will not detach from the end of the tube into the water, but will rather form a concave interface from the perspective of the water and remain in the tube: So for successful bubble detachment in this system, we require p_g,A ≥ p_atm + 2γ/R. Is this correct, at least in the ideal case? In practice would mean that the tubing and vessels upstream of the inverted collection cylinder will be left with residual gas at non-zero pressure even after bubbling has ceased.

 

[eljose79]

Thank you for your question regarding the minimum gas input pressure required for bubbles to detach from the input tube in the evolved gas volume measurement method using water displacement. Your understanding of Laplace's law and its application in this system is correct. The minimum required pressure for successful bubble detachment can be estimated using the equation pg,A ≥ patm + 2γ/R, where pg,A is the gas pressure at the base of the water column, patm is the atmospheric pressure, γ is the interfacial tension, and R is the bubble radius.

In the ideal case, this equation holds true and the gas pressure must be greater than the sum of atmospheric pressure and the pressure due to surface tension in order for bubbles to detach from the tube and rise to the top of the column. However, in practice, there may be other factors at play that could affect the minimum required pressure. For example, the presence of impurities or surfactants in the gas or water could alter the interfacial tension and therefore affect the minimum pressure required. Additionally, the size and shape of the gas input tube and the water column could also play a role.

It is possible that residual gas may remain in the tubing and vessels upstream of the inverted collection cylinder even after bubbling has ceased. This could be due to factors such as gas solubility in water, fluctuations in atmospheric pressure, or variations in the surface tension. However, these factors can be minimized by ensuring that the system is properly calibrated and free from impurities.

I hope this helps to clarify your understanding of the evolved gas volume measurement method using water displacement. If you have any further questions, please do not hesitate to ask.