If you take an ordinary sized plastic water bottle full of water and pour a packet of powdered (or liquid) drink mix into it, Will shaking the bottle with the cap screwed on to dissolve the mix into the water work better if there is a small amount of air inside the bottle? Or does it not make a difference at all and will shaking work just as well with no air bubbles at all in the bottle?
[Physics] Fluid Dynamics – Water Bottle Drink Mix: Air or No Air
airentropyfluid dynamicswater
Related Solutions
There are various factors influencing how fast CO2 comes out of solution, but the two dominant ones are temperature and pressure.
Keeping the bottle closed allows the partial pressure of CO2 to build up above the liquid. Eventually a balance is reached so that there is no net releasing or dissolving of CO2 from/to the liquid. This is why soda lasts a long time between manufacture and when you first open the bottle. This is also why there is pressure in the bottle when you first open it. Keeping the bottle closed between sips will allow some pressure to build up, which will slow the rate of CO2 release from the liquid.
In general, gasses dissolve better in liquid at lower temperature. CO2 in water is no exception. Keeping the bottle cold will lower the partial pressure of the CO2, reducing the amount that bubbles out.
The roughness of the surface and the like are secondary effects at best. Probably the third strongest effect beyond pressure and temperature is mechanical disturbance. This allows a quicker interchange between the dissolved CO2 and the gas above the liquid. Since a recently-open bottle will probably not be in equilibrium, mechanical disturbance, like shaking, will cause CO2 to be released more quickly.
As I understand it, you have two theories.
- the leak is caused by the drop in temperature of the liquid
- the leak is caused by the increase in pressure when screwing on the top
I think we can probably rule out both of these, and here's why:
Temperature drop
As we know from the ideal gas law, $$\displaystyle \frac{P_i V_i}{T_i} = \frac{P_f V_f}{T_f}$$ Where $P$ is pressure, $V$ is volume and $T$ is temperature. Subscripts are $i$ for initial and $f$ for final. This suggests that as the liquid cools (assuming constant volume) the pressure would also go down. So that would seem to eliminate the possibility of the temperature change causing a pressure increase and therefore a leak.
Placement of cap
If we suppose that the volume of the cap is $A \text{cm}^3$ and that the cap effectively seals completely when the threads first engage (which is unlikely, practically speaking), and the volume of air within the bottle is originally $B \text{cm}^3$, then the increase in pressure would be $(A+B)/B$. So if the volume of the cap is significant compared to the volume of air within the filled bottle and if the cap seals immediately, then it could be a factor. However, it seems unlikely to me that the screw top seals immediately on engagement of the threads. This is easily checked -- screw on the top half-way and squeeze the bottle. If gas escapes, it's not that tightly sealed and we can probably eliminate this theory.
Reduction in ambient pressure
An altitude change, especially if the product is shipped by air, would have the effect of reducing the external (ambient) pressure outside the bottles which could indeed cause a leak.
Best Answer
Within reasonable limits, the more air in the bottle the better the mixing.
This isn't to do with air bubbles. To get good mixing you need turbulent flow, and for that you need high flow velocities. If the bottle is completely full it's hard to get a high flow velocity started because for water to move it has to push other water out of the way. If the bottle is only half full the moving water only has to push air out of the way, and air is both much more compressible and much less viscous than water. For any given amount of shaking effort you'll get higher flow velocities and therefore better mixing if there is a substantial amount of air in the bottle.