A bubble of air is suspended in clean water at ambient pressure and temperature.
ID: 1939149 • Letter: A
Question
A bubble of air is suspended in clean water at ambient pressure and temperature.a) How do the pressure and the number of molecules inside the bubble scale with the radius R of the bubble? Assume the surface tension gamma is known and you can model the bubble/air as an ideal gas.
b) Find the minimum radius of the bubble so that the ideal gas law is valid? (hint: the volume fraction of the atoms inside the bubble has to be << 1)
c) If we add surfactant to water to lower the surface tension to 0.5* gamma, identify the minimum radius of the bubble in part (b)
Explanation / Answer
P in = P amb + P surf = P amb + 2 ? r (1a) r Radius of the bubble in m ? Surface tension in joule/m2 of N/m. The surface tension of water at 273 K is 0.073 N/m. Pin Pressure inside the bubble in N/m2=10-5bar Pamb Ambient pressure in N/m2=10-5bar Psurf Pressure due to the surface tension in N/m2=10-5bar From this equation we learn that the smaller the bubble, the higher the pressure inside. You can experience the radius dependency of the pressure by trying to blow a balloon (bubble principles perfectly apply to a balloon up to the point where the balloon explodes). To get the first blow of air into the balloon (small radius) is a hell of a job, whereas it becomes easier if the balloon becomes larger. next contents previous Bubbles and diffusion When we have a bottle of beer things get a bit more complicated (Usually the opposite holds, but when we look at the bubbles it might be). Bubbles in beer contain Carbon Dioxide. There is also Carbon Dioxide in solution in the beer. Carbon Dioxide can diffuse from the solution into the bubble or vice versa, depending on the partial pressure of the Carbon Dioxide in solution and in the bubble. If we assume that the bubble consist of only Carbon Dioxide, the Carbon Dioxide pressure in the bubble is given by equation (1) and depends on the radius of the bubble. We define the partial pressure of the Carbon Dioxide in solution in the beer to be Pt. (If we regard the bottle of beer as a primitive model for a diver, we could call it 'tissue tension'). If the bottle is closed, the partial pressure of the Carbon Dioxide in solution Pt is in equilibrium with the ambient pressure Pamb. If we assume there is only Carbon Dioxide gas in the (closed) beer bottle, the beer is saturated with Carbon Dioxide and Pt will be equal to Pamb (we can neglect hydrostatic pressure). The pressure in the bubble Pin will be higher than Pt due to the surface tension. Gas from within the bubble will diffuse into solution and the bubble will collapse. So every bubble will collapse eventually due to this gradient Pin-Pt. This is why in a closed bottle of beer there are no bubbles and there is no foam. However, if we open the bottle things will be different. The ambient pressure will drop, whereas the value of Pt remains the same, at least for the moment. In this case Pt is larger than Pamb: the beer is supersaturated with Carbon Dioxide. Given an ambient pressure Pamb and the partial pressure Pt of the Carbon Dioxide in solution, there is a critical bubble radius rmin at which the pressure inside the bubble Pin equals Pt. The critical radius can be found by substituting Pin by Pt in equation (1): r min = 2 ? P t - P amb
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