Answer to Question #83978 in Molecular Physics | Thermodynamics for Hyhy

Question

A diver observes a bubble of air rising from the bottom of a lake (where the absolute pressure is 3.50 atm) to thesurface (where the pressure is 1.00 atm). The temperature at the bottom is 4.0°C, and the temperature at the surface is 19.0C° higher. Find the ratio of the volume of the bubble as it reaches the surface to its volume at the bottom? Doesthe bubble expand or compress in volume? Assume that the number of moles of gas in the bubble remains constant.

Answer 1

Treating the air in the bubble as an ideal gas, we use the equation of state for ideal gas in the form pV = νRT, where p is the pressure, V is the volume, ν is the number of moles, and T is the absolute temperature of the gas, with R being the universal gas constant. Since the number of moles ν remains constant, we have the ratio of volumes at different levels in the form

V_1/V_2 =T_1/T_2 p_2/p_1

Let index “1” refer to the surface, and index “2” to the bottom. For the corresponding absolute temperatures, we have T2 = (273.15 + 4.0) K = 277.15 K, T1 = T2 + 19.0 K = 296.15 K. Thus, T1/T2 = 1.07. We also have p2/p1 = 3.5. Hence, V1/V2 = 1.07 × 3.5 = 3.74. The bubble expands in volume 3.74 times.