Abubble of co2 forms at the bottom of a lake and rises to the surface. the pressure p below the surface of a lake increases with depth h as ap = pgah, where p is the density of the water. as the bubble rises, the pressure decreases and the bubble expands. if the bubble rises slowly, the process is isothermal. but if the bubble rises rapidly, the process is adiabatic. you can assume f = 6 for co2 gas near room temperature (3 translational modes and 2 rotational modes and one active bending mode.) a) compare two initially identical bubbles a and b: bubble a rises adiabatically, and bubble b rises isothermally. let pi be the pressure at the bottom of the lake and p2 be the pressure at the top of the lake. for each of the bubbles, a and b, derive a formula for the ratio of the volumes v2/v1 , and predict which bubble will expand more as it rises. b) for each bubble, a and b, by what factor will the volume increase if the bubble starts at a depth of 100 m and rises to the surface? c) for each bubble, a and b, derive an expression for dv/dp. your answers should depend only on v, p, and perhaps other constants like g or y=(f +2)/f.