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A gas bubble rises through a liquid column. When the bubble is at height y measured from the bottom (liquid depth H, surface pressure P0, liquid density rho0), its temperature T is given by (expansion treated adiabatically):

1. T0 * ((P0 + rho0*g*H)/(P0 + rho0*g*y))^(2/5)
2. T0 * ((P0 + rho0*g*(H - y))/(P0 + rho0*g*H))^(2/5)
3. T0 * ((P0 + rho0*g*H)/(P0 + rho0*g*y))^(3/5)
4. T0 * ((P0 + rho0*g*(H - y))/(P0 + rho0*g*H))^(3/5)

Correct answer: T0 * ((P0 + rho0*g*(H - y))/(P0 + rho0*g*H))^(2/5)

Solution

The depth of the bubble below the free surface when it is at height y from the bottom is (H - y), so the pressure there is P(y) = P0 + rho0*g*(H - y). At the bottom (y = 0) the pressure is P0 + rho0*g*H with temperature T0. For an adiabatic process T ~ P^((gamma-1)/gamma). As the bubble rises the surrounding pressure decreases, so it cools. The matching option uses the (H - y) pressure in the numerator and the given exponent, giving option with exponent 2/5.

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