Two identical bubbles of gasform at the bottom of a lake, then rise to the surfa

Question

Two identical bubbles of gasform at the bottom of a lake, then rise to the surface. Because thepressure is much lower at the surface that at the bottom, bothbubbles expand as tehy rise. However, bubble A rises very quickly,so that no heat is exchanged between it and the water. Meanwhile,bubble B rises slowly (impeded by a tangle of seaweed), so that italways remains in thermal equilibrium with the water (which has thesame temperature everywhere). Which of the two bubbles is larger bythe time they reach the surface? Explain.

Explanation / Answer

This a problem regarding adiabatic expansion of the first bubble.

Since dU = dQ - dW,

dU = change in internal energy

dQ = heat added to the system

dW = work done by the system

In case of the first bubble, dQ = 0,

dU = -dW

Since internal energy is a function of temperature, dU = n.Cv.dT,

and the bubble expansion makes work done by the bubble positive (dV > 0), dU < 0 for first bubble.

So, the first bubble cools down as it rises.

PdV = work done by first bubble = -dU = -n.Cv.dT > 0

So, dT < 0.

Since temperature of the second bubble remains same as the constant temperature of water everywhere,

for the second bubble, dU = 0. It keeps absorbing heat to compensate for the work it does in expanding against the water pressure.

So, for the second bubble:

dQ = dW.

Since final volume will depend on the internal pressure and temperature of the bubbles,

and pressure must be same for both as the bubbles will expand till internal pressure is balanced by external water pressure,

Using PV = nRT, P1=P2, n1=n2, R = constant.

T1/V1 = T2/V2

T1/T2 = V1/V2

Since T1 < T2 as first bubble cools down, V1 < V2

So, the second bubble is larger.

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