Paragraph: A small spherical monoatomic ideal gas bubble ( γ = 3 5 ) is trapped inside a liquid of density ρ 1 (see figure). Assume that the bubble does not exchange any heat with the liquid. The

Question

Paragraph: A small spherical monoatomic ideal gas bubble ( γ = 3 5 ​ ) is trapped inside a liquid of density ρ 1 ​ (see figure). Assume that the bubble does not exchange any heat with the liquid. The bubble contains n moles of gas. The temperature of the gas when the bubble is at the bottom is T 0 ​ , the height of the liquid is H and the atmospheric pressure is p 0 ​ (Neglect surface tension). Question: When the gas bubble is at height y from the bottom, its temperature is

TestHub · Accepted Answer

As there is no exchange of heat. Therefore, process is adiabatic. Applying, ∴ or ​ T p γ 1 − γ ​ = constant T 2 ​ p 2 γ 1 − γ ​ ​ = T 1 ​ p 1 γ 1 − γ ​ ​ T 2 ​ ​ = T 1 ​ ( p 2 ​ p 1 ​ ​ ) γ 1 − γ ​ = T 1 ​ ( p 1 ​ p 2 ​ ​ ) γ γ − 1 ​ ​ Substituting the values we have, T 2 ​ ​ = T 0 ​ [ p 0 ​ + ρ l g H p 0 ​ + ρ l g ( H − y ) T ​ ] 5/3 5/3 − 1 ​ = T 0 ​ [ p 0 ​ + ρ l g H p 0 ​ + ρ l g ( H − y ) ​ ] 2/5 ​ ∴ correct option is (b).

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