Today we will discuss three linked comprehension type multiple choice questions which appeared in IIT-JEE 2008 question paper:

**Paragraph for Question Nos. 1 to 3**

A small spherical monoatomic ideal gas bubble (γ = 5/3) is trapped inside a liquid of density *ρ*_{ℓ} (see figure). Assume that the bubble does not exchange any heat with the liquid. The bubble contains *n* moles of gas. The *T*_{0}, the height of the liquid is *H* and the atmospheric pressure is *P*_{0} (Neglect surface tension).

**1. **As the bubble moves upwards, besides the buoyancy force the following forces are acting on it:

(A) Only the force of gravity

(B) The force due to gravity and the force due to the pressure of the liquid

(C) The force due to gravity, the force due to the pressure of the liquid and the force due to viscosity of the liquid

(D) The force due to gravity and the force due to viscosity of the liquid

The force due to the pressure of the liquid is the buoyancy force. Therefore, besides the buoyancy force the forces acting on the bubble are the force due to gravity and the force due to viscosity of the liquid [Option (D)].

**2.** When the gas bubble is at height *y* from the bottom, its

(A) *T*_{0}[(*P*_{0 }+ *ρ*_{ℓ}* gH*) /(*P*_{0 }+ *ρ*_{ℓ}* gy*)]^{2/5}

(B) *T*_{0}[{*P*_{0 }+ *ρ*_{ℓ}* g*(*H *–*y*)}/(*P*_{0 }+ *ρ*_{ℓ}* gH*)]^{2/5}

(C) *T*_{0}[(*P*_{0 }+ *ρ*_{ℓ}* gH*) /(*P*_{0 }+ *ρ*_{ℓ}* gy*)]^{3/5}

(D) *T*_{0}[{*P*_{0 }+ *ρ*_{ℓ}* g*(*H *–*y*)}/(*P*_{0 }+ *ρ*_{ℓ}* gH*)]^{3/5}

Since there is no heat exchange, the process is adiabatic for which

*T P*^{[(1-}^{ }** ^{γ)/ γ]}** = constant where

*γ*is the ratio of specific heats.

[Many of you might be remembering this as *T*^{ γ}* **P*^{(1-}^{ }^{γ)} = constant].

At the bottom pressure = *P*_{0 }+ *ρ*_{ℓ}* gH* and *T*_{0}

At height *y* from the bottom pressure = *P*_{0 }+ *ρ*_{ℓ}* g*(*H *–*y*) and *T* (let us say)

Therefore, *T*_{0}(*P*_{0 }+ *ρ*_{ℓ}* gH*)^{–2/5 }= *T*[*P*_{0 }+_{ }*ρ*_{ℓ}* g*(*H *–*y*)]^{–2/5} so that

*T =** T*_{0}[(*P*_{0 }+ *ρ*_{ℓ}* gH*) /{*P*_{0 }+ *ρ*_{ℓ}* g*(*H *–*y*)}]^{–2/5}

Or, *T =** T*_{0} [{*P*_{0 }+ *ρ*_{ℓ}* g*(*H *–*y*)}/(*P*_{0 }+ _{ }*ρ*_{ℓ}* g*(*H *)]^{2/5}

So option (B) is correct.

**3.** The buoyancy force on the gas bubble is (Assume R is the universal gas constant)

(A) *ρ*_{ℓ}* nRgT*_{0 }[(*P*_{0 }+ *ρ*_{ℓ}* gH*)^{ 2/5}/(*P*_{0 }+ *ρ*_{ℓ}* gy*)^{ 7/5}]

(B) *ρ*_{ℓ}* nRgT*_{0} /[(*P*_{0 }+ *ρ*_{ℓ}* gH*)^{ 2/5}{*P*_{0 }+ *ρ*_{ℓ}* g*(*H *–*y*)}^{3/5}]

(C) *ρ*_{ℓ}* nRgT*_{0 }[(*P*_{0 }+ *ρ*_{ℓ}* gH*)^{ 3/5}/(*P*_{0 }+ *ρ*_{ℓ}* gy*)^{ 8/5}]

(D) *ρ*_{ℓ}* nRgT*_{0} /[(*P*_{0 }+ *ρ*_{ℓ}* gH*)^{ 3/5}{*P*_{0 }+ *ρ*_{ℓ}* g*(*H *–*y*)}^{2/5}]^{}

Force of buoyancy, F= *Vρ*_{ℓ }*g*

But *V = nRT/P *so that F = *nRT**ρ*_{ℓ }*g /P *

Here *P = **P*_{0 }+ *ρ*_{ℓ}* g*(*H *–*y*) and *T =* *T*_{0 }[{*P*_{0 }+ *ρ*_{ℓ}* g*(*H *–*y*)}/(*P*_{0 }+ _{ }*ρ*_{ℓ}* g*(*H *)]^{2/5}

Substituting, F = *nR**ρ*_{ℓ }*g T*_{0 }[{*P*_{0 }+ *ρ*_{ℓ}* g*(*H *–*y*)}/(*P*_{0 }+ _{ }*ρ*_{ℓ}* g*(*H *)]^{2/5}/* *[*P*_{0 }+ *ρ*_{ℓ}* g*(*H *–*y*)]

Or, F = *ρ*_{ℓ}* nRgT*_{0} /[(*P*_{0 }+ *ρ*_{ℓ}* gH*)^{ 2/5}{*P*_{0 }+ *ρ*_{ℓ}* g*(*H *–*y*)}^{3/5}]

So option (B) is correct.

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