MHT CET202616 April 2026Evening ShiftPhysicsMechanical Properties of FluidsActual
Under isothermal condition, two soap bubbles of radii r₁ and r₂ combine to form a single soap bubble of radius R. The surface tension of the soap solution is (P = outside pressure)
Options
- AP(R^3 - r₁^3 - r₂^3) 4(r₁^2 + r₂^2 - R^2)
- BP(R^3 + r₁^3 + r₂^3) 2(r₁^2 - r₂^2 + R^2)
- CP(r₁^3 - r₂^3 - R^3) (R^2 + r^2 + r^2)
- DP(R^3 - r₁^3 + r₂^3) 2(r₁^2 + r₂^2 - R^2)
Correct answer
A. P(R^3 - r₁^3 - r₂^3) 4(r₁^2 + r₂^2 - R^2)
Step-by-step solution
Let T be the surface tension of the soap solution. The pressure inside a soap bubble of radius r is given by P_i = P + 4T r , where P is the outside pressure. The volume of the bubble is V = 4 3 r^3 . Under isothermal conditions, the temperature remains constant. The total number of moles of air inside the bubbles is conserved, which gives P₁ V₁ + P₂ V₂ = P_f V_f . Substituting the values for the two initial bubbles and the final bubble: (P + 4T r₁ ) 4 3 r₁^3 + (P + 4T r₂ ) 4 3 r₂^3 = (P + 4T R ) 4 3 R^3 Dividing t