A soap bubble is a system consisting of two subsystems. Subsystem (f) is the thin film and subsystem (g) is the gas enclosed inside the film. The surrounding air is a thermal bath. The equilibrium is characterised by the minimum of the free energy F of the system. The differential of the free

Question

A soap bubble is a system consisting of two subsystems. Subsystem (f) is the thin film and subsystem (g) is the gas enclosed inside the film. The surrounding air is a thermal bath. The equilibrium is characterised by the minimum of the free energy F of the system. The differential of the free energy dF reads,

[latex] dF = − (S_g + S_f) dT + 2 γ dA − (p − p_0) dV [/latex].

where A is the surface area of the soap film and V the volume of the bubble. The parameter γ is called the surface tension. It characterises the interactions at the interface between the liquid and the air. Since the soap film has two such interfaces, there is a factor 2 in front of the parameter γ. The surface tension γ is an intensive variable that plays an analogous role for a surfacic system as the pressure p for a volumic system. However, the force due to pressure of a gas is exerted outwards whereas the force due to the surface tension is exerted inwards. This is the reason why the signs of the corresponding two terms in dF differ. The term [latex]p − p_0[/latex] is the pressure difference between the pressure p inside the bubble and the atmospheric pressure [latex] p_0[/latex]. Consider the bubble to be a sphere of radius r and show that,

[latex] p − p_0 = \frac{4 γ}{r} [/latex].

Accepted Answer

Since the surrounding air is a thermal bath, the temperature is constant, i.e. dT = 0. For a spherical bubble, the area differential is given by,

[latex] dA = 4 π (r + dr)^2 − 4 π r^2 = 4 π (2 rdr + dr^2)≈ 8π r dr [/latex].

where we neglect the second-order term in [latex]dr^2[/latex]. The volume differential is given by,

[latex] dV = \frac{4π}{3} (r + dr)^3 − \frac{4π}{3} r^3 = \frac{4π}{3} (3 r^2 dr + 3 r dr^2 + dr^3) ≈ 4π r^2d [/latex].

where we neglect the second-order term in [latex] dr^2[/latex] and the third-order term in [latex] dr^3[/latex]. At equilibrium, the free energy F is minimum. Thus,

[latex] dF = 16π γ r dr − 4π (p − p_0) r^2dr = 0[/latex].

which implies that the pressure difference is given by,

[latex] p − p_0 = \frac{4γ}{r} [/latex].

Question 4.7: A soap bubble is a system consisting of two subsystems. Subs...

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