A soap bubble of radius `r` is inflated with an ideal gas. The atmospheric pressure is `p_0`, the surface tension of the soap water solution is `alpha`. Find the difference between the molar heat capacity of the gas during its heating inside the bubble and the molar heat capacity of the gas under constant pressure, `C - C_p`.

Explanation

When heat is given to a soap bubble the temperature of the air inside rises and the bubble expands but unless the bubble bursts, the amount of air inside does not change. Further we shall neglect the variation of the surface tension with temperature. Then from the gas equations
`(p_0 + (4 alpha)/(r))(4 pi)/(3) r^3 = v R T, v = Constant`
Differentiating
`(p_0 + (8 alpha)/(3 r))4 pi r^2 dr = v R dT`
or `dV = 4 pi r^2 dr = (vRdT)/(p_0 + (8 alpha)/(3 r))`
Now from the first law
`₫ Q = v CdT = v C_V dT + (v R dT)/(p_0 + (8 alpha)/(3 r)).(p_0 + (4 alpha)/( r))`
or `C = C_V + R (p_0 + (4 alpha)/(r))/(p_0 (8 alpha)/(3 r))`
using `C_p = C_V + R, C = C_p + ((1)/(2) R)/(1 + (3 p_0 r)/(8 alpha))`.