Question

Two soap bubbles of radii a and b combine to form a single bubble of radius c. If P is the external pressure then surface tension of the soap bubble is:

A. $\dfrac{P({{c}^{3}}+{{a}^{3}}+{{b}^{3}})}{4({{a}^{2}}+{{b}^{2}}+{{c}^{2}})}$

B. $\dfrac{P({{c}^{3}}-{{a}^{3}}-{{b}^{3}})}{4({{a}^{2}}+{{b}^{2}}-{{c}^{2}})}$

C. $P{{c}^{3}}-4{{a}^{2}}-4{{b}^{2}}$

D. $P{{c}^{3}}-2{{a}^{2}}+3{{b}^{2}}$

Answer 1

Hint: Use ideal gas law for the combination of two soap bubbles and the formula for surface tension to calculate the surface tension of the soap bubble. Get the pressure inside each soap bubble as a function of surface tension and use the gas law for the combination of two bubbles to get surface tension as a function of P, c, a, b. 

Step by step solution:

We have the ideal gas law given by $PV = nRT$

where $P$ is the internal pressure of the gas and $V$ is the volume of the gas n is the number of moles is the absolute temperature and $R$ is a constant.

For the combination of two soap Bubbles we have $P_1 V_1 + P_2 V_2 = P_f V_f$

Where the $P_1$, $P_2$ are internal pressure of the two gas bubbles and $V_f$ is the internal pressure of the big bubble which is the combination of the two bubbles

We know the surface tension for the soap bubble can be calculated as:

$P_{in}-P_{ext} = \dfrac{4T}{r}$ 

where $r$ is the radius of the bubble and the $T$ is the surface tension of the bubble.

We are given external pressure as $P_{ext} = P$

So we get for the combination of the two bubbles

$(P+ \dfrac{4T}{a})\dfrac{4}{3}\pi a^3 +(P+ \dfrac{4T}{b})\dfrac{4}{3}\pi b^3 = (P+ \dfrac{4T}{c})\dfrac{4}{3}\pi c^3$

$P(a^3+b^3-c^3) = 4T(c^2-a^2-b^2)$

$T = \dfrac{P(c^3 -a^3 -b^3)}{4(a^2+b^2-c^2)}$

In conclusion, we have used the ideal gas law for the combination of two soap Bubbles and we have used the expression for the surface tension of a soap bubble to calculate the surface tension in terms of the external pressure acting on the soap bubbles.

Hence, the correct answer is option B.

Additional information: We know ideal gas law is given by $PV = nRT$. When two soap bubbles combine, the number of moles in the final gas bubble will be the sum of the number of moles in the individual gas bubbles. And at the same time, we get

$n_1 + n_2 = n_f$,

$\Rightarrow \dfrac{P_1 V_1}{RT} + \dfrac{P_2 V_2}{RT} = \dfrac{P_f V_f}{RT}$

$\Rightarrow  P_1 V_1 + P_2 V_2 = P_f V_f$

Thus we derived expression used the combination of two soap Bubbles in the above problem

Note: The most common mistake that can be done in this kind of problem is that one may take the external pressure as the value of p in the ideal gas law. one should be clear about the terms in the gas law. P in the ideal gas refers to the internal pressure in case of the soap bubble.