The radii of two soap bubbles are r_1 and r_2 | vedclass

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(C) Under isothermal conditions,the temperature $T$ remains constant. Since the process is isothermal,the total number of moles of air inside the bubb

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$R=\sqrt{r_1^2+r_2^2}$

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(C) Under isothermal conditions,the temperature $T$ remains constant. Since the process is isothermal,the total number of moles of air inside the bubbles remains constant. Assuming the pressure inside a soap bubble of radius $r$ is $P = P_0 + \frac{4S}{r}$,where $P_0$ is atmospheric pressure and $S$ is surface tension. For small bubbles,$P \approx \frac{4S}{r}$. Using $PV = nRT$,for constant $T$ and $n$,$PV$ is constant. $P_1 V_1 + P_2 V_2 = P_R V_R$ $\left(\frac{4S}{r_1}\right) \left(\frac{4}{3} \pi r_1^3\right) + \left(\frac{4S}{r_2}\right) \left(\frac{4}{3} \pi r_2^3\right) = \left(\frac{4S}{R}\right) \left(\frac{4}{3} \pi R^3\right)$ $\frac{16}{3} \pi S r_1^2 + \frac{16}{3} \pi S r_2^2 = \frac{16}{3} \pi S R^2$ $r_1^2 + r_2^2 = R^2$ $R = \sqrt{r_1^2 + r_2^2}$

The radii of two soap bubbles are $r_1$ and $r_2$. In isothermal condition,they combine with each other to form a single bubble. The radius of the resultant bubble is

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