The excess pressure inside a soap bubble A in air is half the excess pressure inside another soap bubble B in air. If the volume of the bubble A is $$n$$ times the volume of the bubble B, then the value of n is ________.

For a soap bubble in air, the excess pressure inside it is given by $$\Delta P = \frac{4T}{r}$$ where $$T$$ is the surface tension of the soap film and $$r$$ is the radius of the bubble.

Let the radii of bubbles A and B be $$r_A$$ and $$r_B$$ respectively.
Then $$\Delta P_A = \frac{4T}{r_A}$$ $$\Delta P_B = \frac{4T}{r_B}$$

The statement “excess pressure inside bubble A is half the excess pressure inside bubble B” translates to $$\Delta P_A = \frac{1}{2}\,\Delta P_B$$ Substituting from above, $$\frac{4T}{r_A} = \frac{1}{2}\left(\frac{4T}{r_B}\right)$$

Simplifying (the factor $$4T$$ cancels out): $$\frac{1}{r_A} = \frac{1}{2}\,\frac{1}{r_B}$$ which gives $$r_B = \frac{1}{2}\,r_A$$ or equivalently $$r_A = 2\,r_B$$.

The volume of a spherical bubble is $$V = \frac{4}{3}\,\pi r^{3}$$. Therefore, the ratio of the volumes of bubbles A and B is $$\frac{V_A}{V_B} = \left(\frac{r_A}{r_B}\right)^3 = \left(2\right)^3 = 8$$.

Given that $$V_A = n\,V_B$$, we have $$n = 8$$.

Hence the required value of $$n$$ is $$8$$.