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__________.

<p>The excess pressure inside a soap bubble is determined by the formula:</p> <p>$ \Delta \mathrm{P} = \frac{4 \mathrm{~T}}{\mathrm{R}} $</p> <p>where $ \Delta \mathrm{P} $ is the excess pressure, $ \mathrm{T} $ is the surface tension of the soap film, and $ \mathrm{R} $ is the radius of the bubble.</p> <p>Given that the excess pressure inside bubble A is half that inside bubble B, we have:</p> <p>$ \Delta \mathrm{P}_{\mathrm{A}} = \frac{1}{2} \Delta \mathrm{P}_{\mathrm{B}} $</p> <p>This implies:</p> <p>$ \frac{R_{\mathrm{A}}}{R_{\mathrm{B}}} = \frac{\Delta \mathrm{P}_{\mathrm{B}}}{\Delta \mathrm{P}_{\mathrm{A}}} = 2 $</p> <p>Now, considering the volumes of the bubbles, the relationship between volume and radius for a sphere is given by:</p> <p>$ V = \frac{4}{3} \pi R^3 $</p> <p>Thus, the ratio of the volumes of bubbles A and B is:</p> <p>$ \frac{V_{\mathrm{A}}}{V_{\mathrm{B}}} = \left( \frac{R_{\mathrm{A}}}{R_{\mathrm{B}}} \right)^3 = 2^3 = 8 $</p> <p>Therefore, the volume of bubble A is $ n = 8 $ times the volume of bubble B.</p>