1. Excess pressure in a soap bubble

For a soap bubble, the excess pressure is $\Delta P = \frac{4T}{R}$ where $T$ is surface tension and $R$ is the radius of the bubble.

2. Given condition

Let bubble A have radius $R_A$ and bubble B have radius $R_B$.

Given: $\Delta P_A = \frac{1}{2}\Delta P_B$

Using the formula: $\frac{4T}{R_A} = \frac{1}{2}\cdot \frac{4T}{R_B}$

Cancel $4T$: $\frac{1}{R_A} = \frac{1}{2R_B}$

So, $R_A = 2R_B$

3. Relate volume of bubbles

Volume of a sphere is $V = \frac{4}{3}\pi R^3$

Hence, $\frac{V_A}{V_B} = \left(\frac{R_A}{R_B}\right)^3 = 2^3 = 8$

Therefore, $V_A = 8V_B$

So, the value of $n$ is: $\boxed{8}$