The excess pressure inside a soap bubble is thrice the excess pressure inside a second soap bubble. The ratio between the volume of the first and the second bubble is:

<p>To find the ratio between the volumes of the first and the second soap bubble, we need to understand the relation between the excess pressure inside a soap bubble and its volume.</p> <p>The excess pressure ($P$) inside a soap bubble is given by the formula:</p> <p>$P = \frac{4T}{r}$</p> <p>where $T$ is the surface tension of the soap solution, and $r$ is the radius of the soap bubble. For a soap bubble, the factor of 4 comes from having two surfaces (inner and outer), each contributing $2T/r$ to the pressure.</p> <p>Given that the excess pressure inside the first soap bubble ($P_1$) is thrice the excess pressure inside the second soap bubble ($P_2$), we can write:</p> <p>$P_1 = 3P_2$</p> <p>Substituting the formula for excess pressure, we get:</p> <p>$$\frac{4T}{r_1} = 3 \times \frac{4T}{r_2} \Rightarrow \frac{1}{r_1} = 3 \times \frac{1}{r_2} \Rightarrow \frac{r_2}{r_1} = 3$$</p> <p>Next, we calculate the ratio between their volumes. The volume ($V$) of a sphere (or a bubble) is given by:</p> <p>$V = \frac{4}{3}\pi r^3$</p> <p>Thus, the volume ratio of the first bubble ($V_1$) to the second bubble ($V_2$) is:</p> <p>$$\frac{V_1}{V_2} = \frac{\frac{4}{3}\pi r_1^3}{\frac{4}{3}\pi r_2^3} = \left(\frac{r_1}{r_2}\right)^3$$</p> <p>Since we found that $\frac{r_2}{r_1} = 3$, it then follows that:</p> <p>$\frac{V_1}{V_2} = \left(\frac{1}{3}\right)^3 = \frac{1}{27}$</p> <p>Therefore, the correct option is:</p> <p>Option B: $1: 27$</p>