Question 37: Properties of Matter

Question

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:

Accepted Answer

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.
The excess pressure ($P$) inside a soap bubble is given by the formula:
$$P = \frac{4T}{r}$$
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.
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:
$$P_1 = 3P_2$$
Substituting the formula for excess pressure, we get:
$$\frac{4T}{r_1} = 3 	imes \frac{4T}{r_2} \Rightarrow \frac{1}{r_1} = 3 	imes \frac{1}{r_2} \Rightarrow \frac{r_2}{r_1} = 3$$
Next, we calculate the ratio between their volumes. The volume ($V$) of a sphere (or a bubble) is given by:
$$V = \frac{4}{3}\pi r^3$$
Thus, the volume ratio of the first bubble ($V_1$) to the second bubble ($V_2$) is:
$$\frac{V_1}{V_2} = \frac{\frac{4}{3}\pi r_1^3}{\frac{4}{3}\pi r_2^3} = \left(\frac{r_1}{r_2}ight)^3$$
Since we found that $\frac{r_2}{r_1} = 3$, it then follows that:
$$\frac{V_1}{V_2} = \left(\frac{1}{3}ight)^3 = \frac{1}{27}$$
Therefore, the correct option is:
Option B: $$1: 27$$

Answer

$$1: 9$$

Answer

$$1: 27$$

Answer

$$1: 81$$

Answer

$$1: 3$$

Properties of Matter

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:

Choose the correct answer: