Pressure inside two soap bubbles are 1.01 and 1.02 atmosphere, respectively. The ratio of their volumes is:

Let us denote the atmospheric (external) pressure by $$P_{0}$$. For a thin soap bubble the excess pressure inside the bubble is given by the well-known formula

$$\Delta P \;=\; P_{\text{inside}} - P_{0} \;=\; \dfrac{4T}{r}$$

Here $$T$$ is the surface tension of the liquid film and $$r$$ is the radius of the spherical bubble. This expression contains the factor $$4T$$ (and not $$2T$$) because a soap bubble has two liquid surfaces in contact with air, one on the inside and one on the outside.

We are told that the absolute internal pressures of the two bubbles are

$$P_{1}=1.01\;\text{atm},\qquad P_{2}=1.02\;\text{atm}$$

and, naturally, both bubbles are in the same atmosphere so

$$P_{0}=1.00\;\text{atm}$$

Using the formula for each bubble, we first calculate their respective excess pressures:

For the first bubble $$\Delta P_{1}=P_{1}-P_{0}=1.01-1.00=0.01\;\text{atm}$$

For the second bubble $$\Delta P_{2}=P_{2}-P_{0}=1.02-1.00=0.02\;\text{atm}$$

Now we substitute these excess pressures into the relation $$\Delta P=\dfrac{4T}{r}$$ to connect each pressure to the corresponding radius.

For the first bubble $$\dfrac{4T}{r_{1}}=0.01\;\text{atm}\qquad\Longrightarrow\qquad r_{1}=\dfrac{4T}{0.01\;\text{atm}}$$

For the second bubble $$\dfrac{4T}{r_{2}}=0.02\;\text{atm}\qquad\Longrightarrow\qquad r_{2}=\dfrac{4T}{0.02\;\text{atm}}$$

To compare the radii, we take the ratio of the two expressions:

$$\dfrac{r_{2}}{r_{1}} =\dfrac{\dfrac{4T}{0.02}}{\dfrac{4T}{0.01}} =\dfrac{0.01}{0.02} =\dfrac{1}{2}$$

So we find $$r_{2}=\dfrac{r_{1}}{2}$$ meaning the second bubble (with the higher internal pressure) has half the radius of the first bubble.

Each bubble is spherical, so its volume is given by the standard geometric formula

$$V=\dfrac{4}{3}\pi r^{3}$$

Therefore, the ratio of their volumes becomes

$$\dfrac{V_{1}}{V_{2}} =\dfrac{\dfrac{4}{3}\pi r_{1}^{3}}{\dfrac{4}{3}\pi r_{2}^{3}} =\left(\dfrac{r_{1}}{r_{2}}\right)^{3} =\left(\dfrac{r_{1}}{\,r_{1}/2}\right)^{3} =(2)^{3}=8$$

Thus

$$V_{1}:V_{2}=8:1$$

The option that matches this ratio is Option C.

Hence, the correct answer is Option C.