Step-by-step Derivations

1. Understand the Physical Principles

   • The pressure inside a soap bubble is greater than the pressure outside due to surface tension. A soap bubble has two surfaces (inner and outer), so the excess pressure inside is given by the formula: $\Delta P = \frac{4T}{r}$ where $T$ is the surface tension and $r$ is the radius of the bubble.
   • The total pressure inside the bubble ($P_{in}$) is the sum of the outside pressure ($P_{out}$) and the excess pressure (\\[\Delta P\\]): $P_{in} = P_{out} + \Delta P$ In this problem, the outside pressure is the chamber pressure, $P_0 = 8 \text{ N/m}^2$.
   • The air inside the bubble is assumed to behave as an ideal gas, following the Ideal Gas Law: $PV = nRT$ where $P$ is the pressure, $V$ is the volume, $n$ is the number of moles, $R$ is the universal gas constant, and $T$ is the absolute temperature. Since both bubbles are in the same chamber, the temperature $T$ is the same for both.

2. Calculate the Pressure inside Bubble A

   • Given: Radius of bubble A, $r_A = 2 \text{ cm} = 0.02 \text{ m}$.
   • Surface tension, $T = 0.04 \text{ N/m}$.
   • Excess pressure in bubble A: $\Delta P_A = \frac{4T}{r_A} = \frac{4 \times 0.04}{0.02} = \frac{0.16}{0.02} = 8 \text{ N/m}^2$
   • Total pressure inside bubble A: $P_A = P_0 + \Delta P_A = 8 \text{ N/m}^2 + 8 \text{ N/m}^2 = 16 \text{ N/m}^2$

3. Calculate the Pressure inside Bubble B

   • Given: Radius of bubble B, $r_B = 4 \text{ cm} = 0.04 \text{ m}$.
   • Excess pressure in bubble B: $\Delta P_B = \frac{4T}{r_B} = \frac{4 \times 0.04}{0.04} = 4 \text{ N/m}^2$
   • Total pressure inside bubble B: $P_B = P_0 + \Delta P_B = 8 \text{ N/m}^2 + 4 \text{ N/m}^2 = 12 \text{ N/m}^2$

4. Apply the Ideal Gas Law to find the Ratio of Moles

   • For bubble A: $P_A V_A = n_A R T$
   • For bubble B: $P_B V_B = n_B R T$
   • To find the ratio $n_B/n_A$, we divide the second equation by the first: $\frac{P_B V_B}{P_A V_A} = \frac{n_B R T}{n_A R T}$ $\frac{n_B}{n_A} = \frac{P_B V_B}{P_A V_A}$

5. Calculate the Ratio of Volumes

   • The volume of a spherical bubble is $V = \frac{4}{3} \pi r^3$.
   • The ratio of volumes is: $\frac{V_B}{V_A} = \frac{\frac{4}{3} \pi r_B^3}{\frac{4}{3} \pi r_A^3} = \left(\frac{r_B}{r_A}\right)^3 = \left(\frac{0.04}{0.02}\right)^3 = (2)^3 = 8$

6. Calculate the Final Ratio $n_B/n_A$

   • Substitute the values of pressures and the volume ratio into the equation for the ratio of moles: $\frac{n_B}{n_A} = \frac{P_B}{P_A} \times \frac{V_B}{V_A} = \frac{12}{16} \times 8$ $\frac{n_B}{n_A} = \frac{3}{4} \times 8 = 3 \times 2 = 6$

Thus, the ratio $n_B/n_A$ is 6.