A bubble has surface tension S. The ideal gas inside the bubble has ratio of specific heats $\gamma =\frac{5}{3}$. The bubble is exposed to the atmosphere and it always retains its spherical shape. When the atmospheric pressure is $P_{a_1}$, the radius of the bubble is found to be r₁ and the temperature of the enclosed gas is T₁. When the atmospheric pressure is $P_{a_2}$, the radius of the bubble and the temperature of the enclosed gas are r₂ and T₂, respectively. Which of the following statement(s) is (are) correct ?

The correct statements are:
1. If the surface of the bubble is a perfect heat conductor and the change in atmospheric temperature is negligible, then ${\left(\frac{r_1}{r_2}\right)}^3=\frac{P_{a_2}+\frac{4S}{r_2}}{P_{a_1}+\frac{4S}{r_1}}$
2. If the surface of the bubble is perfect heat insulator, then ${(\frac{T_2}{T_1}}^{\frac{5}{2}}=\frac{P_{a_2}+\frac{4S}{r_2}}{P_{a_1}+\frac{4S}{r_1}}$

Step 1: Pressure inside the bubble
A spherical bubble has two free surfaces (inner and outer), so the excess pressure inside the bubble due to surface tension S and radius r is given by:
$\Delta P=\frac{4S}{r}$
Therefore, the total absolute pressure P of the gas inside the bubble when the atmospheric pressure is P_a is:
$P=P_a+\frac{4S}{r}$

For state 1:
$P_1=P_{a_1}+\frac{4S}{r_1}$

For state 2:
$P_2=P_{a_2}+\frac{4S}{r_2}$

Step 2: Analysis when the bubble surface is a perfect heat conductor (Isothermal Process)
If the surface is a perfect heat conductor and the ambient atmospheric temperature remains constant, heat exchange occurs instantaneously, keeping the temperature of the gas inside the bubble constant (T₁ = T₂).

Using Boyle's Law for an ideal gas at constant temperature:
$P_1{V}_1=P_2{V}_2$

Substituting the volume of a sphere $V=\frac{4}{3}\pi r^3$:
$(P_{a_1}+\frac{4S}{r_1})\cdot \frac{4}{3}\pi r_1^3=(P_{a_2}+\frac{4S}{r_2})\cdot \frac{4}{3}\pi r_2^3$

Simplifying the above equation:
${\left(\frac{r_1}{r_2}\right)}^3=\frac{P_{a_2}+\frac{4S}{r_2}}{P_{a_1}+\frac{4S}{r_1}}$
Thus, this statement is correct.

Step 3: Analysis when the bubble surface is a perfect heat insulator (Adiabatic Process)
If the surface is a perfect heat insulator, no heat is exchanged between the enclosed gas and the environment (Q = 0). Thus, the process is adiabatic.

For an adiabatic process of an ideal gas with ratio of specific heats $\gamma =\frac{5}{3}$, the relationship between pressure P and temperature T is given by:
$T^{\gamma }{P}^{1-\gamma }=\text{constant}$

Substituting $\gamma =\frac{5}{3}$ into the equation:
$T^{\frac{5}{3}}{P}^{1-\frac{5}{3}}=T^{\frac{5}{3}}{P}^{-\frac{2}{3}}=\text{constant}$

Taking both sides to the power of $\frac{3}{2}$ gives:
$\frac{T^{\frac{5}{2}}}{P}=\text{constant}$

Equating the initial and final states:
$\frac{T_1^{\frac{5}{2}}}{P_1}=\frac{T_2^{\frac{5}{2}}}{P_2}$

Rearranging terms to find the ratio of temperatures:
${\left(\frac{T_2}{T_1}\right)}^{\frac{5}{2}}=\frac{P_2}{P_1}=\frac{P_{a_2}+\frac{4S}{r_2}}{P_{a_1}+\frac{4S}{r_1}}$
Thus, this statement is also correct.