A bubble has surface tension S . The ideal gas inside the bubble has ratio of specific heats γ = 5 3 .…

A bubble has surface tension S. The ideal gas inside the bubble has ratio of specific heats γ=53. The bubble is exposed to the atmosphere and it always retains its spherical shape. When the atmospheric pressure is Pa1, the radius of the bubble is found to be r1 and the temperature of the enclosed gas is T1. When the atmospheric pressure is Pa2, 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?
  1. If the surface of the bubble is a perfect heat insulator, then r1r25=Pa2+2Sr2Pa1+2Sr1
  2. If the surface of the bubble is a perfect heat insulator, then the total internal energy of the bubble including its surface energy does not change with the external atmospheric pressure.
  3. If the surface of the bubble is a perfect heat conductor and the change in atmospheric temperature is negligible, the r1r23=Pa2+4Sr2Pa1+4Sr1
  4. If the surface of the bubble is a perfect heat insulator, then T2T152=Pa2+4Sr2Pa1+4Sr1

Solution

When, 

PressureRadiusTemperaturePa1→r1→T1Pa2→r2→T2

For adiabatic process

P1V1γ=P2V2γ

Pa1+4Tr143πr1353=Pa2+4Tr243πr2353

r1r25=Pa2+4Tr2Pa1+4Tr1

Therefore, option A is incorrect.

Now,

T1V1γ-1=T2V2γ-1

T2T1=V1V2γ-1=r1r2323

T2T1=Pa2+4Tr2Pa1+4Tr125

Therefore, option D is correct.

For option B Total internal energy + surface energy will not be same as work done by gas will be there. Option B is incorrect.

For option C change in temperature is negligible. Therefore, the process is isothermal.

P1V1=P2V2

Pa1+4Tr143πr13=Pa2+4Tr243πr23

r1r23=Pa2+4Tr2Pa1+4Tr1

Option C is correct.

.

Asked in: JEE Advanced 2022 (Paper 2)

Practice more Thermodynamics questions on Aicharya