arthham

JEE Advanced 2022 · Thermodynamics

A bubble has surface tension SS. The ideal gas inside the bubble has ratio of specific heats γ=53\gamma = \frac{5}{3}. The bubble is exposed to the atmosphere and it always retains its spherical shape. When the atmospheric pressure is Pa1P_{a1}, the radius of the bubble is found to be r1r_1 and the temperature of the enclosed gas is T1T_1. When the atmospheric pressure is Pa2P_{a2}, the radius of the bubble and the temperature of the enclosed gas are r1r_1 and T2T_2, respectively. Which of the following statement(s) is(are) correct?

Answers: C. If the surface of the bubble is a perfect heat conductor and the change in atmospheric temperature is negligible, then (r1r2)3=Pa2+4Sr2Pa1+4Sr1\left(\frac{r_1}{r_2}\right)^3 = \frac{P_{a2} + \frac{4S}{r_2}}{P_{a1} + \frac{4S}{r_1}}. · D. If the surface of the bubble is a perfect heat insulator, then (T2T1)52=Pa2+4Sr2Pa1+4Sr1\left(\frac{T_2}{T_1}\right)^{\frac{5}{2}} = \frac{P_{a2} + \frac{4S}{r_2}}{P_{a1} + \frac{4S}{r_1}}.

Watch this solved on the board The full walkthrough, written out a line at a time with narration explaining each step as it appears. Pause anywhere and ask why a line is there. Free to watch

Or read the worked solution below. Jump to the steps ↓

Written solution

Step 1: Determine the pressure inside the soap bubble.

A soap bubble has two surfaces (inner and outer), so the excess pressure is 4Sr\frac{4S}{r}.

Pin=Pa+4SrP_{\text{in}} = P_a + \frac{4S}{r}

Step 2: Analyze the isothermal case (perfect heat conductor).

For a perfect heat conductor, the process is isothermal, meaning the temperature remains constant.

PinV=constantP_{\text{in}} V = \text{constant}

Since the volume of a spherical bubble is V=43πr3V = \frac{4}{3}\pi r^3, we have:

(Pa+4Sr)r3=constant\left(P_a + \frac{4S}{r}\right) r^3 = \text{constant}

Applying this relation to states 1 and 2:

(Pa1+4Sr1)r13=(Pa2+4Sr2)r23\left(P_{a1} + \frac{4S}{r_1}\right) r_1^3 = \left(P_{a2} + \frac{4S}{r_2}\right) r_2^3 (r1r2)3=Pa2+4Sr2Pa1+4Sr1\left(\frac{r_1}{r_2}\right)^3 = \frac{P_{a2} + \frac{4S}{r_2}}{P_{a1} + \frac{4S}{r_1}}

This shows that Statement 3 is correct.

Step 3: Analyze the adiabatic case (perfect heat insulator).

For a perfect heat insulator, the process is adiabatic.

PinVγ=constantP_{\text{in}} V^{\gamma} = \text{constant}

Given γ=53\gamma = \frac{5}{3} and V∝r3V \propto r^3, the volume term becomes:

Vγ∝(r3)5/3=r5V^{\gamma} \propto (r^3)^{5/3} = r^5

Thus, the pressure relation is:

(Pa+4Sr)r5=constant\left(P_a + \frac{4S}{r}\right) r^5 = \text{constant}

Applying this to states 1 and 2:

(r1r2)5=Pa2+4Sr2Pa1+4Sr1\left(\frac{r_1}{r_2}\right)^5 = \frac{P_{a2} + \frac{4S}{r_2}}{P_{a1} + \frac{4S}{r_1}}

Statement 1 uses 2Sr\frac{2S}{r} instead of 4Sr\frac{4S}{r}, which makes Statement 1 incorrect.

Step 4: Determine the temperature relation in the adiabatic case.

For an adiabatic process, the temperature and volume are related by:

TVγ−1=constantT V^{\gamma - 1} = \text{constant}

Using γ−1=53−1=23\gamma - 1 = \frac{5}{3} - 1 = \frac{2}{3} and V∝r3V \propto r^3:

T(r3)2/3=Tr2=constantT (r^3)^{2/3} = T r^2 = \text{constant} T1r12=T2r22T_1 r_1^2 = T_2 r_2^2 T2T1=(r1r2)2\frac{T_2}{T_1} = \left(\frac{r_1}{r_2}\right)^2

Raising both sides to the power of 52\frac{5}{2}:

(T2T1)5/2=(r1r2)5\left(\frac{T_2}{T_1}\right)^{5/2} = \left(\frac{r_1}{r_2}\right)^5

Substituting the pressure relation from Step 3:

(T2T1)5/2=Pa2+4Sr2Pa1+4Sr1\left(\frac{T_2}{T_1}\right)^{5/2} = \frac{P_{a2} + \frac{4S}{r_2}}{P_{a1} + \frac{4S}{r_1}}

This shows that Statement 4 is correct.

Step 5: Analyze the total internal energy in the adiabatic case.

By the first law of thermodynamics, the change in internal energy is:

ΔU=Q−W\Delta U = Q - W

Since the bubble is insulated, Q=0Q = 0, but it performs work against the external atmospheric pressure:

ΔU=−PaΔV≠0\Delta U = -P_a \Delta V \neq 0

Since the volume changes, the total internal energy is not constant, making Statement 2 incorrect.

Answer: Correct statements are (3) and (4).

More from Thermodynamics

All 26 Thermodynamics questions →