JEE Advanced · Physics · 13. Thermodynamics

A bubble has surface tension $S$ . The ideal gas inside the bubble has ratio of specific heats $γ = \frac{5}{3}$ . 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 $r_{1}$ and the temperature of the enclosed gas is $T_{1}$ . When the atmospheric pressure is $P_{a 2}$ , the radius of the bubble and the temperature of the enclosed gas are $r_{2}$ and $T_{2}$ , respectively. Which of the following statement(s) is(are) correct?

A If the surface of the bubble is a perfect heat insulator, then ${(\frac{r_{1}}{r_{2}})}^{5} = \frac{P_{a 2} + \frac{2 S}{r_{2}}}{P_{a 1} + \frac{2 S}{r_{1}}}$
B 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.
C If the surface of the bubble is a perfect heat conductor and the change in atmospheric temperature is negligible, the ${(\frac{r_{1}}{r_{2}})}^{3} = \frac{P_{a 2} + \frac{4 S}{r_{2}}}{P_{a 1} + \frac{4 S}{r_{1}}}$
D If the surface of the bubble is a perfect heat insulator, then ${(\frac{T_{2}}{T_{1}})}^{\frac{5}{2}} = \frac{P_{a 2} + \frac{4 S}{r_{2}}}{P_{a 1} + \frac{4 S}{r_{1}}}$

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Answer & Solution

Correct Answer

(D) If the surface of the bubble is a perfect heat insulator, then ${(\frac{T_{2}}{T_{1}})}^{\frac{5}{2}} = \frac{P_{a 2} + \frac{4 S}{r_{2}}}{P_{a 1} + \frac{4 S}{r_{1}}}$

Step-by-step Solution

Detailed explanation

When,

\begin{matrix} Pressure & Radius & Temperature \\ P_{a 1} & \to & r_{1} & \to & T_{1} \\ P_{a 2} & \to & r_{2} & \to & T_{2} \end{matrix}

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

Same subject

Match the temperature of a black body given in List-

I

with an appropriate statement in List-

II

, and choose the correct option.
[Given: Wien’s constant as

2.9 \times 10^{- 3} m - K

and

\frac{h c}{e} = 1.24 \times 10^{- 6} V - m

]

	List- $I$		List- $II$
$P$	$2000 K$	$1$	The radiation at peak wavelength can lead to emission of photoelectrons from a metal of work function $4 eV$
$Q$	$3000 K$	$2$	The radiation at peak wavelength is visible to human eye
$R$	$5000 K$	$3$	The radiation at peak emission wavelength will result in the widest central maximum of a single slit diffraction
$S$	$10000 K$	$4$	The power emitted per unit area is $\frac{1}{16}$ of that emitted by a blackbody at temperature $6000 K$
		$5$	The radiation at peak emission wavelength can be used to image human bones

JEE Advanced 2023 Medium

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From JEE Advanced

A right-angled prism of refractive index

μ_{1}

is placed in a rectangular block of refractive index

μ_{2}

, which is surrounded by a medium of refractive index

μ_{3}

, as shown in the figure. A ray of light

e

enters the rectangular block at normal incidence. Depending upon the relationships between

μ_{1}, μ_{2}

and

μ_{3}

, it takes one of the four possible paths

e f, e g, e h

e i

Match the paths in List

I

with conditions of refractive indices in List

II

and select the correct answer using the codes given below the lists:

	List I		List II
A.	$e \to f$	P.	$μ_{1} > \sqrt{2} μ_{2}$
B.	$e \to g$	Q.	$μ_{2} > μ_{1} and μ_{2} > μ_{3}$
C.	$e \to h$	R.	$μ_{1} = μ_{2}$
D.	$e \to i$	S.	$μ_{2} < μ_{1} < \sqrt{2} μ_{2} and μ_{2} > μ_{3}$

JEE Advanced 2013 Medium

Match the gases under specified conditions listed in Column I with their properties/laws in Column II. Indicate your answer by darkening the appropriate bubbles of $4 \times 4$ matrix given in the ORS.

Column I	Column II
A. Hydrogen gas $(P=200 atm,$ $T=273 K)$	p. Compressibility factor $\neq 1$
B. Hydrogen gas $(P \sim 0,$ $T=273 K)$	q. Attractive forces are dominant
C. $CO _2(P=1 atm,$ $T=273 K)$	r. $P V=n R T$
D. Real gas with very large molar volume	s. $P(V-n b)=n R T$

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Column I	Column II
A. Hydrogen gas \((P=200 atm,\) \(T=273 K)\)	p. Compressibility factor \(\neq 1\)
B. Hydrogen gas \((P \sim 0,\) \(T=273 K)\)	q. Attractive forces are dominant
C. \(CO _2(P=1 atm,\) \(T=273 K)\)	r. \(P V=n R T\)
D. Real gas with very large molar volume	s. \(P(V-n b)=n R T\)