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Relation between Cp and Cv

Consider a cl...

Question

Consider a closed system (cylinder) whose walls are adiabatic. The cylinder lies in a horizontal position & is divided into three parts $A_{1}$ , $A_{2}$ and $A_{3}$ by means of partions $S_{1}$ and $S_{2}$ which can move along the length of the cylinder without friction, $S_{1}$ is adiabatic while $S_{2}$ is conducting. Initially each of the parts $A_{1}$ , $A_{2}$ & $A_{3}$ contains one mole of He gas at pressure $P_{0}$ , temperature $T_{0}$ & volume $V_{0}$ .
( $C_{v} = \frac{3}{2} R, C_{p} = \frac{5}{2} R, γ = \frac{5}{3}$ , take He as an ideal mono atomic gas)
Now heat is supplied slowly to the gas in part $A_{1}$ , till the Temperature in part $A_{3}$ becomes $T_{3} = \frac{9 T_{0}}{4}$ .
Which statement(s) is/are correct?

P_{1}

P_{2}

P_{3}

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V_{1}

V_{2}

V_{3}

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T_{1}

T_{2}

T_{3}

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V_{1} = \frac{65}{27} V_{0}

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Solution

The correct option is D $V_{1} = \frac{65}{27} V_{0}$
Given,
Partitions are free to move. Hence pressure in all the parts are equal
Hence A is correct.
$P_{1}$ = $P_{2}$ = $P_{3}$

partion $S_{1}$ is not conducting heat. Hence heating will increase the temperature of $A_{1}$ and hence volume $A_{1}$ increases at constant pressure.

$V_{1}$ > $V_{2}$ = $V_{3}$
Hence B is correct.
Also,
$T_{1}$ > $T_{2}$ = $T_{3}$
Hence C is incorrect.

$A_{2}$ & $A_{3}$ undergo adiabatic compression
Using $T V^{γ - 1}$ =constant,
Vf final volume of $A_{2}$ and $A_{3}$ together.
Then,
$T_{0} (2 V_{0})^{\frac{2}{3}} = \frac{9}{4} T_{o} (V_{f})^{\frac{2}{3}}$
$V_{f} = \frac{8}{27} {2 V}_{0}$

$V_{2}$ = $V_{3} = \frac{8}{27} V_{0}$

$V_{1} = 3 V_{0} - V_{f}$
$V_{1} = \frac{65}{27} V_{0}$
Hence D is correct

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