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In the following P-V diagram of an ideal gas, AB and CD are isothermal whereas BC and DA are adiabatic process. The value of $V_{B} / V_{C}$ is

= V_{A} / V_{D}

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< V_{A} / V_{D}

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> V_{A} / V_{D}

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Solution

The correct option is A $= V_{A} / V_{D}$
For isothermal process,
$P V = c o n s t a n t$
$f o r A B :$
$∴ P_{A} V_{A} = P_{B} V_{B}$
$\frac{P_{A}}{P_{B}} = \frac{V_{B}}{V_{A}} ⟶ (1)$
$f o r B C :$
$P_{B} V_{B}^{Y} = P_{C} V_{C}^{Y} ⟶ (2)$
Substituting value of $P_{B}$ from $(1) & (2)$
$\frac{P_{A} V_{A}}{V_{B}} V_{B}^{Y} = P_{C} V_{C}^{Y}$
$P_{A} V_{A} V_{B}^{Y - 1} = P_{C} V_{C}^{Y} ⟶ (3)$
$f o r C D :$
$P_{C} V_{C} = P_{D} V_{D}$
$\frac{P_{C}}{P_{D}} = \frac{V_{D}}{V_{C}}$
$f o r D A :$
$P_{D} V_{D}^{Y} = P_{A} V_{A}^{Y}$
$\frac{V_{C} P_{C}}{V_{D}} V_{D}^{Y} = P_{A} V_{A}^{Y}$
$P_{C} V_{C} V_{D} = P_{A} V_{A}^{Y} ⟶ (4)$
$P_{A} V_{A}^{Y} = P_{C} V_{C} V_{D}^{Y - 1}$
$P_{A} V_{A} V_{B}^{Y - 1} = P_{C} V_{C}^{Y}$
Dividing the two we get,
$\frac{P_{A} V_{A}^{Y}}{P_{A} V_{A} V_{B}^{Y - 1}} = \frac{P_{C} V_{C} V_{D}^{Y - 1}}{P_{C} V_{C}^{Y}}$
$\frac{V_{A}^{Y - 1}}{V_{B}^{Y - 1}} = \frac{V_{D}^{Y - 1}}{V_{C}^{Y - 1}}$
$\frac{V_{B}^{Y - 1}}{V_{C}^{Y - 1}} = \frac{V_{A}^{Y - 1}}{V_{D}^{Y - 1}}$
${(\frac{V_{B}}{V_{C}})}^{Y - 1} = {(\frac{V_{A}}{V_{D}})}^{Y - 1}$
$\frac{V_{B}}{V_{C}} = \frac{V_{A}}{V_{D}}$
$(A) = \frac{V_{A}}{V_{D}}$
ANSWER : A.

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In the following P-V diagram of an ideal gas, AB and CD are isothermal whereas BC and DA are adiabatic process. The value of VB/VC is

In the following P-V diagram of an ideal gas, AB and CD are isothermal whereas BC and DA are adiabatic process. The value of $V_{B} / V_{C}$ is