An ideal gas undergoes an expansion from a state with temperature $T_{1}$ and volume $V_{1}$ through three different polytropic process A, B and C as shown in the $P - V$ diagram. If $| Δ E_{A} |, | Δ E_{B} |$ and $| Δ E_{C} |$ be the magnitude of changes in internal energy along the three paths respectively, then

| Δ E_{A} | < | Δ E_{B} | < | Δ E_{C} |

if temperature in every process decreases

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| Δ E_{A} | > | Δ E_{B} | > | Δ E_{C} |

if temperature in every process decreases

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| Δ E_{A} | > | Δ E_{B} | > | Δ E_{C} |

if temperature in every process increases

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| Δ E_{B} | < | Δ E_{A} | < | Δ E_{C} |

if temperature in every process increases

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Solution

The correct option is C $| Δ E_{A} | > | Δ E_{B} | > | Δ E_{C} |$ if temperature in every process increases
Initial state is same for all the three processes (say initial internal energy $= E_{0})$ .
In the final state, $V_{A} = V_{B} = V_{C}$
and $P_{A} > P_{B} > P_{C}$
$P_{A} V_{A} > P_{B} V_{B} > P_{C} V_{C}$
$E_{A} > E_{B} > E_{C}$
If temperature in every process decreases, i.e. $T_{1} > T_{f}$ , then $E_{0} > E_{f}$ for all the three process and hence $(E_{0} - E_{A}) < (E_{0} - E_{B}) < (E_{0} - E_{C})$
$| Δ E_{A} | < | Δ E_{B} | < | Δ E_{C} |$
If temperature in every process increases, i.e. $T_{1} < T_{2}$ , then $E_{0} < E_{f}$ for all the three process and hence $(E_{A} - E_{0}) > (E_{B} - E_{0}) > (E_{C} - E_{0})$
$| Δ E_{A} | > | Δ E_{B} | > | Δ E_{C} |$

Why this Question ?

Key Concept: Conservation of energy using first law of thermodynamics.

Importance for JEE: Graphical analysis of work done in relation with Internal energy is important in JEE

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