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Units and Dimensions

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A sample of ideal gas undergoes isothermal expansion in a reversible manner from volume $V_{1}$ to volume $V_{2}$ .
The initial pressure is $P_{1}$ and the final pressure is $P_{2}$ . The same sample is then allowed to undergo reversible expansion under adiabatic conditions from volume $V_{1}$ to $V_{2}$ . The initial pressure being same but final pressure is $P_{2}$ .
Which relation is correct $(\begin{matrix} Υ = \frac{C_{P}}{C_{V}} \end{matrix}) ?$

A

P_{1} V_{1} = P_{2} V_{3}

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B

P_{2} V_{1} = P_{3} V_{2}

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C

P_{1} V_{1}^{Υ} = P_{2} V_{2}^{Υ}

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D

\frac{P_{1}}{P_{2}} = {(\begin{matrix} \frac{V_{2}}{V_{1}} \end{matrix})}^{Υ - 1}

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Solution

The correct option is C $P_{1} V_{1}^{Υ} = P_{2} V_{2}^{Υ}$
Since it undergoes a reversible adiabatic expansion, we have $P_{1} V_{1}^{γ} = P_{2} V_{2}^{γ}$ .
Hence option C is the right answer.

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