બંધ પાત્રમાં \(27^{\circ} \mathrm{C}\) તાપમાને રહેલો એક વાસ્તવિક વાયુ આકૃતિમાં દર્શાવ્યા મુજબ ચક્રીય પ્રક્રિયામાંથી પસાર થાય છે. વાયુ માર્ગ \(A\) થી \(B\) માટે \(P V^3=\mathrm{RT}\) સમીકરણનું પાલન કરે છે. સંપૂર્ણ ચક્રમાં ચોખ્ખું કાર્ય _______ છે. (ધારી રહ્યા છીએ કે \(R=8 \mathrm{~J} / \mathrm{molK}\) )

A \(225 \mathrm{~J}\)
B \(205 \mathrm{~J}\)
C \(20 \mathrm{~J}\)
D \(-20 \mathrm{~J}\)

Verified Solution

Answer & Solution

Correct Answer

(B) \(205 \mathrm{~J}\)

Step-by-step Solution

Detailed explanation

\(\mathrm{W}_{\mathrm{AB}} =\int \mathrm{PdV}\) \(\quad\) (T ને અચળ ધારીએ) \(=\int \frac{\mathrm{RTdV}}{\mathrm{V}^3}\) \(=\mathrm{RT} \int_2^4 \mathrm{~V}^{-3} \mathrm{dV}\) \(=8 \times 300 \times\left(-\frac{1}{2}\left[\frac{1}{4^2}-\frac{1}{2^2}\right]\right)\)…

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.

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