એક આદર્શ વાયુનું દબાણ અને કદ \(\mathrm{PV}^{3 / 2}=\mathrm{K}\) (અચળ) તરીકે સંબંધિત છે. જ્યારે વાયુને A \(\left(P_1, V_1, T_1\right)\) સ્થિતિમાંથી B \(\left(P_2, \mathrm{~V}_2, \mathrm{~T}_2\right)\) સ્થિતિમાં લઈ જવામાં આવે ત્યારે થયેલું કાર્ય _______ છે.

A \(2\left(\mathrm{P}_1 \mathrm{~V}_1-\mathrm{P}_2 \mathrm{~V}_2\right)\)
B \(2\left(\mathrm{P}_2 \mathrm{~V}_2-\mathrm{P}_1 \mathrm{~V}_1\right)\)
C \(2\left(\sqrt{\mathrm{P}_1} V_1-\sqrt{\mathrm{P}_2} V_2\right)\)
D \(2\left(\mathrm{P}_2 \sqrt{\mathrm{V}_2}-\mathrm{P}_1 \sqrt{\mathrm{V}_1}\right)\)

Verified Solution

Answer & Solution

Correct Answer

(A) \(2\left(\mathrm{P}_1 \mathrm{~V}_1-\mathrm{P}_2 \mathrm{~V}_2\right)\)

Step-by-step Solution

Detailed explanation

\(\mathrm{PV}^{\mathrm{x}}=\) અચળ માટે જો વાયુ દ્વારા થયેલ કાર્ય પૂછવામાં આવે તો \({W}=\frac{\mathrm{nR} \Delta \mathrm{T}}{1-\mathrm{x}}\) અહીં x \(=\frac{3}{2}\) \( W =\frac{\mathrm{P}_2 \mathrm{~V}_2-\mathrm{P}_1 \mathrm{~V}_1}{-\frac{1}{2}}\)…

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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