એક લાક્ષણિક દહનશીલ એન્જીન (કંબશન એન્જીન) માં વાયુનાં અણુ દ્વારા થયેલ કાર્યને \(W=\alpha^{2} \beta e^{\frac{-\beta x^{2}}{k T}}\) દ્વારા આપવામાં આવે છે જ્યાં \(x\) સ્થાનાંતર, \(k\) બોલ્ટ્ઝમેન અચળાંક અને \(T\) તાપમાન દર્શાવે છે. જો \(\alpha\) અને \(\beta\) અચળાંકો હોય, તો \(\beta\) નું પરિમાણ ......... હશે.

A \(\left[ MLT ^{-2}\right]\)
B \(\left[ M ^{0} LT ^{0}\right]\)
C \(\left[ M ^{2} LT ^{-2}\right]\)
D \(\left[ MLT ^{-1}\right]\)

Verified Solution

Answer & Solution

Correct Answer

(B) \(\left[ M ^{0} LT ^{0}\right]\)

Step-by-step Solution

Detailed explanation

\(kT\) has dimension of energy \(\frac{\beta x ^{2}}{ kT }\) is dimensionless \([\beta]\left[ L ^{2}\right]=\left[ ML ^{2} T ^{-2}\right]\) \([\beta]=\left[ MT ^{-2}\right]\) \(\alpha^{2} \beta\) has dimensions of work…

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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