સમોષ્મિ પ્રક્રિયા દરમિયાન આદર્શ વાયુનું કદ બમણું થાય છે. બે વાયુના અણું વચ્ચેનો સરેરાશ અથડામણ સમય \(\tau_{1}\) થી \(\tau_{2}\) થાય છે. જો \(\frac{C_{p}}{C_{v}}=\gamma\) હોય તો આ વાયુ માટે \(\frac{\tau_{2}}{\tau_{1}}\) કેટલો થશે?

A \(\left(\frac{1}{2}\right)^{\frac{\gamma+1}{2}}\)
B \(2\)
C \(\frac 12\)
D \(\left(\frac{1}{2}\right)^{\gamma}\)

Verified Solution

Answer & Solution

Correct Answer

(A) \(\left(\frac{1}{2}\right)^{\frac{\gamma+1}{2}}\)

Step-by-step Solution

Detailed explanation

\({\lambda \propto V}\) The average time between the collisions of the gas molecules is nothing but the mean free path divided by the root mean square speed of the gas molecules. \(\mathrm{So}\) \(\Longrightarrow\left[\text { Time }=t=\frac{\lambda}{v_{R M S}}\right]\) Now we…

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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