\(T _{1}\) અને \(T _{2}\) તાપમાને રહેલ બે આદર્શ બહુ પરમાણ્વીય વાયુને મિશ્ર કરતાં ઉર્જાનો વ્ય્ય થતો નથી. જો \(F _{1}\) અને \(F _{2}, m _{1}\) અને \(m _{2}, n _{1}\) અને \(n _{2}\) અનુક્રમે મુક્તતાના અંશો, દળ અને અણુની સંખ્યા હોય તો તેમના મિશ્રણનું તાપમાન કેટલું થાય?

A \(\frac{ n _{1} T _{1}+ n _{2} T _{2}}{ n _{1}+ n _{2}}\)
B \(\frac{ n _{1} F _{1} T _{1}+ n _{2} F _{2} T _{2}}{ n _{1} F _{1}+ n _{2} F _{2}}\)
C \(\frac{ n _{1} F _{1} T _{1}+ n _{2} F _{2} T _{2}}{ F _{1}+ F _{2}}\)
D \(\frac{ n _{1} F _{1} T _{1}+ n _{2} F _{2} T _{2}}{ n _{1}+ n _{2}}\)

Verified Solution

Answer & Solution

Correct Answer

(B) \(\frac{ n _{1} F _{1} T _{1}+ n _{2} F _{2} T _{2}}{ n _{1} F _{1}+ n _{2} F _{2}}\)

Step-by-step Solution

Detailed explanation

Let the final temperature of the mixture be \(T\). Since, there is no loss in energy. \(\Delta U =0\) \(\Rightarrow \frac{ F _{1}}{2} n _{1} R \Delta T +\frac{ F _{2}}{2} n _{2} R \Delta T =0\)…

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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