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Mechanical Properties Of Fluids

Question
CBSEENPH11019533

nmoles of gas A are mixed with nmoles of gas B whose ratio of specific heats are γ1 and γ2. What are the values of specific heats Cp and Cv of the mixture?

Solution
The ratio of specific heat of gas A is γ1 and that of gas B is γ2
The specific heat of gas A and gas B at constant volume is, 
straight C subscript straight v 1 end subscript equals fraction numerator straight R over denominator straight y subscript 1 minus 1 end fraction space and space straight C subscript straight v 2 end subscript equals fraction numerator straight R over denominator straight y subscript 2 minus 1 end fraction respectively
 
The specific heat of mixture at constant volume is, 
space space space space space space space
space space space space space space space space space space space straight C subscript straight v equals fraction numerator straight n subscript 1 straight C subscript straight v 1 end subscript plus straight n subscript 2 straight C subscript straight v 2 end subscript over denominator straight n subscript 1 plus straight n subscript 2 end fraction space space

space space space space space space space space space space space space space equals fraction numerator straight n subscript 1 begin display style fraction numerator straight R over denominator straight y subscript 1 minus 1 end fraction end style plus straight n subscript 2 begin display style fraction numerator straight R over denominator straight y subscript 2 minus 1 end fraction end style over denominator straight n subscript 1 plus straight n subscript 2 end fraction space

rightwards double arrow space space space straight C subscript straight v equals fraction numerator straight R over denominator straight n subscript 1 plus straight n subscript 2 end fraction open parentheses fraction numerator straight n subscript 1 over denominator straight y subscript 1 minus 1 end fraction plus fraction numerator straight n subscript 2 over denominator straight y subscript 2 minus 1 end fraction close parentheses
Similarly, specific heat of mixture at constant pressure is, 

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