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Thermodynamics

Question
CBSEENPH11020493

Two thermally insulated vessels 1 and 2 are filled with air at temperatures (T1, T2), volume (V1, V2) and pressure (P1, P2) respectively. If the valve joining two vessels is opened, the temperature inside the vessel at equilibrium will be

  • T1 + T2

  • T1 + T2/2

  • fraction numerator straight T subscript 1 straight T subscript 2 space left parenthesis straight P subscript 1 straight V subscript 1 plus straight P subscript 2 straight V subscript 2 right parenthesis over denominator straight P subscript 1 straight V subscript 1 straight T subscript 2 space plus space straight P subscript 2 straight V subscript 2 straight T subscript 1 end fraction
  • fraction numerator straight T subscript 1 straight T subscript 2 space left parenthesis straight P subscript 1 straight V subscript 1 space plus straight P subscript 2 straight V subscript 2 right parenthesis over denominator straight P subscript 1 straight V subscript 1 straight T space plus space straight P subscript 2 straight T subscript 2 straight T subscript 2 end fraction

Solution

C.

fraction numerator straight T subscript 1 straight T subscript 2 space left parenthesis straight P subscript 1 straight V subscript 1 plus straight P subscript 2 straight V subscript 2 right parenthesis over denominator straight P subscript 1 straight V subscript 1 straight T subscript 2 space plus space straight P subscript 2 straight V subscript 2 straight T subscript 1 end fraction

There will be no change in number of moles if the vessels are joined by the valve. Therefore, from gas equation
PV space equals space nRT
fraction numerator straight P subscript 1 straight V subscript 1 over denominator RT subscript 1 end fraction space plus space fraction numerator straight P subscript 2 straight V subscript 2 over denominator RT subscript 2 end fraction space equals space fraction numerator straight P left parenthesis straight V subscript 1 space plus straight V subscript 2 right parenthesis over denominator RT end fraction space
rightwards double arrow space fraction numerator straight P subscript 1 straight V subscript 1 straight T subscript 2 space plus space straight P subscript 2 space straight V subscript 2 straight T subscript 1 over denominator straight T subscript 1 straight T subscript 2 end fraction space equals fraction numerator space straight P left parenthesis straight V subscript 1 space plus straight V subscript 2 right parenthesis over denominator straight T end fraction
space straight T space equals space fraction numerator straight P left parenthesis straight V subscript 1 space plus straight V subscript 2 right parenthesis straight T subscript 1 straight T subscript 2 over denominator left parenthesis straight P subscript 1 straight V subscript 1 straight T subscript 2 space plus space straight P subscript 2 straight V subscript 2 straight T right parenthesis end fraction
Now, according to Boyle's law (pressure = constant) P1 V1 + P2 V2 = P(V1 + V2 )
Hence comma space straight T equals space fraction numerator left parenthesis straight P subscript 1 straight V subscript 1 space plus straight P subscript 2 straight V subscript 2 right parenthesis straight T subscript 1 straight T subscript 1 over denominator straight P subscript 1 straight V subscript 1 straight T subscript 1 space plus space straight P 2 straight V subscript 2 straight T subscript 1 end fraction