States of Matter

States of Matter

Question

Two closed bulbs of equal volume (V) containing an ideal gas initially at pressure pi and temperature T1 are connected through a narrow tube of negligible volume as shown in the figure below. The temperature of one of the bulbs is then raised to T2. The final pressure pf is:

  • 2 straight p subscript straight i open parentheses fraction numerator straight T subscript 1 over denominator straight T subscript 1 plus straight T subscript 2 end fraction close parentheses
  • 2 straight p subscript straight i open parentheses fraction numerator straight T subscript 2 over denominator straight T subscript 1 plus straight T subscript 2 end fraction close parentheses
  • 2 straight p subscript straight i space open parentheses fraction numerator straight T subscript 1 straight T subscript 2 over denominator straight T subscript 1 plus straight T subscript 2 end fraction close parentheses
  • straight p subscript straight i open parentheses fraction numerator straight T subscript 1 straight T subscript 2 over denominator straight T subscript 1 plus straight T subscript 2 end fraction close parentheses

Answer

B.

2 straight p subscript straight i open parentheses fraction numerator straight T subscript 2 over denominator straight T subscript 1 plus straight T subscript 2 end fraction close parentheses

Initially,
Number of moles of gases in each container = fraction numerator straight p subscript straight i straight V over denominator RT subscript straight i end fraction
Total number of moles of gases in both containers
equals space 2 fraction numerator straight p subscript straight i straight V over denominator RT subscript straight i end fraction
After mixing, number of moles in left chamber
fraction numerator straight p subscript straight f straight V over denominator RT subscript straight i end fraction
Number of moles in right chamber = fraction numerator straight p subscript straight f straight V over denominator RT subscript 2 end fraction
Total number of moles
=   fraction numerator straight p subscript straight f straight V over denominator RT subscript 1 end fraction space plus space fraction numerator straight p subscript straight f straight V over denominator RT subscript 2 end fraction
equals space fraction numerator straight p subscript straight f straight V over denominator straight R end fraction open parentheses 1 over straight T subscript 1 plus 1 over straight T subscript 2 close parentheses
As the total number of moles remains constant. 
Hence,

fraction numerator 2 straight p subscript straight i straight V over denominator RT subscript straight i end fraction space equals space fraction numerator straight p subscript straight f straight V over denominator RT subscript straight i end fraction space plus space fraction numerator straight p subscript straight f straight V over denominator RT subscript 2 end fraction space rightwards double arrow space straight p subscript straight f space equals space 2 straight p subscript straight i open parentheses fraction numerator straight T subscript 1 over denominator straight T subscript 1 plus straight T subscript 2 end fraction close parentheses

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