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Mechanical Properties Of Fluids

Question
CBSEENPH11020267

Consider an ideal gas confined in an isolated closed chamber. As the gas undergoes an adiabatic expansion the average time of collision between molecules increases as Vq , where V is the volume of the gas. The value of q is: open parentheses straight gamma space equals space straight C subscript straight p over straight C subscript straight v close parentheses

  • fraction numerator 3 straight gamma plus 5 over denominator 6 end fraction
  • fraction numerator 3 straight gamma minus 5 over denominator 6 end fraction
  • fraction numerator straight gamma plus 1 over denominator 2 end fraction
  • fraction numerator straight gamma minus 1 over denominator 2 end fraction

Solution

C.

fraction numerator straight gamma plus 1 over denominator 2 end fraction

For an adiabatic process TVγ-1 = constant
We know that average time of collision between molecules
straight tau space equals space fraction numerator 1 over denominator nπ square root of 2 vrms end root straight d squared end fraction
Where n= number of molecules per unit volume
vrms = rms velocity of molecules
straight n space proportional to 1 over straight V space and space space straight v subscript rms space proportional to space square root of straight T
straight tau space proportional to fraction numerator straight V over denominator square root of straight T end fraction
Thus, we can write
n =K1V-1 and Vrms  = K2T1/2
Where K1 and K2 are constants.
For adiabatic process TVγ-1 = constant. Thus we can write
straight tau space proportional to space VT to the power of negative 1 divided by 2 end exponent space proportional to space straight V space left parenthesis straight V to the power of 1 minus straight gamma end exponent right parenthesis to the power of negative 1 divided by 2 end exponent
straight tau proportional to space straight V to the power of fraction numerator straight gamma plus 1 over denominator 2 end fraction end exponent