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what do you mean by adiabatic and isothermal elasticities?
what is the ratio of adiabatc to isothermal elasticity and why?

Asked by imtiyazmulla68 10th July 2017, 9:13 PM
Answered by Expert
Answer:
Due to high compressibility, gases possess volume elasticity. However, the magnitude of the volume elasticity depends on conditions under which it is compressed.
If the gas is compressed such that the temperature is not allowed to change, i.e. under isothermal conditions than the corresponding volume elasticity is known as isothermal elasticity represented as KT
If the gas is compressed such that no heat is allowed to enter or leave the system, i.e. under adiabatic conditions than the corresponding elasticity is known as adiabatic elasticity and is represented as KΦ.

Isothermal elasticity: For a perfect gas at constant temperature,
pV = constant
Differentiating the above, we get
begin mathsize 12px style straight p space plus space straight V fraction numerator dp over denominator dV space end fraction space equals space 0
straight p space equals space minus space fraction numerator dp over denominator begin display style bevelled dV over straight V end style space end fraction
Here comma space space minus space fraction numerator dp over denominator begin display style bevelled dV over straight V space end style end fraction space equals space Measure space of space volume space elasticity.
Under space isothermal space conditions
straight K subscript straight T space equals space straight p
straight i. straight e. space the space isothermal space elasticity space for space perfect space gas space under space isothermal space conditions space is
straight K subscript straight T space equals space straight p end style
 
Adiabatic elasticity: 
For a perfect gas under adiabatic elasticity,
pVγ = constant
Differentiating the above wrt volume, we get
begin mathsize 12px style straight p. space γV to the power of straight gamma minus 1 end exponent space plus space straight V to the power of straight gamma fraction numerator dp over denominator begin display style bevelled dV over straight V end style end fraction space equals space 0
γp space equals space minus fraction numerator dp over denominator bevelled dV over straight V end fraction space
Here space minus fraction numerator dp over denominator bevelled dV over straight V end fraction space space measures space the space volume space elasticity
straight K subscript straight ϕ space equals space γp
where space straight gamma space equals space straight C subscript straight p over straight C subscript straight v end style
 
Ratio of the two is begin mathsize 12px style straight K subscript straight ϕ over straight K subscript straight T space equals space straight gamma space equals space straight C subscript straight p over straight C subscript straight v end style
Answered by Expert 12th July 2017, 12:15 PM
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