Question

What is Isothermal elasticity and Adiabatic elasticity ? Give their derivation. ​

Answer 1

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

Explanation:

pls mark as brainliest.... bro✌✌