Question

Which of the following best doseribe
, the, relation
between
isothermal Compressibility(Kt) and adiabatic compressibility (Ks)is?
​

Answer 1

Explanation:

Compressibility β is defined as the reciprocal of bulk modulus (k) and is equal to the ratio of the fractional change in volume to the stress (pressure) applied on the body.

β=1k=−1V∂V∂p … … … (1)

Since the question is about isothermal and adiabatic compressibility, take the example of ideal gas.

We have equation for isothermal process (for one mole of gas for simplicity)

pV= constant

We differentiate this to get

p∂V+V∂p=0

⇒−1V∂V∂p=1p

The left hand side of this expression is isothermal compressibility (see equation (1)). Thus

βiso=1p … … … (2)

Now we have the equation for adiabetic process

pVγ = constant

We differentiate this to have

pγVγ−1+Vγ∂p∂V=0

A little rearrangement yields

−1V∂V∂p =1γp

⇒βadia=1γp … … … (3)

Since γ is always greater than 1, therefore from equations (2) and (3) it is clear that

βiso>βadia