Question

Derive the isentropic and isothermal compressibility terms (for an ideal gas) in the most simplified form and compare them.

Answer 1

The isentropic compressibility of an ideal gas is κ_s = 1/(y*p) while the isothermal compressibility is κ_T = 1/p. Since y > 1 for real gases with degrees of freedom, isentropic compressibility is less than isothermal compressibility for an ideal gas.

Step-by-step explanation:

To derive the isentropic and isothermal compressibility of an ideal gas, we use thermodynamic relationships. Isentropic compressibility (also known as adiabatic compressibility) is defined under conditions where entropy remains constant (a reversible adiabatic process). On the other hand, isothermal compressibility is defined for processes that occur at a constant temperature.

The isentropic compressibility (κ_s) is given by:

κ_s = -µdV/Vdp|_s

For an ideal gas, using the adiabatic condition pV^y = constant, where y is the heat capacity ratio (Cp/Cv), we derive the expression for κ_s as:

κ_s = 1/(γ*p)

The isothermal compressibility (κ_T) is defined as:

κ_T = -µ1/V(∂V/∂p)_T

For an ideal gas, pV = nRT (Ideal Gas Law), thus we can derive:

κ_T = 1/p

Comparing both, we see that κ_s < κ_T for ideal gases since γ > 1 for real gases with degrees of freedom.

Answer 2

Isothermal compressibility is different from isentropic compressibility by temperature instead of entropy.

Step-by-step explanation:

Isothermal compressibility refers to that type of compressibility where volume change takes place at constant temperature whereas isentropic compressibility refers to that in which volume change takes place at constant entropy. Entropy is the measure of a thermal energy of the system per unit temperature that is unavailable for doing useful work.