Thermodynamic property relation derivation

[Breeder18]

Homework Statement

Derive an expression for the isentropic compressibility of a solid (BETA sub S) in terms of isothermal compressibility (beta sub T) and other properties normally tabulated.

Homework Equations

β_T≝-1/V (∂V/∂P)_T

β_S≝-1/V (∂V/∂P)_S

The Attempt at a Solution

I may be going in circles...
I realize some of my work, if not most, is redundant but I was hoping to see a possible method. Any and all help would be greatly appreciated.

P.S. I tried using the latex equation editor, but after 15 mins, wanted to shoot myself.

 

[NateD]

Using the definition of isothermal compressibility, β_T, and isentropic compressibility, β_S, the following can be derived. β_T≝-1/V (∂V/∂P)_Tβ_S≝-1/V (∂V/∂P)_SBased on the definition of compressibility, we can see that it is related to the change in volume with respect to pressure. We can also see that the pressure and volume are functions of temperature. Based on this, we can derive the following equation:β_S = -1/V * (∂V/∂P)_T * (∂P/∂T)_VWe can then simplify the equation by substituting the definition of isothermal compressibility into the equation. β_S = -1/V * β_T * (∂P/∂T)_VTherefore, an expression for the isentropic compressibility of a solid in terms of isothermal compressibility and other properties normally tabulated can be derived as follows:β_S = -1/V * β_T * (∂P/∂T)_V