Thermal Physics adiabatic and isothermal compressibilty

[mmedrano8]

Ok so I found something online but I need to understand this problem

Prove that the ratio of the adiabatic compressibilty k_s to the isothermal compressibility k_r is equal to the ratio of the specific heat at constant colume, C_v, to that at constant pressure, C_p

Definitions of the compressibility

adiabatic compressibilty k_s = (-1/v)(∂V/∂p)_s

isothermal compressibilty k_T = (-1/V)(∂V/∂P)_T

I think we need to start with dS = (∂S/∂T)_pdT + (∂s/∂p)_TdP

 

[Chestermiller]

Ok so I found something online but I need to understand this problem

Prove that the ratio of the adiabatic compressibilty k_s to the isothermal compressibility k_r is equal to the ratio of the specific heat at constant colume, C_v, to that at constant pressure, C_p

Definitions of the compressibility

adiabatic compressibilty k_s = (-1/v)(∂V/∂p)_s

isothermal compressibilty k_T = (-1/V)(∂V/∂P)_T

I think we need to start with dS = (∂S/∂T)_pdT + (∂s/∂p)_TdP

The isothermal compressibility can be obtained directly from the ideal gas law. The adiabatic compressibility can be obtained directly from PV^γ=constant.

 

[Oxvillian]

Or to show it in general (without assuming an ideal gas) you can use the cyclic relation for partial derivatives.