Prove expansibility / isothermal compressibility

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Expansibility (β) is defined as (1/V)(∂V/∂T) at constant temperature, while isothermal compressibility (κ) is defined as (-1/V)(∂V/∂P) at constant pressure. The discussion focuses on proving the relationship (∂P/∂T) at constant volume equals (β/κ). To derive this, the equation dV = (∂V/∂T)PdT + (∂V/∂P)TdP is used, setting dV to zero to isolate (∂P/∂T)V. This approach leads to the conclusion that the relationship between expansibility and compressibility can be established through thermodynamic principles.
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β is expansibility , (1/V)(∂V/∂T) at constant temperature
κ is isothermal compressibility , (-1/V)(∂V/∂P) at constant pressure

How to prove (∂P/∂T) at constant V = (β/κ)
Thank you
 
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Outrageous said:
β is expansibility , (1/V)(∂V/∂T) at constant temperature
κ is isothermal compressibility , (-1/V)(∂V/∂P) at constant pressure

How to prove (∂P/∂T) at constant V = (β/κ)
Thank you

dV = (∂V/∂T)PdT + (∂V/∂P)TdP

set dV = 0 to evaluate (∂P/∂T)V
 
Chestermiller said:
dV = (∂V/∂T)PdT + (∂V/∂P)TdP

set dV = 0 to evaluate (∂P/∂T)V


Understand. Thank you
 

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