Thermodynamic property relation derivation

In summary, the expression for the isentropic compressibility of a solid (BETA sub S) in terms of isothermal compressibility (beta sub T) and other properties normally tabulated can be derived by using the definition of compressibility and substituting the definition of isothermal compressibility into the equation. This results in the equation β_S = -1/V * β_T * (∂P/∂T)_V.
  • #1
Breeder18
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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.
 

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  • #2
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
 

What is a thermodynamic property relation derivation?

A thermodynamic property relation derivation is a mathematical process used to derive relationships between different thermodynamic properties, such as temperature, pressure, and volume. These relationships allow scientists to understand and predict the behavior of materials under different thermodynamic conditions.

Why is the thermodynamic property relation derivation important?

The thermodynamic property relation derivation is important because it helps scientists understand how different thermodynamic properties are related to each other. This understanding is crucial for predicting the behavior of materials in various thermodynamic processes, such as heat transfer, chemical reactions, and phase changes.

What are the basic steps involved in a thermodynamic property relation derivation?

The basic steps involved in a thermodynamic property relation derivation include identifying the thermodynamic properties involved, setting up equations using the laws of thermodynamics, manipulating the equations to eliminate any redundant terms, and solving for the desired relationship between the properties.

What are some common thermodynamic property relations that are derived?

Some common thermodynamic property relations that are derived include the ideal gas law, the van der Waals equation, and the Maxwell relations. These relations are used to describe the behavior of gases, liquids, and solids under different thermodynamic conditions.

What are the limitations of thermodynamic property relation derivations?

Thermodynamic property relation derivations are based on certain assumptions and idealized conditions, which may not always accurately reflect real-world situations. Additionally, these derivations may become more complex for systems with multiple phases or non-ideal behavior. Therefore, it is important to consider the limitations and applicability of these relations when using them in scientific research.

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