# Understanding Expansion, Compression and Entropy Coefficients

• tsuwal
In summary: T)v=-a/k (helmoltz)(∂T/∂p)s=a*T/cp*k*(-a/k*dV)(∂T/∂p)s=-a^2*T/Cp*dVIn summary, the conversation discusses the equations (DV/DS)p=(DT/Dp)s=a*T/cp*(rho) (enthalpy), (Dp/DT)v=(DS/DV)t=-a/k (helmoltz), and (DS/Dp)t=-(DV/DT)p=-Va (gibbs) and their corresponding coefficients a, k, and cp. The individual wants to deduce the derivative (DT/DV
tsuwal
So, until now I know:
(DV/DS)p=(DT/Dp)s=a*T/cp*(rho) (enthalpy)
(Dp/DT)v=(DS/DV)t=-a/k (helmoltz)
(DS/Dp)t=-(DV/DT)p=-Va (gibbs)

a=expansion coefficient
k=isothermal compression coefficent
cp=heat capacity at constante pressure

I want to deduce DT/DV at constant entropy=(DT/DV)s. BUT HOW?
Let me try to write S(T,V), then,
dS=Cv/T*dT-a/k*dV
putting S=0, i get,
a/k*dV=Cv/T*dT <=> (DT/DV)s=a*T/Cv*k

am I right?

Is this the one you want?

$$\begin{array}{l} T = {\left( {\frac{{\partial U}}{{\partial S}}} \right)_V} \\ {\left( {\frac{{\partial T}}{{\partial V}}} \right)_S} = \left[ {\frac{\partial }{{\partial V}}{{\left( {\frac{{\partial U}}{{\partial S}}} \right)}_V}} \right] = \frac{{{\partial ^2}U}}{{\partial V\partial S}} \\ \end{array}$$

and

$$\begin{array}{l} P = - {\left( {\frac{{\partial U}}{{\partial V}}} \right)_S} \\ {\left( {\frac{{\partial P}}{{\partial S}}} \right)_V} = - \left[ {\frac{\partial }{{\partial S}}{{\left( {\frac{{\partial U}}{{\partial S}}} \right)}_V}} \right] = - \frac{{{\partial ^2}U}}{{\partial S\partial V}} \\ \end{array}$$

Therefore

$${\left( {\frac{{\partial T}}{{\partial V}}} \right)_S} = - {\left( {\frac{{\partial P}}{{\partial S}}} \right)_V}$$

Hey, thanks for worring so much, but until there I knew...
I want to evaluate that derivative further and write in terms of a,k,Cv,Cp,T,p,... as I did
(∂T/∂p)s=a*T/cp*(rho)

## What is the meaning of expansion coefficient?

The expansion coefficient, also known as thermal expansion coefficient, is a measure of how much a material expands or contracts in response to changes in temperature. It is represented by the Greek letter alpha (α) and is typically given in units of per degree Celsius (or Kelvin).

## How is the compression coefficient defined?

The compression coefficient, also known as bulk modulus, is a measure of a material's resistance to compressive stress. It is represented by the Greek letter kappa (κ) and is typically given in units of pressure such as Pascals (Pa) or Gigapascals (GPa).

## What is the relationship between expansion and compression coefficients?

Expansion and compression coefficients are inversely related to each other. This means that as one increases, the other decreases. This is because materials that expand easily when heated tend to compress easily when subjected to pressure, and vice versa.

## How does entropy coefficient relate to expansion and compression?

The entropy coefficient, also known as entropy change coefficient, is a measure of how much the entropy (disorder or randomness) of a material changes in response to changes in temperature or pressure. It is represented by the Greek letter delta (Δ) and is typically given in units of per degree Celsius (or Kelvin) or per Pascals (or Gigapascals). Entropy coefficient is related to expansion and compression coefficients as it describes the changes in the physical properties of a material due to changes in temperature or pressure.

## Why are expansion, compression, and entropy coefficients important in science?

Expansion, compression, and entropy coefficients are important in science because they help us understand how materials behave under different conditions. They are used in various fields such as materials science, thermodynamics, and engineering to predict and control the behavior of materials. These coefficients are also essential in the design of structures and devices that are subjected to temperature and pressure changes.

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