# I Change in Entropy of a Solid or Liquid

#### Philip Koeck

What about if we allow for a temperature and volume change in a solid or a liquid?
Would the entropy change still only depend on the temperature change or also on the volume change.
For a solid I would think that the volume change doesn't matter since it doesn't change the "amount of disorder", but for a liquid the volume change should matter.

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#### Chestermiller

Mentor
What about if we allow for a temperature and volume change in a solid or a liquid?
Would the entropy change still only depend on the temperature change or also on the volume change.
For a solid I would think that the volume change doesn't matter since it doesn't change the "amount of disorder", but for a liquid the volume change should matter.
For a single phase pure substance or a constant composition mixture, the variation in entropy can be determined from $$dS=\frac{C_p}{T}dT+\left(\frac{\partial S}{\partial P}\right)_TdP$$It follows from the equation $$dG=-SdT+VdP$$ that the partial derivative of entropy with respect to pressure is given by:$$\left(\frac{\partial S}{\partial P}\right)_T=-\left(\frac{\partial V}{\partial T}\right)_P$$
For a liquid or solid, the equation of state is $$dV=V(\alpha dT-\beta dP)$$where $\alpha$ is the volumetric coefficient of thermal expansion and $\beta$ is the bulk compressibility. So, $$\left(\frac{\partial V}{\partial T}\right)_P=\alpha V$$So, we have:$$dS=\frac{C_p}{T}dT-\alpha VdP$$
Because the specific volume and coefficient of thermal expansion of solids and liquids are very small, in virtually all practical situations, the second term is negligible.

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• Philip Koeck

#### Philip Koeck

For a single phase pure substance or a constant composition mixture, the variation in entropy can be determined from $$dS=\frac{C_p}{T}dT+\left(\frac{\partial S}{\partial P}\right)_TdP$$It follows from the equation $$dG=-SdT+VdP$$ that the partial derivative of entropy with respect to pressure is given by:$$\left(\frac{\partial S}{\partial P}\right)_T=-\left(\frac{\partial V}{\partial T}\right)_P$$
For a liquid or solid, the equation of state is $$dV=V(\alpha dT-\beta dP)$$where $\alpha$ is the volumetric coefficient of thermal expansion and $\beta$ is the bulk compressibility. So, $$\left(\frac{\partial V}{\partial T}\right)_P=\alpha V$$So, we have:$$dS=\frac{C_p}{T}dT-\alpha VdP$$
Because the specific volume and coefficient of thermal expansion of solids and liquids are very small, in virtually all practical situations, the second term is negligible.
I just quickly checked what that would give for an ideal gas (by replacing α and dP from the ideal gas law) and I get dS = n CV dT / T + n R dV / V, just like it should be. Very nice!

I'm wondering a bit about solids versus liquids.
For liquids I can understand that entropy changes with volume since a liquid can arrange itself in more different ways if it has more space.
For a solid, however, I don't see that. In a perfect crystal every atom is in its spot no matter how big the distance between atoms is. How can one explain the volume dependence of entropy then?

#### Chestermiller

Mentor
I just quickly checked what that would give for an ideal gas (by replacing α and dP from the ideal gas law) and I get dS = n CV dT / T + n R dV / V, just like it should be. Very nice!

I'm wondering a bit about solids versus liquids.
For liquids I can understand that entropy changes with volume since a liquid can arrange itself in more different ways if it has more space.
For a solid, however, I don't see that. In a perfect crystal every atom is in its spot no matter how big the distance between atoms is. How can one explain the volume dependence of entropy then?
Sorry, I'm a continuum mechanics guy, so analyzing it in terms of atoms and molecules is not part of my expertise.

#### Lord Jestocost

Gold Member
2018 Award
For a solid, however, I don't see that. In a perfect crystal every atom is in its spot no matter how big the distance between atoms is. How can one explain the volume dependence of entropy then?
The main reason are anharmonic effects as the phonons have, for example, frequencies that depend on volume.
[PDF]Vibrational Thermodynamics of Materials - Caltech

• Chestermiller and Philip Koeck

"Change in Entropy of a Solid or Liquid"

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