# Change in Entropy of a Solid or Liquid

• Philip Koeck
In summary, for a liquid or solid, the volume dependence of entropy can be explained by the anharmonic effects of the phonons.
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.

Last edited by a moderator:
Philip Koeck said:
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.

Last edited:
Philip Koeck
Chestermiller said:
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?

Philip Koeck said:
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.

Philip Koeck said:
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

## 1. How is the change in entropy of a solid or liquid calculated?

The change in entropy of a solid or liquid can be calculated using the formula ΔS = mCln(T₂/T₁), where ΔS is the change in entropy, m is the mass of the substance, C is the specific heat capacity, T₂ is the final temperature, and T₁ is the initial temperature.

## 2. What factors affect the change in entropy of a solid or liquid?

The change in entropy of a solid or liquid is affected by the temperature change, the mass of the substance, and the specific heat capacity of the substance. Other factors such as pressure and volume may also play a role in certain scenarios.

## 3. Can the change in entropy of a solid or liquid be negative?

Yes, the change in entropy of a solid or liquid can be negative if the system loses energy and becomes more ordered. This is known as a decrease in entropy.

## 4. How does the change in entropy of a solid compare to that of a liquid?

The change in entropy of a solid is typically smaller than that of a liquid, as solids have a more ordered structure and less freedom of movement for their particles. Liquids, on the other hand, have a more disordered structure and more freedom of movement for their particles, resulting in a larger change in entropy.

## 5. What is the significance of the change in entropy of a solid or liquid?

The change in entropy of a solid or liquid is a measure of the disorder or randomness of the particles in the system. It is an important concept in thermodynamics and is used to understand and predict the behavior of substances during physical and chemical processes, such as phase changes and reactions.

• Thermodynamics
Replies
58
Views
4K
• Thermodynamics
Replies
19
Views
2K
• Thermodynamics
Replies
5
Views
1K
• Thermodynamics
Replies
12
Views
2K
• Thermodynamics
Replies
1
Views
778
• Thermodynamics
Replies
4
Views
1K
• Thermodynamics
Replies
15
Views
1K
• Thermodynamics
Replies
16
Views
932
• Thermodynamics
Replies
9
Views
1K
• Thermodynamics
Replies
2
Views
3K