The Basic Version

In the case of bottles or vessels of constant cross-section, the cross-sectional area A always determines the capacity of such bottles / vessels to a certain filling height. Transferring this comparison to the storage of thermal energy means:

The entropy determines the thermal energy stored up to a certain temperature.

The analogy of the fluid model and the defintion of temperature show that in the energy storing process entropy assisted by thermal capacity plays the role of the storage size. This may be surprising at first because the molar thermal capacity* is often seen in this role. However, thermal capacity as the first derivative of entropy recieves all properties from its parent function. Numerous experimental findings confirm this interpretation of the entropy (see below 2. Refinement and see 5.2.). The three rules of the shelf model already indicated that the entropy is a thermal property that describes the size of the thermal energy storage system.
As is known, solids typically have small entropy values, liquids larger and gases particularly large values of entropy (cross sections) when all substances are judged at the same temperature region. The following figure shows, in a simple cross-sectional drawing of a fluid model, a cooling process of a solid substance in an air-filled space. If the tap is opened and thus the thermal contact is established, the thermal energy flows as with connected vessels until the fill level is equal in both parts of the model apparatus. The solid material cools, the room air heats up. The final temperature, that is the filling level, is not closer to the mean value of the two temperatures, but closer to the initial temperature of the cold room, i. e. the system with the larger storage size. The total energy has remained constant in this example.

The above image visualizes calculations based on one of the relations:

ΔQ is the thermal work done by the hot material on cold material in order to bring about the temperature change. In addition to the thermal material properties (thermal capacity or entropy), the respective portion sizes, ie the masses or the substance quantities of the substances involved, are also included in this approximate calculation. The physics prefers rather the specific, thus related to the mass material properties, while the chemistry uses the molar, related to the amount of substance variants. The simple fluid model copes with both versions as mass and amount of substance are proportional to each other.
The quality of the approximation depends on the temperature region in which the process takes place and according to this either the mean value of thermocapacity (left eq.) or of temperature (right eq.) has to be used. In everyday processes entropy changes are usually negligible and the thermal capacity is assumed to be constant. Since temperature changes are easier to measure rather than entropy changes, the first equation is used more frequently.
This basic version of the fluid model can also be used to describe very complex thermal processes, such as those in a refrigerator (see 6.4.2.).

In the following, we show that through different stages of the refinement of this model, the quite complex thermodynamic properties of real substances can be visualized very differentiated and made intelligible.

*(Depending on the experimental context, the thermocapacity and entropy of an examined material portion are alternatively referred to the mass instead of the amount of substance, so that the respective specific values of entropy and thermocapacity are then used.)

The 1. Refinement

Entropy and thermal capacity contribute both to the storage size for thermal energy of material systems, but in a different way. The difference, however, can be very well integrated into the fluid model:
As the fill level (temperature) increases, the ground pressure (temperature) as well as the pressure on the lateral surfaces rises. Depending on the design of the bottle (substance system), the vertical walls yield more or less, which increases the cross-sectional area (entropy). This increase represents the molar thermal capacity in this model, which - as shown above - changes logarithmically with the temperature (filling level). The different size of the hydrostatic pressure at different heights below the surface of the medium would lead to a "bulging" deformation of the side walls. Whereas in the basic model the walls were assumed to be rigid, the first refined version of the fluid model takes this phenomenon into account: Instead of bulging, the deformation of the walls is averaged over all heights, as shown in the next computer animation.

Animation: Thermal Work in Fluid Model

Entropy and thermal capacity together describe the size of a thermal energy store, ie, the energy that can be stored at the quantized energy levels of a substance system at a particular temperature. The entropy contains the statement about the total energy, which is stored from the absolute zero point to the current temperature of the experimental context, while the thermal capacity indicates the change of the entropy in the course of the temperature scale or in the range of the current current temperature. The following animation shows that the fluid model is transferring the quantum theoretical results to this model. During the temperature increase shown, the stored thermal energy triples, but the entropy increases only to 1.5 times. The radius of the model cylinder is only around 1.2 times:

Animation: Thermal Work in Shelf and Fluid Model

The 2. Refinement

When looking at the previous animation, the color change of the fluid medium is noticed, and it requires an explanation as to which phenomenon is referred to by this.

We examine the above graph, which shows the entropy and thermal capacity values of silver [8.2/12].
The enumerated "energy house numbers" on the entropy curve are obviously getting closer together. The first kilojoule of stored thermal energy - starting at absolute zero - yields a temperature of 98 K with an entropy of S = 16.7 J/Kmol (S/R = 2.0). The second kilojoule does not double this temperature, but leads only to 145 K with an entropy of S = 25.0 J/Kmol (S/R = 3.0), the third only to 187 K with the entropy of S = 31.1 J/Kmol (S/R = 3.7), and so on ...

With increasing temperature the quotient in the last column becomes smaller and we realize, that:

The product of entropy and temperature increases with increasing temperature more than the stored energy.

Since the axes in the above diagram correspond exactly to the filling height and the cross-sectional area in the fluid model we are able to understand this phenomenon of chemical substances by the help of the fluid model

For normal liquids, we assume that the density is not dependent on the fill level. The situation is different for the gases, which as well are fluid: d = f(h), which means that density decreases with increasing height according to the barometric formula. If we now imagine a dense gas under the fluid model medium, we can also correspond to the actual thermodynamic behavior in the model. However, we simplify this again by means of averaging the density over the entire fill level and representing this mean (storage) density by different color intensities in the fluid model.
Since we developed the fluid model in connection with the shelf model, we were able to calculate and draw the fill height, the cross-sectional area as well as the stored energy quantitatively in our last animation (see above). Therefore, the decrease of the color intensity was not arbitrary, but corresponded exactly to the quantumtheoretical basis.

We have developed the fluid model from the thermodynamic behavior of silver. It is to be shown now that this is not a special case of the silver, but that it is a general, thermal phenomenon.

The diagram shows a whole range of solids: a polar molecular compound (H₂O), a salt-like ionic compound (NaCl), a non-polar molecular element (Br) and a metal (Pb). Here as well, you can see that the "energy house numbers" increasingly close together. We have now selected the axes in this diagram so that the temperature appears as a dependent variable. This makes it clear that in the case of substances with large entropies, it requires more thermal energy in order to bring them to correspondingly high temperatures, ie that such substances are thermally inert substances. Thus, ice (1 mol of atoms) is heated with 1 kJ thermal work to 200 K, while 1 mol of lead atoms can be brought only to about 70 K with the same amount of work. To reach the near of 200K more than 4 kJ thermal work is needed for 1 mol lead atoms.

The 3. Refinement

The cylinder of our model storage system has two inflow options. We need these to reproduce the two fundamentally different types of temperature changes:
According to the first law, the internal energy U can be changed by thermal work and/or mechanical work. Because in either case we are dealing with work performed either on the system or by the system, we have replaced the shut-off valve with a symbol which is intended to represent a generator or a pump motor. In the upper case, we were dealing with thermal work. The thermal energy was introduced into the system by the side inflow as the thermal contact was established. Let us now turn to the mechanical work.

The abovementioned temperature increase without entropy increase can also be represented by the fluid model. If the entropy remains constant despite the supply of energy, this means that the cross-sectional area remains the same. Although the hydrostatic pressure still rises with the filling height, the walls do not recede in this case. They become more rigid and thus more stable. This is symbolized by the increasing intensity of the wall color from light gray to dark gray.

Please also look at the visualization of this issue in the fluid model:

Animation: Mechanical Work in Shelf- and Fluid Model

The animation shows that all levels are shifted horizontally parallel, so that all occupation numbers remain the same and only the level intervals increase. As a result of these higher level distances, the blue half-value energy line becomes longer, i.e. the temperature becomes higher. If no particle changes the energy level, this indicates that no electrical alternating fields occur during this process. The temperature change by means of mechanical work, i.e. volume work, therefore, has different physical causes. This different background makes it necessary to distinguish thermodynamically between volume work and all kinds of non-volume work.

The 4. Refinement

This refinement leads us to a level appropriate to the universities. According to classical interpretation, atomic gases can only carry out translation movements in three spatial directions. Each gas atom provides three degrees of freedom for the storage of thermal energy, one each in the x, y and z directions. Quantitatively, a degree of freedom represents a eigenvalue system with an infinite number of quantized energies. The energetic distances of these levels follow certain laws, into which the masses of the gas atoms, the forces acting between them, and the size of the space enter. (See Section 2.4) A single gas atom is located at a certain time in one of these states of the x, y, and z eigenvalue systems. The states in the different spatial directions can be excited equally or differently.
Each state is linked to a space function, which describes how the corresponding atom is distributed over the entire space. Since this function is determined from the time-independent Schrödinger equation, it follows that during the lifespan of this state, it has different values in different spatial areas, but these are all constant over time. Such states are therefore called stationary states. Each particle can switch to another stationary state with a different space function when either a photon is emitted or absorbed.
The behavior of particles in such a state makes it clear that this state is not compatible with a state of motion during a location change of a particle. An illustrative idea of the different translational behavior of an atom in differently excited translational states does not result from this. However, the term "translational state" has been retained.

In the case of the molecular gases, there are two further types of storages*, which differ markedly in their thermal properties from the translational states. The denoting of these thermal storages is based on the classical idea of rotational and vibrational motion.

If the gas forms diatomic molecules, the number of degrees of freedom, ie the number of eigenvalue systems for the storage of thermal energy, is retained (see the following table). However, the dependence of the energy level distances on the particle mass, the interatomic forces and the size of molar volume changes.

In the classical concept, diatomic molecules can rotate about two axes of rotation and they can vibrate with each other. The two axes of rotation are perpendicular to each other and perpendicular to the molecular axis in which the oscillation takes place.
On the quantum level, the states of the corresponding eigenvalue systems have been retained as rotational and vibrational states, although the measurable behavior of these states is not well compatible with the classical ideas of rotation and vibration:
A rotational frequency does not result from the behavior in one of the stationary rotational states, but from the transitions between two such states. The same applies to the vibrational states. Here, too, these transitions are associated with the emission and absorption of photons. The photon frequencies can be interpreted as rotational frequencies or vibrational frequencies. For every eigenvalue there is also a temporally constant space function, which can not be interpreted as a standing wave:
Except at the nodes, standing waves in the entire space have no temporally constant amplitudes.

* The implementation of a fourth storage, namely the storage of thermal energy in the electron states, is dispensed with here since it does not contribute to the entropy and therefore to the thermal behavior of the most substances.

Thus, we have got to know three different types of thermal energy storage, and we now turn to the interaction of these storages in different cases, that is, we are now concentrating on the behavior of the molecular gases first in thermal work and afterwards in volume work. As an example of a molecular gas we take gaseous bromine and heat it with thermal radiation (from a torch or from an electrically heated wire) from the standard temperature to 500K. The following two figures show the three storage systems in the fluid model, left at standard temperature and right at 500K. The base line connects the three reservoirs so that they behave physically as well as classical vessels.

Heating of Br₂(g) by Thermal Work

It is easy to see that the level, ie the temperature, as well as the amount of fluid medium, ie thermal energy, has become larger in all three storages. It is more difficult to see that the entropy, that is, the cross-sectional areas, have also increased. At best it is to recognize this from the vibration storage, since here the area is very small and thus the percental increase turns out to be quite large, about 75%. (Translation + 5.4%, rotation + 5.9%) The display was calculated by the Thermulation-II program and you have to take into account that due to the resolution of the screen, the percentage change may remain below one pixel, so the screen can not display it.

Heating of Br₂(g) by Volume Work

Here, as well, you can see that temperature and thermal energy have increased in all three storages. It would actually be expected that the volume work only affects the translation store, because only these eigenvalues depend on the volume. Then, however, the temperature in the translation storage should be different from that in the two other storages. Since this is difficult to imagine, we have to look for a mechanism by which the thermal energy is transferred from the translation storage to the other two.
In detail, we will address this question only in the next chapter on spontaneous emission. But we have pointed out here the importance of the photons, which always occur when particles change their levels, so we can here already understand the fundamental:
If the temperature rises in the translation storage, an increased emission of thermal radiation begins there. This stronger photon current passes through the base line to the rotation and vibration storages in the fluid model and causes the particles to reach higher energy levels until there the same temperature is reached as in the translation storage, which was the point of attack of the volume work. The changes in the thermal energy due to the volume work are the same in the rotation and vibration storages as in the thermal case, apart from rounding deviations.
Here, we see a difference between ideal and real gases: While the entropy in the ideal gas remains constant despite temperature and energy increase, the entropy of the real gas increases as well. Due to the "fluidity" of the photons thermal energy is transmitted from the translation storage to the other storages and causes there the increase of entropy. The internal degrees of freedom make the bromine a real gas.
As a result of the thermal work, the entropy increased by 7.5% while in the case of volume work the entropy increase was limited to only 3.2%