The second law of thermodynamics and the uniqueness of heat flow?

Question

From wiki:

A simple statement of the law is that heat always flows spontaneously from hotter to colder regions of matter (or 'downhill' in terms of the temperature gradient).

I have a question:

Isn't it true for all the gradients (gravitational, electrical, chemical, pressure, concentration etc.) in physics? Energy flows to close gradients in particular direction (to even out). Evergy gradients all act like that. I mean,

• water spontaneously flows from high pressures to low pressures,
• stones spontaneously fall from higher elevation to lower elevations,
• electrons spontaneously flow from high potential to low potential (yeah, there is a convenction for the direction of the flow +-, but you know what I mean: gradient spontaneously close, just like the temperature gradient),
• etc.

1. Why speak of "heat" only in "heat always flows spontaneously from hotter to colder".
2. Why don't we speak of "energy always flows spontaneously to close a corresponding gradient to spread out more", why talk only about "heat" like it's special when there are examples above.

Don't get me wrong, I know that "heat" is special. But I don't see how this formulation of the second law renders it special among other gradients and energy fluxes.

P.S. I guess, we rather need to speak of "thermal energy" and not of "heat" but the quote is the quote.

Answer 1

Would "because we're talking about thermodynamics" be too glib an answer?

Note also that water, stones, and electrons are not energy. There are gradients and corresponding "flows" in the scenarios you describe, but they all occur for different reasons (molecular collisions, gravity, and the electromagnetism, respectively). In that sense they are analogous, and the flow of heat does fit the analogy.

But there are a thousand and one ways to make a plausible sounding analogy which isn't actually true. What if I said that heat flows from high thermal energy density to low thermal energy density? That fits the template of the analogy, but is completely wrong, so it's rather important to include the correct assertion as part of the foundation of thermodynamics.

Answer 2

Just to add something to this discussion, I'd like to say that I've found other people who are also bothered by similar thoughts:

However, it is strange that the second law of thermodynamics is quite different from other laws, like Newton’s law of motion, Ohm’s law of electric conduction, and Fourier’s law of heat conduction, etc. There are differences such as;

(1) there are many different statements for the same law;

(2) it is only a qualitative description of a physical phenomenon, rather than a quantitative relationship between different physical quantities; and

(3) some phenomena similar to the Clausius Statement exist in other disciplines. For example, an object can never move from low to high location in a gravity field without some other change, and electrical charges can never move from low to high potential in an electrostatic field without some other change.

However, none of them has been accepted as the statement of a certain law. With that in mind, we have reviewed Clausius’ mechanical theory of heat, published in the nineteenth century [14,15], and indeed found that he himself only called “Heat can never pass from a colder to a warmer body without some other change, connected therewith, occurring at the same time” a fundamental principle rather than the statement of the second law of thermodynamics.

source: "What Is the Real Clausius Statement of the Second Law of Thermodynamics?"

another quote by the authors:

Considering the natural tendency that heat always passes from a warmer to a colder body for eliminating the temperature difference, Clausius presented a well-known proposition that “Heat can never pass from a colder to a warmer body without some other change, connected therewith, occurring at the same time”.

... this proposition had only been laid down as a fundamental principle by Clausius.

In spite of this, regrettably, ever since then, this fundamental principle was usually mistaken for the Clausius Statement of the second law, probably because of its well-understood empirical property.

So, it looks like, the authors agree that "heat flowing down gradient" is not a rendition of the 2nd law, and thus has little to do with entropy...

Answer 3

Isn't it true for all the gradients (gravitational, electrical, chemical, pressure, concentration etc.) in physics? Energy flows to close gradients in particular direction (to even out). Energy gradients all act like that.

You are correct.

However, outside the realm of statistical mechanics, the calculation of the entropy change, if any, for any process will generally require that the change in entropy be expressed in terms of heat and temperature, and not the other physical phenomena you've mentioned and there associated gradients. Consider the following:

Heat always generates entropy for any finite temperature difference (any gradient) making it inherently irreversible. Only when the gradient approaches zero will heat approach being reversible.

On the other hand, your other examples may or may not generate entropy. For example, an object falling in a vacuum may generate no entropy. If the object (the system) happens to collide with an ideal spring (the surroundings) causing it to rebound, it can theoretically return to its original state with no change to either the system or surroundings. Unlike heat, the process can theoretically be made reversible regardless of the "steepness" of the gradient.

But when these other examples do generate entropy, it is most often the result of the dissipative effects of some form of friction. That results in, or is generally associated with, the cause and/or effect of heat. While the study of mechanics, fluids and electromagnetism accounts for the dissipative effects as a loss of some form of mechanical energy or electrical energy, that is where they generally stop, with thermodynamics taking over. Thermodynamics then defines entropy change as a function of a reversible transfer of heat, $\delta Q_{rev}$ at a boundary temperature of $T$:

$$dS=\frac{\delta Q_{rev}}{T}$$

For a total entropy change of the system between two equilibrium states of:

$$\Delta S_{12}=\int_1^2\frac{\delta Q_{rev}}{T}$$

So the @J.Murray statement "because we're talking about thermodynamics" really isn't too glib.

Hope this helps.

Answer 4

I’d like to address one of your comments, because I think it provides the best summary of your question:

I really don't understand why other types of gradients seem to behave according to the second law, but the law doesn't mention them and only focuses on the heat flow.

Note that the text you quote in your question states “A simple statement of the law” (emphasis added). This version focuses on spontaneity of heating. More generally, the Second Law says that total entropy tends to increase.

One example of this is heat flow down a temperature gradient. Explanation: Infinitesimal heat flow $q$ from (into) a small region at approximately constant temperature $T$ removes (adds) infinitesimal entropy $q/T$. From this we find that heat flow down a temperature gradient generates entropy (compare denominators of $T_1>T_2$ on the outcome), whereas heat flow up a temperature gradient—all else equal—destroys entropy. Thus, only the former is permitted. (You asked about the specific dependence on the gradient. The volumetric entropy generation rate for one-dimensional conductive heat flow in some region at temperature $T$ is $\sigma=\frac{k}{T^2}\left(\frac{dT}{dx}\right)^2$, with thermal conductivity $k$; see Balluffi, Allen, and Carter, Kinetics of Materials, for instance, for a derivation.)

But another example of this is minimization of energy potentials (e.g., enthalpy, the chemical potential, the gravitational potential energy) for nonisolated systems. Explanation: derived here. For example, the Gibbs free energy $G$ and thus the chemical potential $\mu$ are minimized for systems in thermal and mechanical contact with their surroundings. For an isothermal pressure gradient, therefore, a chemical potential shift $d\mu=v\,dP$ (molar volume $v$, pressure $P$) tends to eliminate the gradient. In this way, an infinitesimal volume shift $dV=-w/P$ is spontaneous if it increases the volume of the more-compressed system, with more details given in Callen, Section 2-7. This enforces a preferred direction for work $w$ done on a system in this context, analogous to the preferred direction for heating $q$.

So the resolution that best addresses your question is that directional heating spontaneity and energy minimization (in some cases, directional work spontaneity) both follow from the broad tendency of entropy maximization, although only the former is emphasized in the Wikipedia quote.