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First Law of Thermodynamics
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Standard Presentation
Following the detailed discussion of the laws of thermodynamics that can be found, for example, in Chapter I of [C67] we know that, for an infinitesimal quasistatical change of state, the change in the total heat content of a fluid element is given by the,
Fundamental Law of Thermodynamics



[C67], Chapter II, Eq. (2)
[H87], §1.2, Eq. (1.2)
[KW94], §4.1, Eq. (4.1)
[HK94], §1.2, Eq. (1.10)
[BLRY07], §1.6.5, Eq. (1.124)
where, is the specific internal energy, is the pressure, and is the specific volume of the fluid element. Generally, the change in the total heat content can be rewritten in terms of the gas temperature, , and the specific entropy of the fluid, , via the expression,
If, in addition, it is understood that the specified changes are occurring over an interval of time , then from this pair of expressions we derive what will henceforth be referred to as the,
Standard Form
of the First Law of Thermodyamics

[T78], §3.4, Eq. (64)
[Shu92], Chapter 4, Eq. (4.27)
[HK94], §7.3.3, Eq. (7.162)
If the state changes occur in such a way that no heat seeps into or leaks out of the fluid element, then and the changes are said to have been made adiabatically. For an adiabatically evolving system, therefore, the First Law assumes what henceforth will be referred to as the,
Clearly this form of the First Law also may be viewed as a statement of specific entropy conservation.
Entropy Tracer
Initial Recognition
Multiplying the Adiabatic Form of the First Law of Thermodynamics through by and rearranging terms, we find that,












is an equally valid statement of the conservation of specific entropy in an adiabatic flow. In combination, first, with
Form B
of the Ideal Gas Equation
and, second, with the
we may furthermore rewrite this expression as,









This relation has the classic form of a conservation law. It certifies that, within the context of adiabatic flows, the entropy tracer,
is the volume density of a conserved quantity. In this case, that conserved quantity is the entropy of each fluid element.
Substantiation
To further substantiate this claim, we note that,






Now, from the first law, we can write,












which, upon integration, gives,



To within an additive constant, this is precisely the expression for the logarithm of the entropy tracer, as provided immediately above. Hence, we see that,
that is, we see that the variable, , traces the fluid entropy just as traces the fluid mass.
We have found one other instance in the literature — although there are undoubtedly others — where the role of this entropy tracer previously has been identified. In chapter IX of [LL75] we find that, "apart from an unimportant additive constant," the specific entropy is,



[LL75], §80, Eq. (80.12)
Given that , this is clearly the same expression as we have derived for the specific entropy of the fluid.
Incorporation Into the First Law
Multiplying the Standard Form of the First Law of Thermodynamics through by , we can now write,





















Now,









Hence,



Marcello & Tohline (2012), §2.2, Eq. (31)
Notice, as well, that we can write,















Specifically for the case, , this gives,



[Shu92], Chapter 9, Eq. (9.26)
It is fair to say, therefore, that in this specific case [Shu92] also recognized the relevance of and the conservative nature of, what we have referred to as, the entropy tracer.
Nonadiabatic Environments
LeftHand Side (LHS)
There are several potentially useful expressions for the timerate of change of fluid entropy.











Clearly to within an additive constant, an expression for the fluid entropy, itself, is





In optically thick environments where the radiation field is intermixed and in equilibrium with the fluid (gas), the timerateofchange in the entropy of the radiation field is characterized by the expression,





Hence, to within an additive constant, an expression for the entropy of the radiation field is,



RightHand Side (RHS)
Example A
One physically reasonable pair of sources/sinks of entropy in the fluid arise in the context of what [LL75] identify as the general equation of heat transfer, namely,



[LL75], §49, p. 185, Eq. (49.4)
[Shu92], Vol. II, §3, p. 30, Eq. (3.26)
[P00], Vol. I, §8.4, p. 369, Eq. (8.35)
In this expression,



[Shu92], Vol. II, §3, p. 28, Eq. (3.19)
where, is the coefficient of thermal conductivity; and the rate of viscous dissipation,



[Shu92], Vol. II, §3, p. 29, following Eq. (3.25)
where is the "viscous stress tensor," as defined, for example: by equation (15.3) on p. 48 of [LL75]; by equation (44) on p. 52 of [T78]; and by equation (8.34) on p. 369 of [P00]. Note that when [Shu92] defines — see his equations (3.19) and (3.20) on p. 28 — he implicitly zeroes out the coefficient of bulk viscosity component, keeping only the shear viscosity component because it is the piece that is usually of interest in astrophysical discussions. [Shu92] goes on to explain — see on p. 23 immediately following his equation (2.36) — together, the pair of terms on the righthandside express the "time rate of adding heat (through heat conduction and the viscous conversion of ordered energy in differential fluid motions to disordered energy in random particle motions)."
Example B
In addition to the pair of source/sink terms that arise from the general equation of heat transfer, [T78] includes another pair of terms that often arise in discussions of stellar structure and evolution. Specifically, on p. 56, his equation (65) states,
(Note, that [T78] uses the variable notation in place of .) In this expression, expresses the rate at which (specific) energy is released via thermonuclear reactions, and
where [T78] refers to



[T78], §3.4, p. 57, Eq. (68)
as the coefficient of radiative conductivity. The expression for the radiation flux, , presented by [T78] is identical in form to the expression presented above for the flux due to heat conduction, . This highlights the similarities between the manner in which nature handles transport processes ("Fourier's law") — whether by heat conduction (electrons) or radiative diffusion (photons).
Alternatively,^{†} "… recognizing as the energy density of blackbody radiation, we see that [the expression for that appears as equation (2.17) in Volume I of Shu92] has the general form for diffusive fluxes (Fick's law):
where is the diffusivity. Indeed, this comparison allows us to identify the radiative diffusivity as having the characteristic formula,
where is the (Rosseland) meanfree path of the diffusing particles (photons). A 'random walk' slows down the freeflight speed by a typical factor of , so that the time for photons to diffuse to the surfacce of the Sun is roughly times longer than the freeflight time of 2 s. This process prevents the Sun from releasing its considerable internal reservoir of photons in one powerful blast, but instead regulates it to the stately observed luminosity of erg s^{1}."  
^{†}Text in a green font has been taken directly from Volume I, §2, p. 17 of [Shu92]. 
Example C
In astrophysical discussions of the timerateofchange of the fluid entropy, it is not unusual to include a scalar function, , that accounts in a generic manner for volumetric gains of energy due to local sources, and another scalar function, , that accounts in a generic manner for volumetric loses of energy due to local sinks. In place of the above "Example A" righthandside expression, then, we would expect to see,



When, for example, the fluid (gas) is exposed to photon radiation, heating of the fluid by the radiation is handled by setting,
and the fluid cools — returning energy to the radiation field — according to the reciprocating expression,
where, is the StefanBoltzmann constant. In such a case, the righthandside of the equation describing the corresponding timerateofchange of the entropy of the radiation field, , would necessarily contain the same two terms, but in both cases with opposite signs. That is, the entropy of the radiation field sees as a source while it sees as a "sink."
[In addition to and , other terms involving spatial variations in the velocity field and in the radiation energy density also appear on the righthandside of the expression for . For simplicity, and because these other terms are not relevant to the principal point we are making, we have opted not to detail the entire expression for here. The additional terms and details can be found in, for example, J. C. Hayes et al. (2006) or D. C. Marcello & J. E. Tohline (2012).]
First Elaboration
When the expressions for and are added together to obtain a prescription for the timerateofchange of — see, for example, "Example B" above — neither of the functions, or , will appear explicitly because they have opposite signs in the two separate expressions. This will be the case whether the environment is optically thin or optically thick.
Second Elaboration
In an optically thick environment where local thermodynamic equilibrium has been achieved, , so,
In such an environment, we also expect , so the heating and cooling terms will cancel out each other. As a result, the quantity will disappear from the separate expressions for and .
© 2014  2020 by Joel E. Tohline 