Andrew Mason said:
Why would Tb always be the same for both the system and the reservoir? What if there is no direct thermal contact? (eg they are connected via a Carnot device so that all heat flow occurs isothermally at the temperature of the system or reservoir). And how do we maintain the first law if dQsystem = -dQsurr? Are you assuming that no work is being done?
This cannot be true if work is being done. For example, this is not true for a reversible Carnot engine operating between the system and the surroundings. Qsys ≠ -Qsurr and Tsys ≠ TB
AM
In the present discussion of second law concepts, we have subdivided the world into
two separate and distinct entities:
Entity 1 has been loosely referred to as "the system"
Entity 2 (which includes everything not included in Entity 1) has been loosely referred to as "the surroundings."
There is a "boundary" between the system and surroundings which does not permit mass to pass through, so that both the system and surroundings can be regarded as
closed . However, the boundary does permit heat to transfer between the system and surroundings (across its interface), and the boundary can move and deform, so that work is done by the system on the surroundings, and vice versa.
The introduction to this picture of a third entity, say a Carnot device, muddies the waters a little, but only slightly. The third entity can be assimilated into either the system or the surroundings, and then we are back down to 2 entities again. In practice, it might be more convenient to assimilate the third entity into the surroundings, if the focus is more on the system.
In general, both the system and the surroundings can be undergoing irreversible processes, and the two regions will be coupled with one another at the boundary interface. What is the nature of this coupling? From our knowledge of transport phenomena and continuum mechanics, we know that certain parameters must be continuous across the interface: velocity vector, heat flux vector, temperature, and traction stress. The traction stress is equal to the stress tensor dotted with a unit normal vector to the interface. For an inviscid fluid, the traction stress is equal to the pressure times a unit normal, and the work being done by the system on the surroundings per unit area of the interface is equal to the pressure times the normal component of velocity.
Now back to our Entities. When most of us studied thermo, we learned it from a historical perspective. Early 2nd law pioneers worked with multiple entities, including heat reservoirs, Carnot engines, etc. As time progressed, and our understanding evolved, the concept of entropy developed, and the mathematics became more precise and refined. It was recognized that the number of entities could be reduced from many to just two, namely, the system and the surroundings. Then Clausius realized that he did not even need two entities to capture the essence of the 2nd law mathematically; he could do it all with just a single entity, the system. (He regarded the surroundings as just another system). Clausius' work formalized the 2nd law in a concise mathematical fashion.
Here is my attempt to paraphrase Clausius' thinking: I have a closed (constant mass) thermodynamic system that is enclosed within an interface boundary β. The system is initially in thermodynamic equilibrium state A, and I wish to bring it to thermodynamic equilibrium state B. I will consider all processes, both reversible and irreversible, that can accomplish this by applying heat transfer at various locations on β, and by moving and deforming β (i.e., doing work at the interface β). As I test these various processes, I will keep track of the local heat transfer rates I apply at the boundary of β (which can vary with position on β and time), and the corresponding temperatures at these locations along the boundary. Then I will calculate the ratios of the local heat transfer rates to the local temperatures, and sum these quantities over the entire boundary. I will then integrate this sum with respect to time, from time zero (corresponding to equilibrium state A), until the time that system has attained equilibrium state B. I will then compare the values I obtain for this integral for all the possible irreversible and reversible paths that I have imposed on the system. According to my understanding of the 2nd law, this integral will vary from path to path, but will never be greater than a certain limiting value. That limiting value is called the entropy change ΔS for the system. If I represents the value of the integral, then:
ΔS = I
max
The integral I attains this maximum value only along a reversible path. Otherwise I is less than I
max.
ΔS ≥ I
Mathematically, Clausius' representation of the second law constitutes a physical principle expressed using variational calculus.
Unfortunately, the mathematical from of the equation used by Clausius to capture the above powerful ideas is very imprecise and ambiguous, and has confused thermodynamics students over the years:
dS ≥ dQ/T
This equation does not specifically indicate where dQ and T are to be evaluated. For a system experiencing an irreversible path, the temperature (as well as pressure) will vary with spatial location and time within the system. So, what value of T should be used?
Also the heat flux within the system will be non-zero, and will vary with location and time. Our discussion above indicates that dQ and T should be evaluated exclusively at the boundary of the system. Our previous discussion also indicates that there can be multiple locations of heat flow in and corresponding temperatures at the boundary.
The continuum mechanics guys have formalized the mathematics of the Clausius inequality even more precisely by integrating the dot product of heat flux and unit normal divided by temperature over the entire boundary surface, and have used the divergence theorem to show mathematically why the
change in entropy exceeds the integral by a positive definite amount. Google the key words continuum mechanics and Clausius inequality to see this.
