We know that if we brought a body at temperature T' in contact with a heat reservoir at temperature T (where T' < T) entropy of the universe increases. What if we brought a body at temperature T' in contact with a reservoir at temperature T ( where T' > T)? It is supposed also that the entropy of the universe increases. But when am trying to proof it mathematically, I fall into another question: in the first case, the temperature of the heat reservoir remains the same, it is a reservoir after all, but in the second case, does the temperature of the body stay the same? Or it decreases by a certain amount. And the heat reservoir in the second case, does it temperature increases? Or since it is a heat reservoir, it should change the temperature of the body and not vice versa. I should have some misconception going on. Do you have any idea? Thanks in advance.
The following is taken from: “Thermodynamics: Foundations and Applications: by Elias P. Gyftopoulos and Gian Paola Beretta (Dover, 2005). This book is an axiomatic treatment of thermodynamics which avoids statistical mechanics completely. The book is both formal and rigorous. While the mathematics never goes beyond integral calculus, the rigorous proofs have many steps. The total avoidance of statistical mechanics may bother some. Personally, that is my favorite aspect of the book. Statistical mechanics in practice limits the applicability of thermodynamics. Sometimes, statistical arguments fall short because they rely on very specific physical hypotheses. By discarding the statistical mechanics of the book, they broaden the potential use of thermodynamics. The authors present a text which is both logically consistent and practical for applications. For example, when they define reservoir they REALLY define reservoir. There is no ambiguity in how they define it. Definition in Chapter 6.3 (Reservoirs, page 87. “A reservoir R is an idealized kind of system that provides useful reference states both in theory and in applications, and that behaves in a manner approaching the following limiting conditions.” 1. A reservoir passes through stable states only. 2. In the course of finite changes of state at constant or varying values of its amount of constituents and parameters, a reservoir remains in mutual stable equilibrium with a duplicate of itself that experiences no such changes. 3. At constant values of the amounts of constituents and the parameters of each of two reservoirs initially in mutual equilibrium, energy can be transferred reversibly from one reservoir to the other with no net effect on any other system. … Nevertheless, we emphasize that conditions 2 and 3 that define the behavior of a reservoir are so restrictive that the must be regarded as limits that a system obeying the laws of physics can approach but not actually reach.” So if one can change its temperature, it isn't a reservoir. A reservoir is in a perpetual state of thermal equilibrium Systems that approach the limit of being a reservoir have to be very big. The rest of the chapter shows how two reservoirs defined by conditions 1-3 violate at least two laws of physics when joined into a composite system. 1) The uniqueness of a stable equilibrium state. 2) The second law of thermodynamics. So the concept of reservoir as a physical limit has embedded in it a mathematical complication. This may be your conceptual problem: Two reservoirs at different temperatures placed in contact can never come to stable equilibrium. Thus, you can't always make a bigger reservoir by connecting two smaller reservoirs.
Thank you, but this wasn't my question, I was saying how to prove mathematically if a body is placed in contact with a heat reservoir, where the temperature of the heat reservoir is LESS than that of a body, that the entropy of the universe increases!?
M. next, beware, it is not true that entropy of universe increases. Universe does not have entropy. Entropy has meaning only if you refer it to system that can be described by thermodynamic state parameters (p, V...) Entropy is then function of these parameters. You can take as your system the couple body+reservoir. Of course, irrespective of temperatures, the temperature of the body changes to that of reservoir. How to calculate change in entropy of the couple? If the temp. of the body T' is lower than that of the reservoir, the heat will flow from the reservoir to the body. Imagine we perform this process slowly enough so that both the body and the reservoir move close to equilibrium states (quasistatically). Then we can use the formula $$ dS = \frac{dQ}{T} $$ for the increase of entropy for both subsystems. Let ##Q## be total heat transferred to the body so far. For the body, we have $$ dS_B = \frac{dQ}{T_B}, $$ where ##T_B## changes from ##T'## to ##T##. For the reservoir $$ dS_R = \frac{-dQ}{T} $$ In order to calculate this changes for the whole long process, we have to assume something about the dependence of ##Q## on ##T_B##. The simplest thing to do is assume that the body is ideal gas. Try to express ##Q## via ##T_B## and calculate $$ \Delta S_B = \int_T'^T \frac{dQ}{T_B}. $$ The reservoir is simpler: $$ \Delta S_R = -\frac{Q}{T} $$ When you sum the contributions, you should get total entropy increase.
Thanks, but you answered me the other way round, I meant the opposite case. Anyway thank you, I solved the dilemma, and concerning the 'universe' term, it is used in most of books, but I will take what you said into consideration and ask about it. M. Next