# In a Carnot Engine, how does the heat flow from the heat reservoir to the engine if both are at the same temperature?

From what I read about the Carnot Cycle, there are 4 major steps involved, viz. Isothermal Expansion, Adiabatic expansion, Isothermal compression and Adiabatic compression. During the Isothermal expansion phase, the piston-cylinder setup is at a temperature T₁ and a heat reservoir is at the same temperature T₁. When they are brought in contact, heat flows from the hot reservoir to the cylinder and since the temperature of the cylinder remains the same, the cylinder does work (Expands) isothermally. If heat always flows from a hotter body to a colder body, why did heat flow from the hot reservoir to the cylinder given that they are the same temperature?

• Looks like a duplicate of Heat transfer in the isothermal expansion in the Carnot cycle, the first question in the "related questions" sidebar. Commented Oct 23, 2023 at 2:41
• > If heat always flows from a hotter body to a colder body Here is the problem. Thermodynamics theory does not postulate this as a general assumption; it is an approximate empirical law, valid for Newtonian heat transfers. In thermodynamics theory, heat can flow between bodies of same temperature, nothing forbids this. Commented Oct 23, 2023 at 10:00

Allowing the gas to expand a tiny amount (and do a small amount of work on the surroundings) causes the gas temperature to drop a tiny amount relative to the hot reservoir. This then provides the driving force for a small amount of heat to flow from the reservoir to the gas, and restores its original temperature. This is repeated over and over again. The net result is that the gas does work with its temperature essentially equal to the reservoir temperature.

@ChetMiller answer is an excellent answer, present in many textbooks, where one tries to rationalize the apparent contradiction between an isothermal process and the concept of heat transfer as due to a temperature difference. Nothing is wrong with this point of view.

However, I would like to note that in this context, the definition of heat given by Caratheodory and used by other excellent thermodynamicists like Planck and Born appears superior.

Caratheodory definition of heat does not require thermology. It is based on introducing the adiabatic work as one that results in being path-independent. It is an experimental fact that if a thermodynamic system is separated from the external world by particular walls (the adiabatic walls), the work is only a function of the initial and final state. Thus, adiabatic work coincides with the difference of internal energy ($$\Delta U = W_{ad}$$). At this point, heat exchanged in a non-adiabatic process, where work is $$W$$, is defined by the difference $$Q = W_{ad} - W.$$

In the present context, such a definition of heat has the advantage that it doesn't require an additional model of the details of what happens in the quasi-static isothermal process. A model that, in principle, is not unique. One could devise more realistic microscopic mechanisms, not amenable to an increase (even infinitesimal) of the system's temperature.

• According to THOMSEN & HARTKA, "Strange Carnot cycles" Caratheodory's elegant mathematics is bad physics. Here is a quote from Truesdell:"Caratheodory's axioms reflect an error in physics. They presume that the variables sufficient to define mechanical work suffice also to define any thermodynamic system. In the systems treated by the pioneers those variables are pressure and volume. As THOMSEN & HARTKA remarked, Caratheodory's axioms, since they are expressed in terms of these same variables, cannot apply to water in the range of its anomalous behavior.. Commented Oct 23, 2023 at 20:03
• Be it noted that CARNOT, KELVIN, and CLAUSIUS in their basic assumptions had always taken temperature and volume as independent variables upon which the latent and specific heats depend. The difference is not trivial, because the pioneers' choice is general for the fluids they considered, water included; this choice, studies of rational thermodynamics adopt and extend." And here is Thomsen & Hartka "It may be noted that this system does not seem to meet the basic assumptions used in the Carathéodory approach. The state of the system is not uniquely defined by the "deformation coordinate" v ... Commented Oct 23, 2023 at 20:08
• and a single mechanical "nondeformation coordinate"; his assumptions appear valid locally, but not globally. For those who feel that Carathéodory's formulation is the only rigorous approach to thermodynamics, this point would certainly seem to require further investigation." Commented Oct 23, 2023 at 20:08
• @hyportex I do not find Thomsen & Hartka's sentence very clear. They should have been more explicit. Do they refer to the heat definition or the formulation of the second principle? It is also interesting that a comprehensive review of Carathéodory's contributions to Thermodynamics ( Redlich, O. (1968). Fundamental thermodynamics since Carathéodory. Reviews of Modern Physics, 40(3), 556) does not mention Thomsen & Hartka's criticism. Commented Oct 23, 2023 at 21:25
• It started with this Problem I.6 in Sommerfeld: THERMODYNAMICS AND STATISTICAL MECHANICS, page 347: "Imagine a Carnot cycle with water as the working fluid operating between 2C and 6C, so that at 6C there is isothermal expansion and isothermal compression at 2C. It is seen that heat is added during both processes, if the pressure is low enough (cf. (7.10)), and so heat is converted completely into work in violation of the Second Law. How is it possible to resolve this contradiction? Make a qualitative sketch of the isentropes and isotherms in a T , v-diagram in the neighborhood of 4C." Commented Oct 24, 2023 at 22:06