Thermal capacities and polytropic processes

Herein, we analyse polytropic processes for an ideal gas within the wider concept of thermal capacity. To answer the question of whether the thermal capacity is a process, path, or state function, we argue that it should be tentatively set as a path function and if it remains constant along the path, the corresponding process is polytropic. Of all the paths, there are only two, at constant volume and constant pressure, for which the thermal capacities, Cv and Cp, are state functions, i.e., system properties. The discussions herein are valuable both scientifically and instructively because they shed light on issues in undergraduate curricula that are not addressed in sufficient detail in physics textbooks, not even in the most advanced ones.


Thermal capacities: process or path functions?
A typical experimental task in teaching thermodynamics is determining the thermal capacity of a system, for example, a given amount of water. One possible approach is to heat the system using a flame and measure its temperature rise. The infinitesimal heat Q δ required for an infinitesimal temperature change dT is related to the thermal capacity C as [1] d Q C T δ = . (1) Despite the conceptual simplicity of this experiment, it is difficult to measure the heat entering the system from the flame. Therefore, from an experimental perspective, instead of using a flame, it is preferable to heat the system using an electrical resistance r placed inside it and becoming a part of it; the resistance is connected to a battery with an electromotive force e ε , which is placed in the surroundings. In this setup, the energy supplied to the system is not v2-2023-04-11 heat but dissipative (electrical) work, D W , whose differential is given by Comparing (1) and (2), the effects of Q δ and D W δ on the system are indistinguishable [2].
However, although the sequence of system states follows the same path when represented in a diagram, the increase in temperature via heat or dissipative work are two distinct processes because while heat leads to the variation of the entropy of both the system and surroundings, dissipative work leads only to the variation of the entropy of the system.
Path and process are different concepts. A path is defined as the sequence of system states described by a mathematical relation involving only the system variables. In contrast, a process is an interaction between a system and its surroundings, described not only by the system variables but also by those of the surroundings.
Assuming (1) is the definition of C, the thermal capacity is a process function but not a path function because a system following a given path may have different values for C depending on whether Q δ or D W δ , or both, exist. The same is true if (2) is considered as the definition of C. Therefore, neither (1) nor (2) alone are adequate definitions as they are inconsistent with the experimental determination of C, yielding different values based on the process, i.e., depending on whether heat, dissipative work, or both are used in the heating process.
The above considerations lead us to merge the two (inadequate) definitions (1) and (2) into a single definition that is consistent not only with the experimental determination of C but also with the fact that both Q δ and D W δ produce indistinguishable effects on the system (in particular, the same temperature change) [2]. Accordingly, we propose the relation Definition (3), which is virtually absent in the existing literature, establishes C as a path function, which, as discussed earlier, is different from a process function. This definition is more suitable than the previous ones because it does not depend on surroundings variables but instead relies solely on the evolution of the system during the process. The next question we consider is: can C also be a state function? If so, in which cases?

Thermal capacities as state functions
To confirm that the definition of C in (3) is indeed a path function, we consider an equivalent expression where only the system variables appear explicitly. As shown in the Appendix, from the fundamental equation that describes a process [3], we have where S is the system entropy. Inserted (4) in (3), we obtain Thus, at each point on a S-T path, C is determined by the temperature and slope. In another representation, for example, a V-T path, S can be considered as a function of T and V, i.e.,

( )
, ; thus, dS in (5) can be replaced by Consequently, Using the thermal expansion and compressibility coefficients, α and T κ , respectively, which are defined as we obtain ( ) . Therefore, (7) can be rewritten as Both (5) and (10) indicate that C, as defined in (3), is a path function and is expressed only in terms of system variables. This is in contrast to (1) and (2) From (5), for isentropic processes, d 0 S = and 0 C = , and for isothermal processes, d 0 T = and C = ∞ . Among all other possible paths, there are two in which the thermal capacity becomes a state function, i.e., a system property. Using (10), one of these paths is defined by d 0 V = (an isometric process), where C becomes the heat capacity at constant The other path is defined by d 0 P = (an isobaric process), where C becomes the heat capacity at constant pressure P C : Considering (8) and (11), (12) becomes To highlight the ability of the thermodynamics formalism, we consider the path d 0 S = (an isentropic process) and use (5), (10) and (11) to obtain an equivalent expression for V C : This illustrates the importance of V C and P C : they are state functions constituting important system properties, whereas all the other thermal capacities, given by (5) and (10), are path functions. Thus, V C and P C depend only on the system state, and the indices V, P and S on the right-hand sides of (11), (12) and (14) do not refer to specific paths but rather denote the variables to be held constant during partial derivation. This is an important point that is repeatedly misunderstood.
Simple systems (also called PVT systems) are characterised by several properties, only three of which are independent. The easiest properties to measure are P C , α and T κ ; consequently, these are selected as the fundamental properties [4,5] to which all the others can be related; for example, V C can be obtained from (13).

Polytropic processes of an ideal gas
For liquids and solids, 2 0 T α κ ≈ , which implies that V P C C ≈ , based on (13). However, this is not true for gases, for which V C and P C differ significantly. Considering an ideal gas, the v2-2023-04-11 5 state equation is where n is the amount of gas (in mol) and 1 1 8.314 Jmol K R − − = is the universal gas constant.
Applying (15), (8) and (9) for an ideal gas turn into and 1 T P κ = . (17) Inserting these coefficients into (10) and using (11), we obtain Adopting the definition in [6], a polytropic process of an ideal gas is one wherein the thermal capacity remains constant throughout the entire process. Therefore, taking C β as a constant, where β is the polytropic index, (18) can be expressed as Using the state equation (15) we obtain which, when integrated, yields where K is another constant and β is given by Typically (21) is considered the definition of a polytropic process [7]; however, as argued in [6], we consider it to be a result rather than a definition. In a P-V diagram, a polytropic process with an index β follows the path defined by (21) and, by (22), its thermal capacity C β is graph, we conclude that if 1 β γ < < , then 0 C β < ; these are processes for which, by (5), the changes in temperature and entropy of the system have opposite signs [6]. is the adiabatic coefficient. 0 If the system is not an ideal gas, it is difficult to obtain an analytical expression for a polytropic process because not only is V C not constant but also (15)-(17) are no longer valid.
In this case, the curve describing a polytropic process must be determined by solving (10) numerically while imposing a constant value for C, which requires that V C , α and T κ are known throughout the process.

Conclusions
Herein, we analysed the concept of thermal capacity and ascertained the situations in which it is a process, path, or state function. This distinction between process, path, and state is a novel approach that clarifies certain misunderstandings related to this topic, namely, regarding the essential role of dissipative work in conjunction with heat. In general, the thermal capacity is a path function except for two special cases -constant volume and constant pressure processes, wherein their thermal capacities are also state functions, i.e., system properties. Finally, we discussed the polytropic processes of an ideal gas, from a new perspective within the thermal capacity formalism.

Appendix
For reference, (4) is derived here. The application of this formalism to the introductory problem is also discussed. Further details can be found in references [2,3].
A thermodynamic system is characterised by the macroscopic, or state, variables. These comprise temperature T, entropy S, and mechanical variables, the specifics of which depend on the problem under study and are typically referred to as generalised forces Y and generalised displacements X [1]. The conservation of energy principle, in conjunction with the fundamental equation [1,8] applied to the system and surroundings, gives where the subscript 'e' (for exterior) denotes the variables of the surroundings. (A1) describes a thermodynamic process as an interaction between the system and its surroundings, which contains all its physics. Thus, heat and work, which are process functions representing energy crossing the boundary that separates the system from its surroundings, cannot add anything essential to (A1).
Taking the Zemansky definition [1], in line with the reservoir concept [9], work is energy that crosses the boundary and whose overall effect can be described as 'the alteration of the position or configuration of some external mechanical device'. This is the ultimate criterion that determines whether the energy crossing the boundary is work or not [1]. The work is thus given by the term e e d Y X in (A1), i.e. e e d W Y X δ = . (A2) Because each side of (A1) is the internal energy variation of the system, dU , which by the and the difference From the previous equations (A2)-(A5), d T S in (A1) and dU can be written as The brackets in (A6) and (A7) stress that heat, Q δ , and dissipative work, indistinguishable effects on the system. However, they cannot be perceived as the same, because heat changes the entropy of surroundings, given by (A3), but dissipative work does not.
It is instructive to apply this formalism to the problem of determining thermal capacity, presented at the beginning of the paper. The temperature of the system (e.g. a given mass of water) can be raised by heat from a flame (which belongs to the surroundings) or by using an electrical resistance (which belongs to the system) connected to a battery (which belongs to the surroundings).
The system is open to the atmosphere at pressure e P , equal to that of the system, P. Therefore, there are three energy interactions to consider. The heat from the flame is given by (A3 As energy entering the system is taken as positive, and as negative otherwise, the minus signs in (A8) are needed because the decrease in battery charge corresponds to energy entering the system. Lastly, the interaction resulting from the system expansion, in which the volumes