Interrelationship of differential changes of thermodynamic potentials in a system in which a reaction takes place intending to obtain useful work in isentropic conditions–lectures adapted to sensing learners

2.1 Differential change in internal energy

Consider a closed reactive system in which only one reaction takes place to obtain useful work (analogously, the supply of energy to perform some reaction can be observed).

In general, the differential change in the internal energy of a closed system (sy) – which only exchanges energy with the environment (Figure 1) – can be the result of an infinitesimal change in entropy (dS _sy), which is related to the quasi-static transfer of heat between sy and the environment (δq _sy = TdS _sy); an infinitesimal change in the volume of the system (dV _sy) – i.e., the quasi-static differential pressure-volume work exchanged between sy and the environment (δW _V  = -pdV _sy); and the infinitesimal change in the amount of the i-th reactive component ( d n sy i ) due to the quasi-static reaction ⁶  – d W nonV = ∑ μ i d n sy i where W _nonV denotes the change in internal energy in sy due to of a quasi-static chemical reaction, i.e., it represents useful non-pressure-volume work obtained from the system. ^{7 , 8}

In equation (1), T, p, and μ are intensive quantities from the system sy and are, respectively, temperature, pressure, and chemical potential–which is defined for each component from the system. In sy, the change in the amount (mol) of reactive components that belong to the same reaction can be expressed in one extensive quantity (ξ _sy) – the degree of progress of the chemical reaction–extent of the chemical reaction. ^{9 , 10} The extensive quantity ξ _sy as an intensive quantity corresponds to De Donder’s affinity of a chemical reaction (Appendix A). ^{11 , 12}

ν _i denotes the stoichiometric coefficients of the reactive components in the observed reaction (∑_reaction ν _ii = 0) from sy. Equation (1) is the total differential of internal energy in the system sy; therefore, it can be expressed with partial differentials of internal energy:

Figure 1:

A closed system sy in which a chemical reaction takes place: the differential change in the internal energy of the system.

Equations (1) and (3) obtain the definitions of intensive quantities using partial derivatives of internal energy: ∂ U / ∂ S V sy , ξ sy = T , ∂ U / ∂ V S sy , ξ sy = − p and ∂ U / ∂ ξ S sy , V sy = − A . Using the subscript sy for the differential changes of extensive quantities and the partial differentials–especially at the beginning–makes it easier for students to relate the differential changes to physical space (the system where the changes take place) instead of equations (1) and (3) being differential changes of an abstract meaning (Figure 1). Callen ⁷ to equations (1) and (3) as the fundamental thermodynamic equation in differential form and energy representation. As Kandepudi and Prigogine ¹¹ noted, this equation originally came from Gibbs. Equation (1) illustrates the first law of thermodynamics, which students typically recognize through its definition of the products of intensive quantities and the differential changes of corresponding extensive quantities as infinitesimal heat and work.

According to equation (1), the quasi-static exchange of infinitesimal heat between system sy and the environment (δq _sy) results in a differential entropy change in sy (dS _sy):

However, there is also a differential change in entropy that is not covered by equations (1) and (3), which is a differential change in entropy due to the infinitesimal progression of the chemical reaction from the system (dS _ξ ). So, the total differential change of entropy in sy is (Figure 2):

Figure 2:

The change in entropy of a system of constant volume is a consequence of the heat exchange between the system (sy) and the environment or the occurrence of a chemical reaction in the system or both processes simultaneously.

Suppose in sy, i.e., in equations (1) and (3), entropy S _sy (entropy related to heat exchange) and volume V _sy are constant (S _sy,V _sy = const.). In that case, i.e., the system is rigid (isochoric) and adiabatic (the system does not exchange heat with the environment: dS _sy = 0→TdS _sy = δq _sy = 0), the differential change of internal energy in system sy is:

Let reaction ∑ _i ν _ii = 0 from sy occur quasi-statically and isentropically (approximations for a reversible thermodynamic process), ^{7 , 8 , 13 , 14} which implies that the differential change of the total entropy in the system is zero (dS _T = 0). However, as the reaction proceeds in an adiabatic system (dS _sy = 0), it follows that under isentropic conditions–according to equation (5) – the equality dS _ξ  = 0 must also apply (Figure 2). In other words, if equality ⁵ holds, in sy without heat exchange with the environment under the isentropic conditions, the reaction ∑ _i ν _ii = 0 does not change the system’s entropy. Under isentropic conditions in the case of d U S sy V sy sy < 0 , the change in d U S sy V sy sy corresponds to the differential (useful) non pressure-volume work (dW _nonV) – differential work of a reversible process (δW _rev), i.e., the maximum differential work that approaches the value of the differential work of a quasi-static process -

that can be obtained from system sy (Figure 3).

Figure 3:

A closed rigid and adiabatic system sy in which a chemical reaction takes place under isentropic conditions (A) and under non-isentropic conditions (B).

During the lecture, usually, a student, after the introduction of equations (6) and (7), asks the following question: what if there is a change in entropy (dS _ξ ≠0) during the quasi-static reaction ∑ _i ν _ii = 0 in a rigid adiabatic (dS _sy = 0) system? Answer follows from equation (5):

Suppose equation (6) is valid, and there is a chemical reaction–a thermodynamic process–in a system with dS _ξ ≠0. In that case, the thermodynamic process in system sy cannot be isentropic or reversible (in a reversible process, the total entropy does not change). Namely, there is no possibility for dS _ξ compensation due to heat transfer between the system and the environment. However, it should be emphasized to the students that even under conditions dS _ξ ≠0, if the observed reaction from sy is in the quasi-static regime, then equation (6) is still valid, and the same amount of work is obtained as when dS _ξ  = 0, i.e., in the quasi-static regime, energy dissipation into heat is negligible. However, due to equality, ⁸ the reaction generally belongs to an irreversible thermodynamic process in the quasi-static regime. Students from earlier lectures know that if a thermodynamic process is non-isentropic (irreversible), then a smaller amount of useful work is obtained than when the same process is isentropic (reversible), for example, during isothermal irreversible expansion of a system, the environment receives a smaller amount of work in absolute value than during reversible isothermal expansion. This means that if system sy in which the reaction ∑ i ν i i = 0 : d S ξ ≠ 0 takes place would be isentropic (dS _T = 0), then by absolute value, a higher value of useful work should be obtained than d U S sy V sy sy = − A d ξ sy . Which is when the isochoric sy can quasi-statically exchange heat with the environment. Under these conditions, according to equations (1) and (3), the total differential of internal energy for system sy is:

Of course, under isentropic conditions in sy, dS _sy = -dS _ξ applies, so the total differential ⁹ is:

It follows from equation (10) that if it is dS _ξ >0, then under isentropic conditions, the inequality holds:

According to the inequality ¹¹ in the case of an isentropic thermodynamic process (chemical reaction) from sy, under the conditions V _sy = const., a greater amount of useful work is obtained by the absolute value for the quantity |-TdS _ξ |, i.e., for the amount of quasi-static heat required for dS _ξ >0 compensation, than under adiabatic-isochoric (S _sy,V _sy = const.) conditions (Figure 4).

Figure 4:

An isochoric system that is isentropic (A) or not isentropic (B).

If the thermodynamic process at V _sy = const. conditions in sy is not isentropic and quasi-static (reversible), then TdS _sy formally represents the thermal energy supplied to sy and represents the energy loss of the useful work -Adξ _sy (for example, if the useful work from the chemical reaction is used to drive the piston and if friction occurs when the piston moves, then part of |-Adξ _sy| is lost as frictional thermal energy |TdS _sy|, i.e., -Adξ _sy<0 and TdS _sy>0 are valid, Figure 4).

2.2 Differential change of enthalpy

In general, from the function of internal energy (U = f(S,V,ξ)≡U(S,V,ξ)), enthalpy (H) is defined by a partial Legendre transformation ^{7 , 14 , 15 , 16}  – in the state function U, the independent variable (extensive quantity) V is replaced by the intensive quantity pressure p (whose values are easily monitored by manometer):

Differentiation of equation (12) leads to the following expression:

If equation (13) refers to the system sy from Figure 1 (dH = dU _sy + pdV _sy + Vdp _sy), then by introducing the total differential of internal energy ¹ we get:

However, with equation (14), usually a student guesses: professor, you left out the subscript “sy” for the symbol H. Then the professor says: let us wait a while longer; let the subscript be a mystery:

In equation (14), the term TdS _sy comes from equation (1) and, therefore, represents the exchanged heat between isochoric sy and the environment: δ q V sy sy = ∂ U / ∂ S V sy , ξ sy d S sy . However, since equation (14) is the total differential of enthalpy, the following expression applies to TdS _sy: δ q p sy sy + surrounding with  p = c o n s t . = ∂ H / ∂ S p sy , ξ sy d S sy . Therefore, the term TdS _sy simultaneously corresponds to the exchanged heat between sy and the heat source both at a constant volume of sy (V _sy) and at a constant pressure in sy (p _sy):

Based on equation (15), it can be noted that during the heating of isochoric sy, the total supplied thermal energy is spent on increasing the internal energy of the system, while when heating isobaric sy, a part of the heat is spent on pressure-volume work ( p ∂ V / ∂ S p sy , ξ sy d S sy ) and the remaining part of the heat increases the internal energy of the system ( ∂ U / ∂ S p sy , ξ sy d S sy ). By introducing the total differential of the internal energy function U(T,V): d U sy = C V sy d T + π T sy d V ( C V sy  = heat capacity of the system at constant volume and π T sy  = internal pressure in the system) into the partial derivative ∂ U / ∂ S p sy , ξ sy , equation (16) changes to the form (Appendix B):

From the above equation ( α T sy = 1 / V ∂ V / ∂ T p is isobaric expansivity), it can be seen that part of the heat at the constant pressure system (sy), which is not spent on pressure-volume work, remains in the system in the form of the kinetic energy of particles ( C V sy d T ) and the form of the potential energy of intermolecular interactions ( π T sy V α T sy d T = π T sy d V ). If the same amount of heat is supplied at the constant volume system, then the total amount of heat is spent on the kinetic energy of the particles. At the same time, the potential energy of intermolecular interactions does not change since the system’s volume does not change ( π T sy V α T sy d T = π T sy d V = 0 ). Considering equation (16) δ q p sy sy + surrounding   with   p = c o n s t . ≡ δ q p sy sy , expression ¹⁴ for the isobaric condition is often written in the form d H p sy = δ q p sy sy + δ W p sy sy , where δ W p sy sy represents the useful work (non-expansion work) obtained from the reaction (i.e., the maximum work under isobaric and reversible conditions), or the minimum work required for some non-spontaneous reaction to take place–for example, the electrolysis of water. ^{17 , 18 , 19} Of course, under isentropic conditions in the quasi-static regime of the thermodynamic process, i.e., in the case of an infinitely slow reaction, the equalities apply: δ q p sy sy = T d S sy and δ W p sy sy = − A d ξ sy .

If the pressure in sy is constant and if sy does not exchange heat with the environment (sy is adiabatic), then equation (14) is:

However, as -Adξ _sy derives from the expression for the total differential of internal energy ¹ , then the following applies:

In order to understand equation (18) in more detail, it is necessary to return to the differential change in enthalpy from equation (13a) under adiabatic conditions and constant pressure of the system:

If in expression, ¹⁹ the two terms for pressure-volume work are not shortened, then it can be seen that the first and second terms (-pdV _sy-Adξ _sy) refer to the differential change of internal energy in sy, while the last term–as in absolute value is equal to the first term, but has the opposite sign–it represents pressure-volume work in the environment (i.e., the first term represents the work of the system on the environment, while the last term in equation (19) is the work of the environment on the system). Furthermore, the environment can only change volume at the expense of the system volume dV _sy = -dV _surrounding; the environment and system sy have a constant total volume (V _sy+V _surrounding = const.). In a quasi-static change of the state of the system, in order for the system and the environment to be in mechanical equilibrium in each infinitesimal step (volume change), the pressures in sy and the environment must be equal, which is possible if the volume of the environment is multiple (infinitely) more significant than the volume of sy (V _surrounding≫V _sy). In this way, the change in volume in the environment does not affect the pressure of the environment; therefore, the environment behaves as a reservoir of constant pressure. The R symbol will be used as a subscript for physical quantities related to the reservoir (dV _sy = -dV _R). The last term in equation (19), according to the total differential of internal energy, ¹ represents the differential change in the internal energy of the reservoir when its composition does not change (there is no exchange of particles with sy and with the environment of the total system (i.e., composite system: sy + R ≡ Csy); also R is not a reactive system, i.e., no chemical reaction takes place in R). According to equation (19), R does not exchange heat with the system or with the environment of Csy, which means that R is an adiabatic system. Therefore, equation (19) is:

Therefore, d H S sy p sy ? represents the differential change in internal energy in both sy and R, i.e., it defines the differential change of internal energy in the entire composite system (sy + R ≡ Csy), which is adiabatic (isentropic) and isochoric (Figure 5). Therefore, instead of the question mark symbol in d H S sy p sy ? , the composite system symbol follows:

Figure 5:

An isochoric adiabatic system (A) is placed in reservoir R that provides a constant pressure in sy; formally, first the chemical reaction takes place in the adiabatic and isochoric sy (B) and only then the adiabatic volume changes in sy (system expansion, C) and R (reservoir compression, C).

It is important to emphasize that enthalpy as a function H(S _sy,p _sy,ξ _sy) depends exclusively on the parameters of system sy. In contrast, the differential enthalpy change (and the finite change) refers to the sum of the system and the pressure reservoir, i.e., to the composite system. Of course, when students learn the properties of enthalpy, in the course of further lectures, superscripts and subscripts for the symbol H are omitted (this also applies to other thermodynamic potentials). Considering, equation (18) the above equality is:

moreover, it has the following equality: ∂ H / ∂ ξ S sy , p sy = ∂ U / ∂ ξ S sy , V sy . Equations (17)–(22) refer to the adiabatic system (dS _sy = 0); therefore, for the chemical reaction in system sy to be an isentropic thermodynamic process (dS _T = 0), the system’s entropy must not change during the progress of the reaction (dS _ξ  = 0). It follows that the observed reaction from the system is reversible (i.e., an infinitely slow quasi-static process), which means that in the case of an exothermic reaction ( d U S sy V sy sy < 0 ), the differential change in internal energy of sy is equal to the useful work ( d U S sy V sy sy = δ W rev ) obtained from sy (from the chemical reaction). Let the same reaction from the adiabatic isochoric system sy (Figure 5A) take place in the adiabatic isobaric system sy (which is in contact with the pressure reservoir R). If, formally, in the adiabatic isobaric sy, the differential advancement of the chemical reaction occurs first (dξ _sy>0) with a constant volume (Figure 5B) – which is allowed by the property of the total differential, ²⁰ then the differential change of internal energy in sy is d U S sy V sy sy , and after the dξ _sy change, the differential change of volume follows (let dV _sy>0, Figure 5C). After these two infinitesimal steps (which actually take place simultaneously), the differential change in internal energy in sy is:

If the differential changes of internal energy and volume are d U S sy V sy sy < 0 and dV _sy>0, then d U S sy sy > d U S sy V sy sy is valid according to equation (23). However, although the reduction of the internal energy of the adiabatic isobaric system is more significant compared to the adiabatic isochoric system, the absolute value of the useful work is not greater than that obtained from the adiabatic isochoric system. Namely, the pressure-volume work -pdV _sy from equation (23) is the differential internal energy that goes from sy to R (in the case of dV _sy>0). At the same time, -pdV _R = pdV _sy is the differential increase in internal energy in R so that in the composite system (Figure 4C), the pressure-volume works in sy and R are mutually compensated, ²⁰ i.e., in the composite system there has been a redistribution of internal energy (pdV _sy) which remains in the composite system. It follows that the differential change of the total internal energy of the composite system Csy ( d U S Csy V Csy Csy ) – which is equal to the differential change of enthalpy in the adiabatic isobaric system ( d H S sy p sy Csy ) – according to equation (20) is d U S Csy V Csy Csy = d H S sy p sy Csy = δ W rev = d U S sy V sy sy and represents the differential useful work obtained from the adiabatic isobaric system (with an isentropic thermodynamic process) which is in the adiabatic R of constant pressure and is equal to the change in the internal energy of the reaction from the adiabatic isochoric sy (Figure 5).

If the isobaric sy is still located in the adiabatic reservoir of constant pressure R, but sy is no longer adiabatic, then according to equations (14a) and (20) the differential change of enthalpy in Csy is:

of course, as with equation (20), it applies dV _sy = -dV _R. Let the differential change of entropy be dS _ξ >0 during the differential advancement of the reaction from sy (during the reaction, sy is formally adiabatic and isochoric). If the thermodynamic process from sy is isentropic quasi-static (dS _T = 0 – the maximum useful work is obtained), then in each infinitesimal step of the reaction advancement from sy, the change in entropy dS _ξ must be compensated with the departure of heat q = ∂ U / ∂ S V sy ξ sy = T d S sy < 0 from sy (formally, the reaction takes place first at adiabatic and isochoric conditions in sy and then heat exchange follows at isochoric conditions between sy and the environment–the environment of the composite system Csy, Figure 6). The differential change of internal energy in the system due to these two thermodynamic processes (reaction and heat exchange) is d U V sy sy = − T d S ξ − A d ξ sy , so equation (25) is:

Figure 6:

The reaction in the isochoric (A) and isobaric (B) system sy, the isobaric system and the reservoir R form the composite system Csy.

The reaction and heat exchange is followed by a change in the volume of the system and reservoir under formally adiabatic and isobaric conditions. Since the total volume of the composite system (Csy) is constant, when the volume of sy and R changes, the pressure-volume work compensates each other, meaning that equation (26) turns into equality:

As the total entropy change is zero in the isobaric thermodynamic process (27), it follows that the internal energy change of the composite system (enthalpy change at isobaric conditions) is equal to the maximum useful work (δW _rev). From equation (27), it can also be concluded that the same amount of work is obtained (at isentropic conditions) when the reaction takes place at isochoric conditions ( d U V sy sy ) or when the reaction takes place at isobaric conditions ( d H p sy Csy ) (Figure 6).

2.3 Differential change of helmholtz free energy

Legendre’s definition of Helmholtz free energy is ^{7 , 14 , 15 , 16}

Therefore, the Helmholtz free energy is a function F = f(T,V,ξ)≡F(T,V,ξ). By differentiating equation (28) and introducing the total differential of internal energy related to system sy, the following equation is obtained:

and if the volume and temperature in sy are constant, i.e., isochoric and isothermal sy is observed, then it is:

Equation (30) corresponds to a chemical reaction during which the entropy of sy does not change (dS _ξ  = 0). Similar to equation (20), in equation (30), the first and second terms refer to the differential change of internal energy in the isochoric system sy: d U V sy sy = T d S sy − A d ξ sy . In each infinitesimal step of the quasi-static thermodynamic process (reaction) accompanied by thermal fluctuation, the differential heat (TdS _sy) is exchanged with the thermoreservoir (R), which is isothermal, isochoric, and non-reactive. If R exchanges heat exclusively with sy, then heat compensation (-TdS _sy = TdS _R) occurs in the composite system (Csy = sy + R), and Csy remains an isentropic system. Therefore, the last term in equation (30) is the differential change in the internal energy of the thermoreservoir. So, according to equation (30), dF is equal to the differential change of internal energy in the composite system Csy under isothermal and isochoric conditions:

Considering equations (18) and (30), the equality follows:

or

Equation (32) can be interpreted in two ways. Indeed, during the progression of the exothermic reaction (if the goal is to obtain useful work) from sy, the entropy of the isochoric and adiabatic system does not change (dS _ξ =0). In each infinitesimal step of the reaction progress, differential useful work is obtained δ W rev = ∂ U / ∂ ξ S sy , V sy d ξ sy , which is also obtained if sy is in contact with the thermoreservoir or reservoir for pressure–the composite system as the entire system (sy + R) is adiabatic and isochoric. Internal energy equivalent to an infinitesimal amount of heat or infinitesimal pressure-volume work is differently distributed within the composite system. Another interpretation of equality (32) is that during the progress of the reaction, not only does the entropy of sy not change, but neither does the volume change nor does it exchange heat with R, i.e., sy has constant temperature, pressure, and volume.

A reaction whose progress results in a change in the entropy of the system (dS _ξ ≠0). If, for example, an exothermic reaction occurs in an isothermal and isochoric system, then in some infinitesimal step of the quasi-static reaction progress − A d ξ sy = d U S sy V sy sy < 0 is valid, whereby each differential step of the reaction progress can be formally considered to take place in an isochoric and adiabatic system (Figure 7A), the environment of Csy (the global environment) from system sy receives the differential change in the internal energy of the reaction d U S sy V sy sy (in the case of an isentropic reaction, the equality d U S sy V sy sy = δ W rev applies). If during the progress of the reaction is dS _ξ >0 and if the quasi-static thermodynamic process in sy is isentropic (dS _T = 0), then according to equation (5), the equality dS _sy = -dS _ξ is valid, so that equation (30) is:

Figure 7:

A formal illustration of the differential change in Helmholtz free energy: first, under adiabatic and isochoric conditions, the chemical reaction (A) progresses infinitely, and the entropy of the system increases infinitely, which is then compensated by the departure of differential heat (in the form of useful work) from sy to the global environment at isothermal and isochoric conditions (B), the lost heat from sy is compensated from the thermoreservoir (C).

Namely, in equation (30), TdS _sy = ∑δq can represents the exchanged heat between sy and other compartments of the global environment, which means that the first term in equation (33) represents the heat leaving the isochoric and isothermal sy (after the infinitesimal progress of the reaction in a formally adiabatic and isochoric system, Figure 7A) to compensate for the infinitesimal increase in entropy (dS _ξ >0) during the progress of the reaction (Figure 7B). However, since sy is isothermal, heat TdS _ξ enters the system from the thermoreservoir (Figure 7C). At the same time, the same amount of heat leaves R (-TdS _ξ ) so that the heat that compensates for the entropy of the reaction from sy comes from the thermoreservoir. It follows that equation (33) is equal to equation (10), i.e., the differential change d F T sy V sy Csy is equal to the differential change in the internal energy of the isochoric system:

Based on equation (33a), it can be concluded that if dS _ξ is > 0 in the observed reaction from sy, then in each infinitesimal step of the reaction progress, a higher differential useful work is obtained in absolute value than the differential change in internal energy of formally adiabatic and isochoric systems (due to differential progress reactions). On the contrary, if dS _ξ  < 0, the differential useful work is smaller in absolute value than -Adξ _sy since part of -Adξ _sy goes to R as heat to compensate for the decrease in reaction entropy in system sy. ²¹ If expressions (10) and (24)–(27) are compared with expression (33a), then the equality follows:

Namely, if sy is in contact with the corresponding reservoir, then the exchanged energy (heat or pressure-volume work) between the system and the reservoir is mutually compensated, i.e., the net effect is to move some of the internal energy from one part of the composite system to another part. In each infinitesimal step of the progress of the reaction, regardless of the presence of R, the system sy from the composite system behaves as an isochoric system without R and with an infinitesimal change in internal energy: d U V sy sy = − T d S ξ − A d ξ sy . This change in d U V sy sy in the global environment Csy represents the differential useful work.

2.4 Differential change of gibbs free energy

The definition of the Gibbs free energy from the internal energy U(S,V,ξ) based on the Legendre transformation is ^{7 , 14 , 15 , 16}

which means that G is a function: G = f(T,p,ξ)≡G(T,p,ξ). We arrive at the expression (36) by differentiating equation (35) and introducing the total differential of the internal energy related to the system sy where one reaction occurs:

If sy is isotherm and isobar, then the above equation is:

In equation (37), the first three terms represent the differential internal energy change in sy. As the H and F functions (thermodynamic potentials) show, appropriate reservoirs of intensive state quantities were required to maintain the system’s constant pressure and temperature. Similarly, with the thermodynamic potential G, a p, T-reservoir (R) is needed to maintain the constant temperature and pressure in sy. The last two terms in equation (37) correspond to the differential internal energy change in R. Since the first term (TdS _sy) and the fourth term (-TdS _sy) in equation (37) are equal in absolute value, it means that R exchanges heat exclusively with sy (analogously, sy can exchange heat exclusively with R). Similarly, the second term (-pdV _sy) and the last term (pdV _sy) in absolute value are equal to each other and represent the pressure-volume work exchanged between sy and R. It can be concluded that the differential change of Gibbs free energy (37) under isothermal and isobaric conditions represents a differential change of internal energy in the composite system Csy = sy + R ( d G p sy T sy Csy = d U S Csy V Csy Csy ). Suppose the composite system Csy is adiabatic and isochoric. In that case, it follows that the entropy change of the system (sy) during the reaction progress is zero–so that the thermodynamic process as a whole would be isentropic–and the differential change in the internal energy of Csy, according to equation (37), is equal to the change in the internal energy of the system (sy) itself ( d G p sy T sy Csy = d U S Csy V Csy Csy = d U S sy V sy sy ). Energy fluctuations, i.e., thermal and pV fluctuation between sy and R, compensate for each other (if they exist). Therefore, the differential change d G p sy T sy Csy equals the differential change in internal energy of a formally adiabatic and isochoric system. Suppose a change in the volume of sy and thermal fluctuation does not accompany the progress of the reaction from sy. In that case, the isobaric-isothermal system is also isochoric-adiabatic (Figure 8). Considering the expression (37) and equality (32), it follows the expansion of equality (32) with the term related to the Gibbs free energy:

or

Figure 8:

Suppose the reaction progress from sy is not accompanied by a change in the system’s volume and a thermal fluctuation with R. In that case, every system from the composite system is also an isochoric-adiabatic sy and is equivalent to an isochoric-adiabatic system without R.

Let us now study the system (sy) with a p,T-reservoir when the reaction’s progress changes the system’s entropy (dS _ξ ≠0). If, formally, the infinitesimal progression of the exothermic reaction occurs first in a system that is adiabatic and isochoric, then the differential change in internal energy of sy is: δ W rev = d U S sy V sy sy = ∂ U / ∂ ξ S sy , V sy d ξ sy (Figure 9A). For example, during the reaction’s differential progress, the system’s entropy increases infinitesimally (dS _ξ >0). For the thermodynamic process to be isentropic, differential heat (-TdS _ξ ) leaves sy to the global environment (in the form of useful work), whereby the reaction entropy in a formally isochoric and isothermal system is compensated (Figure 9B). Since sy is an isothermal system, the lost heat is compensated by the input of differential heat from the p,T-reservoir (Figure 9C). If, during the differential progress of the reaction, the volume of sy decreases infinitesimally (dV _sy<0) under isobaric and isothermal conditions (formally, after the infinitesimal progress of the reaction and the process of infinitesimal heat exchange), then sy gains differential energy in the form of pressure-volume work (-pdV _sy>0) from R (at the same time, R loses differential energy in the form of pressure-volume work (pdV _sy<0); Figure 9D), but as sy is isothermal, this differential excess of internal energy (dU ^ex = pdV _sy<0) from sy goes to the global environment as useful work (Figure 9E):

In the above expression, the differential enthalpy change appears, which differs from the differential enthalpy changes introduced by equations (22) and (27) (Figure 10):

Figure 9:

A formal illustration of the differential change in equation (39).

Figure 10:

Absolute values ​​of differential changes of thermodynamic potentials.

Based on the expression (39), it can be concluded that the global environment of the composite system from Csy receives the differential useful work partly from the system (-Adξ _sy) and partly from the p,T-reservoir (-TdS _ξ +pdV _sy) (Figure 9). If it is d U S sy V sy sy = ∂ U / ∂ ξ S sy , V sy d ξ sy = − A d ξ sy < 0 ; dS _ξ >0; dV _sy<0 and dS _T = 0, then in relation to the other isochoric composite systems discussed so far (Csy where R is a thermal reservoir or a pressure reservoir and Csy = sy–isochoric system without R, Figure 10), from the composite system with the p,T-reservoir, the highest value of useful (non-pressure-volume) work is obtained by absolute value. Therefore, this work is also called maximum non-expansion work ( δ W rev max ), and it is valid (Figure 10):

and

Let the goal be the carry-out of a non-spontaneous chemical reaction in the isentropic and quasi-static regime, and let the differential changes be: d H p T Csy > 0 , d G p sy T sy Csy > 0 and dS _ξ >0. Then, according to equation (39), the differential change of enthalpy in some infinitesimal step of the progress of a non-spontaneous reaction is: ¹⁷

The differential change in Gibbs free energy represents the minimum work ( δ W rev min ) that must be provided to the system for the infinitesimal progression of non-spontaneous reactions. Because the system is isentropic, the value of the infinitesimal change in enthalpy is compensated by the differential amount of heat from the reservoir (surroundings): TdS _ξ  = -TdS _R.

2.5 Reaction entropy

One more comment remains. Sometimes, during lectures, students ask what precisely the entropy change of a reaction is. It is clear to the students that in exothermic reactions, chemical bonds of different energies are broken and created, and the excess energy can be used as useful work or dissipated in the environment in the form of heat. However, the entropy change during the reaction is quite mysterious for them. In the teaching of physical chemistry and thermodynamics, there is a direction that tends to explain thermodynamic functions (i.e., their changes), if possible (or at least to try) at the molecular (microscopic) level, ^{22 , 23} which we will also try. Let the entropy change in the system (where the reaction occurs) with p,T-reservoir be followed, and for simplicity, let the reaction mixture in the system (sy) behave as an ideal gas (IG). The partial molar entropy of some components in the IG mixture at pressure p and temperature T and composition x = (x ₁…x _i …) - the vector of mole fractions in sy–is ^{24 , 25}

where S _ i p , T corresponds to the molar entropy of the observed component in its pure state at a pressure and temperature identical to the sy mixture’s pressure and temperature; at p,T = const. conditions, the entropy of a mixture containing C components is (n = amount of components):

Since the system contains a reactive mixture, the vector x changes as the reaction progresses. Every component in the system is either a reactant or a product of the studied reaction, and no inert components are present. If the amount (mol) of each component from the reactive mixture is expressed using the stoichiometric coefficient and the degree of progress of the reaction, then taking into account expression (44), equation (45) is (x _i(ξ)≡x _i = f(ξ)):

The last term in the above equation represents the entropy of mixing depending on the degree of progress of the reaction. As the values ​​of the mole fractions in the mixture are less than unity, lnx _i(ξ)<0 follows; therefore, it is:

The second term ( ∑ i C ν i S _ i p , T ξ )) in equation (46a) represents the entropy change at ξ = 1 when the reactants and products are in a pure state, i.e., a hypothetical reaction without mixing. If the constant pressure of the reaction mixture is equal to the standard pressure, then the second term of the equation is the standard change of reaction entropy:

or more generally at some constant pressure:

The first term ( ∑ i C n i 0 S _ i p , T ) from equation (46a) represents the entropy of the reactants of the observed reaction in its pure state when it is ξ = 0. The change in entropy during the progress of the reaction at a particular value ξ relative to the entropy of the state ξ = 0 is (Figure 11):

that is, taking into account expressions (47) and (48a):

Figure 11:

The change in entropy of the system during the reaction progress in case ΔS _ξ (p,T) < 0; reactants at time t = 0 are in stoichiometric amount.

If the first term in equation (50) is zero (ΔS _ξ (p,T) = 0) or |ΔS _ξ (p,T)<0|<ΔS_mix(p,T,x = f(ξ)), then the increase in entropy to the equilibrium state (ξ = ξ _e) in the chemical reaction through a series of quasi-static changes of state between ξ = 0 and ξ = ξ _e (Figure 11) is a consequence of the entropy of mixing (ΔS_mix(p,T,x = f(ξ))>0), which is compensated by the departure of heat from the composite system–an isentropic process–into the global environment as useful work that is superimposed on the useful work that originates from the change in the internal energy of the exothermic reaction (Figure 9). During the quasi-static change of the system’s state during the reaction progress to the equilibrium state, ⁶ in each infinitesimal step of the progress of the exothermic reaction, the infinitesimal change in the system’s internal energy enters the differential useful work–isentropic process. Of course, if ΔS _ξ (p,T)>0 and this is the case when the energy is distributed over a more significant number of the product’s quantum states than there are reactant’s quantum state numbers–the partition function of the product increases concerning the partition functions of the reactants–or by in classical mechanics, energy is distributed over a more significant number of product’s degrees of freedom, then ΔS _ξ (p,T) is superimposed on ΔS_mix(p,T,x = f(ξ)) and the entropy of the reaction increases to ξ = ξ _e>ξ = 0.5ξ = ξ _e>ξ = 0.5 (Figure 12). ^{21 , 26}

Figure 12:

The change in entropy of the system during the reaction progress in case ΔS _ξ (p,T) > 0; reactants at time t = 0 are in stoichiometric amount.