AVAILABILITY LOSSES IN GALVANIC CELLS

Entropy Creation, Waste Work and Thermodynamic Efficiency of Galvanic Cells

As an explanation of the preceding discussion, let us consider a Galvanic Cell in which cell discharge reaction is exothermic and the cell is operating with a constant discharge current (I) through an external load resistance (Rext) which is varied in order to maintain the discharge current (I) fixed during the discharge process. The Galvanic Cell internal resistance is (Rbat) which is assumed to be constant during discharge. The Galvanic Cell open circuit voltage is (Vocv) and its close circuit voltage is (Vccv). The Galvanic Cell discharge current (I) is related to its open circuit voltage (Vocv), internal resistance (Rbat) and external load resistance (Rext) as,

I = Vocv/(Rbat+ Rext) (14)

Now, if one applies Eq.6e for the said Galvanic Cell, following expression can be written for estimation of created entropy σ in the cell for its discharge.

∆Sgc + Q0gc/ T0 = σ (15)

In Eq.15, ∆Sgc is entropy change for process materials employed in the cell, Q0gc is quantity of heat transferred to the environment which is at constant temperature T0 by the process materials in the cell.

In order to evaluate ∆Sgc in Eq.15, following relation can be used for a Galvanic Cell discharge reaction.

∆Sgc = nF (dEgc/dTgc) (16)

In Eq.16, n is stoichiometric number of electrons involved in electrode reaction, F is Faraday’s constant and (dEgc/dTgc) is temperature coefficient of electromotive force (Egc) of the Galvanic Cell. Tgc is operating temperature of the Galvanic Cell. Temperature coefficient of electromotive force (dEgc/dTgc) at a particular cell operating temperature Tgc can be determined experimentally with great accuracy for Galvanic Cells [7]. Q0gc which is magnitude of heat transferred to the environment at constant temperature T0, can be obtained from the equation of first law of thermodynamics. The relevant expression for a Galvanic Cell (in which cell discharge reaction is exothermic) can be given as,

∆Ugc = Q0gc – Wgc (17)

In Eq.17, ∆Ugc is change in internal energy of the process materials employed in the Galvanic Cell and Wgc is total net electrical work (work other than pressure x volume work) obtained from the cell during its discharge process. Total net electrical work output from the Galvanic Cell is also given as,

Wgc = nFEgc = -∆Ggc (18)

In Eq.18, ∆Ggc is Gibbs free energy change for the process materials employed in the Galvanic Cell during discharge. Furthermore, ∆Ugc can be obtained from the following equation for Galvanic Cell discharge process.

∆Ugc = ∆Hgc - P∆Vgc - Vgc∆P (19)

In Eq.19, ∆Hgc is known as heat content change for the process materials employed in the Galvanic Cell during discharge. ∆Vgc is change in volume for the process materials employed in the Galvanic Cell during discharge and ∆P is change in operating pressure on the Galvanic Cell. Since one can reasonably assume that ∆Vgc = 0 and

∆P = 0 for a Galvanic Cell discharge process, ∆Ugc becomes equal to ∆Hgc in a Galvanic Cell during discharge.

∆Hgc possesses a negative value because the discharge reaction is exothermic in the present case of discussion. Heat content change ∆Hgc can be easily evaluated from the relevant Gibbs-Helmholtz relation which is given below.

∆Hgc = -nFEgc + nFTgc(dEgc/dTgc) (20)

Magnitude of heat transferred to the environment Q0gc can then be conveniently evaluated from Eq.17 which will bear a negative value as the discharge reaction is exothermic. Thereafter, created entropy σ can be calculated from Eq.15 and evidently magnitude of Waste Work will hence be estimated as T0σ.

Thermodynamic efficiency for the Galvanic Cell from perspective of second law can then be estimated by evaluating first the availability change ∆Bgc for the Galvanic Cell discharge process. Following Eq.6i, availability change ∆Bgc for the Galvanic Cell discharge process can be given as,

∆Bgc = ∆Ugc + P0∆Vgc - T0∆Sgc (21)

Since it can be assumed that ∆Vgc = 0 for Galvanic Cell discharge process, ∆Bgc can be given as,

∆Bgc = ∆Ugc - T0∆Sgc (22)

Total available work from the perspective of second law of thermodynamics or the “total work recovered” can be obtained from following equation in light of Eq.6k,

Wgca = [T0σ] + [∆Bgc] (23)

In Eq.23, Wgca is total available work or “total work recovered” from the Galvanic Cell during discharge process, [∆Bgc] is availability change for the Galvanic Cell during discharge which is same as “recoverable reversible work” for its discharge and [T0σ] is Waste Work in the Galvanic Cell during its discharge. Thermodynamic Efficiency which is defined as ratio of “total work recovered” to “recoverable reversible work” for a Galvanic Cell discharge process can be estimated as,

It should be pointed out here that [Wgc] which is total net electrical work obtained from the cell during its discharge process need to be equal in magnitude to [Wgca] which is total available work recovered from the cell during its discharge. Thus Mod [Wgc] = Mod [Wgca] although [Wgca] will bear negative sign in contrast to [Wgc] which will have a positive numerical value.

## Effects of Discharge Current (I) and Environment Temperature (To) on σ, T0σ and 𝑬𝑻

In this work, numerical calculations have been carried out to find out how variations in environment conditions and discharge current affect critical performance parameters which are Entropy Creation **σ**, Waste Work **T****0σ **and Thermodynamic Efficiency *ET *of any Galvanic Cell.

### Case: Discharge reaction is Exothermic

As an example, equations and concepts presented in the preceding discussion under section 4 are utilized for evaluation of Entropy Creation **σ**, Waste Work **T****0σ **and Thermodynamic Efficiency *ET *for a Galvanic Cell which has following configuration.

## Al/Al3+, SO42- // Cu2+, SO42-/ Cu (Exothermic) [8]

The Galvanic Cell open circuit voltage (Vocv) at start of discharge = 0.6 V The Galvanic Cell open circuit voltage (Vocv) at end of discharge = 0.001V The Galvanic Cell close circuit voltage (Vccv) at start of discharge = 0.599 V The Galvanic Cell close circuit voltage (Vccv) at end of discharge = 0.0 V

The Galvanic Cell internal resistance is (Rbat) = 0.25 ohms (assumed to be constant during discharge) External load resistance (Rext) at start of discharge = 149.75 ohm (varied during discharge)

Constant discharge current (I) = Vocv/(Rbat+ Rext) = 0.6/(0.25+149.75) = 4 mA Cell operating temperature (Tgc) = 298 K

Environment temperature (T0) = 293 K

Galvanic Cell reaction is: **2Al (s) + 3Cu****2+(aq) = 2Al ****3+(aq) + 3Cu (s)**

Galvanic Cell Cathode Reaction: 3Cu2+(aq) + 6e = 3Cu(s); E0 (standard electrode reduction potential) = +0.337 V against SHE (standard hydrogen electrode).

Galvanic Cell Anode Reaction: 2Al(s) = 2Al3+(aq) + 6e; E0 (standard electrode reduction potential) = -1.662 V against SHE (standard hydrogen electrode).

Galvanic Cell electromotive force at standard state and 298 K (E0gc) = [+0.337-(-1.662)] V ~ 1.999 V = 2.0 V

NERNST Equation for the Galvanic Cell: Egc = E0gc - (RTgc/6F) ln(a2Al3+/a3 ) = Vocv

At equilibrium, Galvanic Cell electromotive force Egc = 0. Hence at equilibrium one can write following.

E0gc = (RTgc/6F) ln(a2Al3+/a3Cu2+) = (RTgc/6F) ln (Keq) (24)

R = universal gas constant, F= Faraday’s constant, aAl3+, aCu2+ are activities of Al3+ and Cu2+ ions respectively, Keq = equilibrium constant for the Galvanic Cell reaction at 298 K.

ln (Keq) = E0gc/(RTgc/6F) = 2.0/ [(8.3144*298)/ (6*96500)] = 467.37

## Keq = 9.4726 x 10202

**Estimation of Entropy Creation σ**

Created entropy can be calculated using Eq.15 which is developed in section 4.

∆Sgc + Q0gc/ T0 = σ

∆Sgc can be obtained using Eq.16 given in section 4.

∆Sgc = nF (dEgc/dTgc)

In case of above-mentioned Galvanic Cell reaction n = 6, F = 96500 C/mole and

∆Hgc = -1271000 J

So, using Gibbs-Helmholtz relation ∆Hgc = -nFEgc + nFTgc(dEgc/dTgc) one can easily estimate [dEgc/dTgc] as, [dEgc/dTgc] = (∆Hgc + nFEgc)/ nFTgc = [(-1271000) + (6*96500*0.6)]/ (6*96500*298)]

= -5.3528 X 10-3 V/K

Therefore, ∆Sgc = nF (dEgc/dTgc) = (6*96500*(-5.3528 X 10-3)) J/K = -3099.33 J/K

Now, ∆Ugc can be obtained from the following equation for Galvanic Cell discharge process.

∆Ugc = ∆Hgc - P∆Vgc - Vgc∆P = -1271000 J (as ∆Vgc and ∆P are assumed to be nil during discharge) Total net electrical work output from the Galvanic Cell is also given as,

Wgc = nFEgc = (6*96500*0.6) J = 347400 J

Therefore, Q0gc which is quantity of heat transferred to the environment at constant temperature T0, can be evaluated as,

Q0gc = ∆Ugc + Wgc = [(-1271000) + (347400)] J = -923600 J. It should be noted here that negative sign indicates heat transferred to the environment from the system which is Galvanic Cell.

Thus, created entropy can be calculated as,

σ = ∆Sgc + Q0gc/ T0 = [-3099.33 + (923600/293)] J/K = 52.88 J/K or **12.64 **Cal/K.

It should be noted here that Q0gc/ T0 is entropy change of the environment which is positive because heat is transferred to the environment from the system (i.e., Galvanic Cell). Moreover, in order to ensure heat flux from the Galvanic Cell to the Environment, it is necessary that Cell operating temperature (Tgc) is greater than Environment temperature (T0).

## Estimation of Waste Work T0σ

Since magnitude of Waste Work is simply given as T0σ, Waste Work can be evaluated as, [T0σ] = 293*52.88 J = 15493.84 J or **3703.1 **Cal

## Estimation of Thermodynamic Efficiency *ET*

Thermodynamic efficiency *ET *for the Galvanic Cell which is second law efficiency can be calculated by comparing the availability change ∆Bgc for the Galvanic Cell discharge process vis-a-vis Wgca which is total available work or “total work recovered” from the Galvanic Cell during discharge process.

Since ∆Bgc = ∆Ugc - T0∆Sgc = [(-1271000)- (293*(-3099.33))] = -362896.31 J and Wgca = [T0σ] + [∆Bgc] = [15493.84 + (-362896.31)] J = -347402.47 J, *ET *can be estimated as,

*ET *= (Wgca/∆Bgc) * 100% = (-347402.47/-362896.31) *100% = **95.73**%

In order to show how variations in environment conditions and discharge current affect critical performance parameters which are Entropy Creation **σ**, Waste Work **T****0σ **and Thermodynamic Efficiency *ET *of any Galvanic Cell, above-described computations are repeated for different discharge currents and environment temperatures. The results are tabulated in Table-1.

### Case: Discharge reaction is Endothermic

As another example, evaluation of Entropy Creation **σ**, Waste Work **T****0σ **and Thermodynamic Efficiency *ET *for a Galvanic Cell whose discharge reaction is endothermic is described below.

The relevant Galvanic Cell will be having following configuration [7].

## Ag/Ag+1, Cl-// Hg+1, Cl- /Hg (Endothermic)

The Galvanic Cell open circuit voltage (Vocv) at start of discharge = 0.04550 V The Galvanic Cell open circuit voltage (Vocv) at end of discharge = 0.001 V The Galvanic Cell close circuit voltage (Vccv) at start of discharge = 0.04542 V The Galvanic Cell close circuit voltage (Vccv) at end of discharge = 0.0 V

The Galvanic Cell internal resistance is (Rbat) = 0.25 ohms (assumed to be constant during discharge) External load resistance (Rext) at start of discharge = 142 ohm (varied during discharge)

Constant discharge current (I) = Vocv/(Rbat+ Rext) = 0.04550/(0.25+142) = 0.32 mA Cell operating temperature (Tgc) = 298 K

Environment temperature (T0) = 303 K

Galvanic Cell reaction is: **Hg****1+ (aq) + Ag (s) = Ag****1+ (aq) + Hg (l)**

Galvanic Cell Cathode Reaction: Hg1+ (aq) + 1 e = Hg (l)**; **E0 (standard electrode reduction potential) = +0.80 V against SHE (standard hydrogen electrode).

Galvanic Cell Anode Reaction: Ag (s) = Ag1+ (aq) + 1 e; E0 (standard electrode reduction potential) = +0.79 V against SHE (standard hydrogen electrode).

Galvanic Cell electromotive force at standard state and 298 K (E0gc) = [(+0.8) -(0.79)] V ~ 0.01 V = 0.01 V NERNST Equation for the Galvanic Cell: Egc = E0gc - (RTgc/1F) ln(aAg+/aHg+) = Vocv

At equilibrium, Galvanic Cell electromotive force Egc = 0. Hence at equilibrium one can write following.

E0gc = (RTgc/1F) ln(aAg+/aHg+) = (RTgc/1F) ln (Keq) (25)

R = universal gas constant, F= Faraday’s constant, aAg+, aHg+ are activities of Ag+ and Hg+ ions respectively, Keq

= equilibrium constant for the Galvanic Cell reaction at 298 K.

ln (Keq) = E0gc/(RTgc/1F) = 0.01/ [(8.3144*298)/ (1*96500)] = 0.3894

## Keq = 0.1476 x 101

**Estimation of Entropy Creation σ**

Created entropy can be calculated using Eq.15 which is developed in section 4.

∆Sgc + Q0gc/ T0 = σ

∆Sgc can be obtained using Eq.16 given in section 4.

∆Sgc = nF (dEgc/dTgc)

In case of above-mentioned Galvanic Cell reaction n = 1, F = 96500 C/mole and

∆Hgc = +5334.6 J (i.e., discharge reaction is endothermic)

So, using Gibbs-Helmholtz relation ∆Hgc = -nFEgc + nFTgc(dEgc/dTgc) one can easily estimate [dEgc/dTgc] as, [dEgc/dTgc] = (∆Hgc + nFEgc)/ nFTgc = [(5334.6) + (1*96500*0.0455)]/ (1*96500*298)]

= 3.38 X 10-4 V/K

Therefore, ∆Sgc = nF (dEgc/dTgc) = (1*96500*(3.38 X 10-4)) J/K = 32.62 J/K ~ **7.8 Cal/K**

Now, ∆Ugc can be obtained from the following equation for Galvanic Cell discharge process.

∆Ugc = ∆Hgc - P∆Vgc - Vgc∆P = 5334.6 J (as ∆Vgc and ∆P are assumed to be nil during discharge)

Total net electrical work output from the Galvanic Cell is also given as, Wgc = nFEgc = (1*96500*0.0455) J = 4390.75 J ~**1049 Cal**

Therefore, Q0gc which is quantity of heat transferred to the Galvanic Cell at constant Environment temperature T0, can be evaluated as,

Q0gc = ∆Ugc + Wgc = [(5334.6) + (4390.75)] J = +9725.32 J ~ **2323 Cal**. It should be noted here that positive sign indicates heat is transferred to the Galvanic Cell from Environment (as the discharge reaction is endothermic).

Thus, created entropy can be calculated as,

σ = ∆Sgc + Q0gc/ T0 = [ 32.62 + (-9725.32/303)] J/K = 0.5232 J/K = **0.1250 **~ **0.13 Cal/K**. It should be noted here that Q0gc/ T0 is entropy change of the Environment which is negative because heat is transferred to the system (i.e., Galvanic Cell) from the Environment as the discharge reaction is endothermic. Moreover, in order to ensure heat flux from the Environment to the Galvanic Cell, it is necessary that Cell operating temperature (Tgc) is less than Environment temperature (T0).

## Estimation of Waste Work T0σ

Since magnitude of Waste Work is simply given as T0σ, Waste Work can be evaluated as, [T0σ] = 303*0.5232 J = 158.53 J ~ **38 **Cal

## Estimation of Thermodynamic Efficiency *ET*

Thermodynamic efficiency *ET *for the Galvanic Cell which is second law efficiency can be calculated by comparing the availability change ∆Bgc for the Galvanic Cell discharge process vis-a-vis Wgca which is total available work or “total work recovered” from the Galvanic Cell during discharge process.

Since ∆Bgc = ∆Ugc - T0∆Sgc = [5334.6 - (303*32.62)] = -4549.26 J ~ **-1087 Cal **and Wgca = [T0σ] + [∆Bgc] = [158.53 + (-4549.26)] J = -4390.73 J ~ **-1049 Cal**, *ET *can be estimated as,

*ET *= (Wgca/∆Bgc) * 100% = (-1049/-1087) *100% = **96.5 **%

In order to show how variations in environment conditions and discharge current affect critical performance parameters which are Entropy Creation **σ**, Waste Work **T****0σ **and Thermodynamic Efficiency *ET *of any Galvanic Cell, above-described computations are repeated for different discharge currents and environment temperatures. The results are tabulated in Table-2.