Cell Thermodynamics: Temperature Coefficient of EMF, Gibbs Free Energy, Entropy Change in Cell, Enthalpy Change for a Cell & Cell Thermodynamics, Practice Problems and FAQs:

Whenever we try to finish a chapter, two things comes in our mind :
How much time it will take to finish?
And at what extent we can understand the chapter?

Imagine if you can get this data. How easy it would have been for you to study.

Similarly in chemistry, whenever we are dealing with a reaction we tend to look into the kinetics (which determines the speed at which reaction takes place) and thermodynamics (which tells us at which extent a reaction takes place). Thus in electrochemistry, we have to look into the thermodynamics of the cells so that we can determine it’s application in various fields.

In the thermodynamics chapter you were introduced to some important terms like Gibbs free energy, work done by the system, entropy, enthalpy, etc. let’s understand how these terms are involved in electrochemical cells.

Table of content

• Cell thermodynamics
• Thermodynamics of Galvanic cell
• Practice problems
• Frequently asked questions (FAQs)

Cell thermodynamics

There are seven important terms which are discussed below:

Temperature coefficient of EMF (T.C)_P

Gibbs Helmholtz law is used to calculate the temperature coefficient of the cell's emf. The emf value decreases as the temperature rises. Internal resistance is also affected by temperature. The resistance decreases as the temperature rises, and the resistance rises as the temperature falls.
In simple terms we can say that the rate of change of EMF with temperature at constant pressure P.
Mathematically,

For single electrode,

$(T.C{)}_P=(\frac{dE}{dT}{)}_p$

For a cell,

$(T.C{)}_P=(\frac{d{E}_{cell}}{dT}{)}_p$

Gibbs Free energy (Δ G)

Gibbs free energy, also known as Gibbs function, Gibbs energy, or free enthalpy, is a quantity used to measure the maximum amount of work done in a thermodynamic system when temperature and pressure remain constant. The symbol 'G' stands for Gibbs free energy. Its energy is typically measured in Joules or Kilojoules. The maximum amount of work that can be extracted from a closed system is defined as Gibbs free energy.

Josiah Willard Gibbs, an American scientist, discovered this property in 1876 while conducting experiments to predict the behaviour of systems when they are combined or whether a process can occur simultaneously and spontaneously. Gibbs free energy was often referred to as available energy. It can be thought of as the amount of useful energy available in a thermodynamic system that can be put to work.

Mathematically,

For single electrode,

$\Delta G= -nFE$

For a cell,

$\Delta G= -nF{E}_{cell}$

For a cell at standard condition that is at 298K and 1 atm pressure,

$\Delta G^0= -nF{E^o}_{cell}$

Entropy change in cell

Entropy is typically referred to as a measurement of a system's randomness or disorder. In the year 1850, a German physicist by the name of Rudolf Clausius first proposed this idea.

Apart from the general definition, there are several definitions that one can find for entropy Let’s discuss two important definitions of entropy

• Thermodynamic definition of entropy
• Statistical definition of entropy

From a thermodynamics viewpoint of entropy, we do not consider the microscopic details of a system. The behaviour of a system is instead described in terms of thermodynamic parameters as temperature, pressure, entropy, and heat capacity using the concept of entropy.This thermodynamic description took into consideration the state of equilibrium of the systems.

Meanwhile, the statistical definition which was developed at a later stage focused on the thermodynamic properties which were defined in terms of the statistics of the molecular motions of a system. The molecular disorder is measured by entropy.

Mathematically,

$$\begin{gathered} \Delta S_{cell}= \frac{-d\left(\Delta G\right)}{dT} \\ \Delta S_{cell}=\frac{-d(-\mathit{n}\mathit{F}{\mathit{E}}_{\mathit{c}\mathit{e}\mathit{l}\mathit{l}})}{dT} =(\frac{d\left(\mathit{n}\mathit{F}{\mathit{E}}_{\mathit{c}\mathit{e}\mathit{l}\mathit{l}}\right)}{dT}{)}_p \\ Or, \\ \Delta S_{cell}=nF(T.C{)}_p \end{gathered}$$

Enthalpy change for a cell

In a thermodynamic system, energy is measured by enthalpy. Enthalpy is a measure of a system's overall heat content and is equal to the system's internal energy plus the sum of its volume and pressure.

From thermodynamics,

Δ G = ΔH - TΔS
ΔH = ΔG + TΔS

ΔH = -nFE + T.nF(T.C)_p or ΔH = nF {T.(T.C)_p - E}

$\Delta H=nFT(\frac{dE}{dT}{)}_p- nFE$

Workdone by the cell

W_cell =  -ΔG_cell
W_cell = -(-nFE_cell)
W_cell =  (nFE_cell)

Maximum work done by the cell

W_max =  -ΔG⁰_cell
W_max = -(-nFE⁰_cell)
W_max = (nFE⁰_cell)

Thermodynamics of Galvanic cell

We know that in thermodynamics we can identify whether a reaction is spontaneous or not on the basis of the value of Gibbs free energy. Thermodynamics of galvanic cell depends upon the Gibbs free energy of a cell reaction and cell potential.

• If there’s a decrease in Gibbs free energy then it is a spontaneous reaction. Whereas if there’s a increase in Gibbs free energy, it will be a non-spontaneous reaction.
• When cell potential of cell is negative, the reaction becomes nonspontaneous whereas when the value of cell potential is positive, the reaction becomes spontaneous.

During working of a galvanic cell work done is the product of charge passing through the cell and the potential difference between the electrodes.

Electrical work = Charge $\times$ E_cell

1 mole of electron = 1F = 96500 C

Electrical work = nFE_cell

Standard cell potential and equilibrium constant

The relationship can be derived from the below equation

ΔG⁰ = nFE⁰_cell

We know that from thermodynamics,

Reaction quotient is defined as the ratio of the activity of the products raised to the power of their stoichiometry to the activity of reactants raised to the power of their stoichiometry.

Reaction Quotient For a general electrochemical reaction, aA + bB + ne- ⇌ cC + dD

$$\begin{gathered} Q=\frac{\left[Product\right]}{\left[Reactant\right]} = \frac{{\left[C\right]}^c{\left[D\right]}^d}{{\left[A\right]}^a{\left[B\right]}^b} \\ {E^0}_{cell}=\frac{RT}{nF} ln K= 2.303 \frac{RT}{nF} logK \\ at 298 k \\ {E^0}_{cell}= \frac{0.0591}{n} log K \end{gathered}$$

Practice problems

Q1. Gibbs free energy change for a single electrode can be represented as

1. ΔG = -nFE
2. ΔG = nFE
3. ΔG = -nFE⁰_cell
4. ΔG = nFE⁰_cell

Answer: (A)

Solution: Gibbs free energy change for a single electrode is

ΔG = -nFE

For a cell it is,

ΔG = -nFE_cell

For a cell at standard condition the value of Gibbs free energy is

ΔG = -nFE⁰_cell

Q2. A reaction is said to be spontaneous if :

1. ΔG = +ve
2. ΔG = -ve
3. E_cell = -ve
4. Both (B) & (C)

Answer: (D)

Solution: If there’s a decrease in Gibbs free energy then it is a spontaneous reaction. Whereas if there’s a increase in Gibbs free energy, it will be a non-spontaneous reaction. When cell potential of cell is negative, the reaction becomes nonspontaneous whereas when the value of cell potential is positive, the reaction becomes spontaneous.

Q3. Equilibrium constant for the below reaction reaction is 1.2 106. What is the standard potential of a cell in which the reaction takes place?

2Cu⁺(aq) -> Cu²⁺(aq) + Cu(s)

Solution:

Given: K = 1.2 $\times$ 10⁶

2Cu⁺(aq) -> Cu²⁺(aq) + Cu(s)

To calculate n let’s write this equation in terms of redox reaction

At cathode: Cu⁺ -> Cu²⁺ + e^-

At anode: Cu⁺ + e^- -> Cu

Here n=1

Now,

$$\begin{gathered} {E^0}_{cell}= \frac{0.0591}{n} log K \\ {E^0}_{cell}= \frac{0.0591}{1} log (1.2\times {10}^6) \\ {E^0}_{cell}=0.36V \end{gathered}$$

Q4. Calculate standard Gibbs free energy change and equilibrium constant at 298K for the below cell reaction. If E⁰_Cd= -0.403 V and E⁰_Sn = -0.136 V. Write the formula of the cell?

Cd(s) + Sn²⁺(aq) --> Cd²⁺ + Sn(s)

Solution:
Given: E⁰_Cd = -0.403 V and E⁰_Sn = -0.136 V
We can see the value of E0Sn is greater than the value of E⁰_Cd
As - 0.136> - 0.403

Thus it implies that Sn ions has a greater reduction potential. Hence, Sn will reduce and work as a cathode and Cd will oxidise and work as anode.
Formula of the cell :

Cd(s)| Cd²⁺(aq) || Sn²⁺(aq) | Sn(s)
E⁰_cell = E⁰_cathode - E⁰_anode = -0.136 V - (-0.403) =0.267 V
ΔG⁰ = -nFE⁰_cell
ΔG⁰ = -2 $\times$ 96500 ^$\times$0.267 = -51531 J = -51.531 kJ
${E^0}_{cell}=\frac{0.0591}{2} log K$

$log K= \frac{{E^0}_{cell}\times 2}{0.0591} = \frac{0.267\times 2}{0.0591}=9.0203$

$K= antilog \left(9.0203\right)=1.05\times {10}^9$

Frequently asked questions (FAQs)

Q1. Why Gibbs free energy change is an extensive property?

Answer: There are properties of matter that are dependent on the quantity or size of the matter, such as length, mass, volume, weight, and so on. These properties are known as extensive properties of matter, and their value changes as the size or quantity of the matter changes. Assume we have two identical boxes, one with a capacity of four litres and the other with a capacity of ten litres. When comparing a ten-litre box to a four-litre box, the ten-litre box will hold more matter.

$$\begin{gathered} \Delta G^0= -nF{E^0}_{cell} \\ {E^0}_{cell}= - \frac{\Delta G^0}{nF} \end{gathered}$$

Let’s consider a reaction

$Zn+C{u}^{2+}\to Z{n}^{2+}+Cu$

No. of electrons involved is 2

Hence,

${E^0}_{cell}= - \frac{\Delta G^0}{2F}$

Now if we increase the quantity of reactant

$2 Zn+2 C{u}^{2+}\to 2 Z{n}^{2+}+2 Cu$

No. of electrons involved is 4

${E^0}_{cell}= - \frac{2 \Delta G^0}{4F} or {E^0}_{cell}= - \frac{\Delta G^0}{2F}$

We can see clearly that even if after changing the value of n , E⁰_cell remains same. Which means E⁰_cell is independent of quantity of substance.

Hence, EMF of a cell is an intensive property.

Q2. Can we have a negative valued EMF for a cell?

Answer: A galvanic or voltaic cell's electromotive force (EMF) is the maximum potential difference between two electrodes. The reaction is reversible if the cell potential is negative.

Q3. From where do we generate EMF in a cell?

Answer: Because there is a charge flow in the cell, battery, and generator. As a result, we can conclude that an emf can be generated in any of the three sources. As a result, the cell, battery, and generator are the sources of emf.

Q4. Why is Gibbs free energy referred to as free energy?

Answer: The effects of energy and entropy are balanced by a composite function known as free energy. The maximum amount of work that a system can perform on its surroundings while operating at constant pressure and temperature is known as the Gibbs energy.