Variations of the Equation

The concept of Variations of the Equation sets a relationship between two variables in such a way that one variable is equal to constant multiplied by the other variable. Thus, the concept tries to create a dependency between two variables. The field of physics uses this concept very frequently. 

For instance, we have the formula s=v*t if there is no acceleration, where s is the displacement, v is the (constant) velocity, and t is the time over which the motion happened.

What Is Variation of the Equation?

Variation of the Equation is a concept that revolves around the derivation of an equation between two variables. The concept tries to derive an equation in such a manner that it sets the two variables in proportion to a constant. Thus, various important equations of physics are based on this concept of Variation of the Equation. Let’s take an example to understand the concept in detail. 

We shall now look at an example of the relationship between Gibbs’s free energy and the electromotive force (EMF) of a cell. 

What is Gibbs Free Energy?

The thermodynamic potential, which represents the reversible or maximum work accomplished by a thermodynamic system under constant pressure and temperature, could be defined as Gibbs free energy. The product of the total charge passed, and the cell’s EMF is used to calculate the work done by electrical power in one second.

What is the Gibbs Free Equation? 

ΔG = ΔH – TΔS

Here, 

G stands for Gibbs free energy. 

The enthalpy change is denoted by H. 

T stands for temperature, and 

The change in entropy is denoted by S. 

When G = 0, the energy reaction is spontaneous. 

When G > 0, energy reactions are more likely to be nonspontaneous. When G = 0, the reaction will be in equilibrium.

Defining the relationship between Gibbs free energy and EMF of a cell notes,

W = nFE (cell)

Here,

W = work done,

nF = total charge passed and;

E(cell) = EMF of the cell

When the charge is reversibly transmitted through the galvanic cell, it is recognised that the galvanic cell completes the greatest amount of work. The drop in Gibbs energy within the reaction is caused by the amount of reversible work done by the galvanic cell.

ΔG = −W

ΔG = −nFEcell

The standard cell potential can also be calculated using the equation above. Ecell = Eocell, while all significant reactive species have the same concentration (that is, the standard cell potential and EMF of a cell are equal).

ΔGo = −nFEo cell

A reaction’s Gibbs energy is classified as an Extensive Thermodynamic Property, indicating that its value is dependent on n. As a result, when two cell processes have different values of n, the Gibbs free energy values also change.

Now, let’s take an example of the Gibbs Free Equation and understand the topic with better clarification. 

Relationship between Gibbs free energy and EMF of a cell examples

Zn(s) + Cu2+(aq) → Zn2+(aq) + Cu(s)

ΔG = −2FEcell

While

After multiplying LHS and RHS by 2, it is seen that,

2Zn(s) + 2Cu2+(aq) → 2Zn2+(aq) +2Cu(s)

ΔG = −4FEcell

According to thermodynamics, a reaction’s Gibbs energy can be connected to a reaction quotient, and when the reaction reaches equilibrium, it can be related to the equilibrium constant. The equilibrium constant of a cell reaction can be connected to the standard cell potential because the Gibbs energy of a reaction is dependent on the EMF of a cell.

When at equilibrium,

Δr Go = −nFEocell

ΔrGo = − RT ln K

−nFEocell = − RT ln K

Eocell = (RT/nF) ln K

Where,

K is the equilibrium constant.

R is the universal gas constant.

We can deduce the following from the equation: 

When K > 1 and Eo(cell) > 0, the reaction is spontaneous and favours the production of the product. 

When K 1 and Eo(cell) 0 are equal, the reaction is nonspontaneous and favours the production of reactants. 

The above explanation is ideal for understanding the relationship between Gibbs’s free energy and a cell’s EMF.

Therefore, the following reaction between zinc and copper on the two electric nodes defines the relationship between the two variables: the EMF of the cell and the amount of charge passed per second under constant temperature. This thermodynamics concept is widely used in physics and happens to be one of the apt examples of Variations of the Equation.

Conclusion 

The phenomenon of variation of the equations is a concept used very often in physics. This relationship between Gibbs free energy and EMF of a cell notes describes the actual working of the Variations of the Equation theory, which sets two variables in a relationship that is affected constant times one variable and in such a way that the result is equal to the other variable. Variations of the Equation theory hold true in setting the relationship between Gibbs free energy and EMF of a cell.