Courses
Courses for Kids
Free study material
Offline Centres
More
Store Icon
Store

Gibbs Duhem Equation

Reviewed by:
ffImage
hightlight icon
highlight icon
highlight icon
share icon
copy icon
SearchIcon

Gibbs Duhem Equation Introduction

Gibbs Duhem Equation was named after Josiah Willard Gibbs and Pierre Duhem.


In thermodynamics, Gibbs Duhem Relation describes the relationship between the changes in any thermodynamic system. In other words, the equation of Gibbs Duhem helps us to calculate the relationship between quantities as a system that remains in equilibrium.


The Gibbs Duhem Equation is as follows:

\[\sum_{i =1}^{I} N_{i} d \mu_{i} = -SdT + VdP....(1)\]


On this page, we will understand the Gibbs Duhem Equation in detail. Also, we will understand how to do the Gibbs Duhem Equation Derivation with an illustrative Gibbs Duhem Equation Example, and the Gibbs Duhem Equation Application.


Gibbs Duhem Equation Definition

Now, let’s describe the parameters involved in equation (1) for Gibbs Duhem Relation:

Here, 

i = the component 

I = the number of different components in a system

\[N_{i}\]=   the numbers of a mole of a component ‘i’

\[d\mu_{i}\] =   the infinitesimal increase in the chemical potential for this component ‘i’

S = the entropy of the system

T =  the absolute Temperature in K

V = Volume

P = Pressure


Point To Note:

The above equation (1), shows that in thermodynamics concentrated properties are not autonomous but rather related, making it a numerical assertion of the state hypothesis. 


At the point when Pressure and Temperature are variable, just I-1 of I parts have independent values for Chemical potential and Gibbs' phase rule follows. 


The Gibbs Duhem Equation can't be utilized for little thermodynamic frameworks because of the impact of surface impacts and other tiny marvels.


Gibbs Duhem Equation Derivation   

 

(Image Will be Updated Soon)


Deriving the Gibbs Duhem Equation from the essential thermodynamic equation is direct. The absolute differential of the broad Gibbs free energy G regarding its characteristic factors is given as;


\[dG = \frac{\partial G}{\partial P} \left |_{T, N}~ dP +  \frac{\partial G}{\partial P}\right |_{P, N} dT + \sum_{i = 1}^{I}\left ( \frac{\partial G}{\partial N_{i}} \right )|_{P, T, Nj \neq i}| d N_{i}\]


As the Gibbs free energy is the Legendre transformation of the internal energy, so the derivatives can be supplanted by its definitions changing the above equation into the accompanying structure:

\[dG = VdP - SdT + \sum_{i = 1}^{I} \mu_{i}dN_{i}....(2)\]


Partial Gibbs Free Energy

The chemical potential is another name for the partial molar Gibbs free energy (or the incomplete Gibbs free energy, contingent upon whether N is in units of moles or particles). 


Hence the Gibbs free energy of a system can be determined by gathering moles together cautiously at a predetermined T, P and at a constant molar ratio synthesis (so the chemical potential doesn't change as the moles are added together). 


The equation for the above statement is given as;

\[dG = \sum_{i = 1}^{I} N_{i} \mu_{i}....(3)\]

The total differentiation of equation (3) is:

\[dG = \sum_{i = 1}^{I} N_{i} \mu_{i} + \sum_{i = 1}^{I} N_{i} d\mu_{i}....(4)\]

Combining the two expressions (equation 3 and 4) for the total differential of the Gibbs free energy gives:

\[VdP - SdT +  \sum_{i = 1}^{I} dN_{i} \mu_{i} = \sum_{i = 1}^{I} dN_{i} \mu_{i} + \sum_{i = 1}^{I} N_{i} d\mu_{i}....(5)\]

Cancelling the common terms, we get our finalized Gibbs Duhem Equation as;

\[\sum_{i = 1}^{I} N_{i} d\mu_{i} = VdP - SdT....(A)^{*}\]

There‘s another way for the Gibbs Duhem Equation Derivation:


Alternative Gibbs Duhem Equation Derivation

Here, will discuss the Gibbs Duhem Equation for Binary Mixture (a Gibbs Duhem Equation Example), at constant P (Isobaric process), at constant T (Isothermal process), the equation so formed is:

\[\cup(\lambda X) = \cup\lambda (X)\]           

Here,  X = all extensive variables of U

\[\cup = TS - PV + \sum_{i = 1}^{I} N_{i} \mu_{i}....(6)\]

Taking the total differentiation of equation (6):

\[d\cup = TdS - SdT - PdV - VdP + \sum_{i = 1}^{I} dN_{i} \mu_{i} + \sum_{i = 1}^{I} N_{i} d\mu_{i}....(7)\]

Since the process is Isobaric and Isothermal, so SdT and VdP becomes zero, so equation (7) can be rewritten as;

\[0 = TdS - PdV + \sum_{i = 1}^{I} N_{i} d\mu_{i}....(7)\]

So, the final Gibbs Duhem Equation is again the same in Equation (A)*:

\[\Rightarrow VdP - SdT = \sum_{i = 1}^{I} N_{i} d\mu_{i}....(B)^{*}\]


Gibbs Duhem Margules Equation

Various applications of Gibbs Duhem Equation are available, we will focus on one Gibbs Duhem Equation Application, so the Application of Gibbs Duhem Equation is given as;


Gibbs Duhem Margules Equation:

The two-parameter Gibbs Duhem Margules Equation is given as;

\[\frac{g^{E}}{RTx_{1}x_{2}} = A_{21}x_{1} + A_{12}x_{2}\]

This equation becomes:

\[In \gamma_{1} = X{_{2}}^{2} (A_{12} + 2(A_{21} - A_{12}))X_{1}...(p)\]

\[In \gamma_{2} = X{_{1}}^{2} (A_{12} + 2(A_{12} - A_{21}))X_{2}...(q)\]


So, we determined the activity coefficient from the Gibbs Free Energy Equation in the above format.


These empirical equations (p and q) are widely used in describing binary mixtures. A knowledge of \[A_{12}~and~A_{21}\] at given absolute temperature “T” is required to calculate the activity coefficients 

\[\gamma_{1}~\gamma_{2}\] for a given solution composition.


Gibbs Duhem Equation Application

For any binary mixture, at constant temperature and pressure, we have;

\[0 = N_{1} \mu_{1} + N_{2} + \mu_{2}\]


Now, normalizing by total number of moles in a system (\[N_{1} + N_{2}\]) and substituting the activity coefficient () with an identity equation (x1 + x2), we get the equation as;

\[0 = X_{1} dln(\gamma_{1}) + X_{2} dln (\gamma_{2})...(C)^{*}\]


Equation (C)*  is instrumental in the calculation of thermodynamically consistent, and precise articulations for the vapour pressure of a fluid mixture from restricted experimental data.

FAQs on Gibbs Duhem Equation

1. What is the definition of Gibbs Free Energy?

In thermodynamics, the Gibbs free energy (or Gibbs energy or G) is a thermodynamic potential that can be utilized to figure out the most extreme reversible work that might be performed by a thermodynamic framework at a consistent temperature and pressure. 


The Gibbs free energy is given as;

  • \[\Delta G =  \Delta H  - T\Delta S\]

Here,

  • \[\Delta H\] = Change in Enthalpy

T = Temperature in K

  • \[\Delta S\]= Change in Entropy

\[\Delta G\], calculated in joules in SI, is the greatest measure of non-extension work that can be removed from a thermodynamically closed system (one that can exchange heat and work with its environmental factors, yet not make any difference). This greatest can be achieved uniquely in a totally reversible process.

2. State Duhem-Margules Equation.

The Duhem–Margules Equation, named after Pierre Duhem and Max Margules, is a thermodynamic assertion of the interaction between the two parts of a single fluid where the vapour mixture is viewed as an ideal gas:

  • \[\frac{(dn~In~P_{A})}{(dn~In~x_{A})_{T,~P}}~=~\frac{(dn~In~P_{B})}{(dn~In~x_{B})_{T,~P}}\]

Here,

  • \[P_{A}~and~P_{B}\] are partial vapour pressures.

  • \[x_{A}~and~x_{B}\] are mole fractions of a liquid.