# How do you calculate Gibbs free energy of mixing?

##### 1 Answer

The Gibbs' free energy of mixing for an ideal binary solution is calculated using this equation:

#\mathbf(Delta_"mix"G^"id" = RT[n_ilnchi_i + n_jlnchi_j])#

It gets more complicated with nonideal solutions though, as you have to incorporate the activity coefficient

Eventually, we would have gotten:

#(Delta_"mix"barG^"real")/(w) = (Delta_"mix"G^"real")/(nw) = (RT)/w[stackrel("ideal solutions")overbrace(chi_ilnchi_i + chi_jlnchi_j)] + stackrel("deviation from ideality")overbrace(chi_ichi_j)# where

#w# is a constant that my textbook states we "don't need to know".

Below, I derive how to get the one for ideal binary solutions, which is easier to understand.

We can start from this equation:

#mu_j^"soln" = mu_j^"*"(l) + RTlnchi_j# (Eq. 1)where:

#mu_j# is thechemical potentialof solvent#j# in the solution, i.e. when solute ispresent. Chemical potential is analogous to potential energy.#mu_j^"*"(l)# is the chemical potential of solvent#j# as a liquid by itself (i.e.pure solvent).#R# and#T# are typical variables that you know from the ideal gas law.#chi_j = (n_j)/(n_i + n_j)# is the#\mathbf("mol")# fractionof solvent#j# in theideal binary solutionat any given moment (with solute if solute is present).

That equation would tell you that the chemical potential decreases upon adding more solute.

The **Gibbs' free energy of mixing** is defined as:

#\mathbf(Delta_"mix"G) = \mathbf(G^"soln"(T,P,n_i,n_j) - [G_i^"*"(T,P,n_i) + G_j^"*"(T,P,n_j)])# (Eq. 2)where:

#G^"soln"# is thefinal Gibbs' free energy statefor the solution as a function of temperature, pressure, and#"mol"# s of solute#i# and solvent#j# .This is after mixing!#G_i^"*"# is thepure initial Gibbs' free energy statefor thesoluteas a function of temperature, pressure, and the#"mols"# ofsolute.This is before mixing!#G_j^"*"# is thepure initial Gibbs' free energy statefor thesolventas a function of temperature, pressure, and the#"mols"# ofsolvent.This is before mixing!

In essence, what this says is that...

The

change in Gibbs' free energy due to mixingthe solute into the solvent,#Delta_"mix"G# , is the difference between the Gibbs' free energy of the solutionbeforemixing (#G_i^"*" + G_j^"*"# ), andafterthe solution has been made (#G^"soln"# ).

Next, we note that **ideal binary solution**, we modify **Eq. 2** to get:

#Delta_"mix"G^"id" = stackrel(G^"soln")overbrace(n_imu_i^"soln" + n_jmu_j^"soln") - stackrel(G^"*")overbrace([n_imu_i^"*" + n_jmu_j^"*"])# (Eq. 3)

Then, note that we use **Eq. 1** to modify **Eq. 3** to get:

#color(blue)(Delta_"mix"G^"id") = n_i(mu_i^"soln" - mu_i^"*") + n_j(mu_j^"soln" - mu_j^"*")#

#= n_iRTlnchi_i + n_jRTlnchi_j#

#= color(blue)(RT[n_ilnchi_i + n_jlnchi_j])#

So, to calculate the Gibbs' free energy of mixing for an ideal binary solution of solute

#n_i# , the#\mathbf("mol")# **s of solute**#\mathbf(i)# .#chi_i = (n_i)/(n_i + n_j)# , the#\mathbf("mol")# **fraction of solute**#\mathbf(i)# .#n_j# , the#\mathbf("mol")# **s of solvent**#\mathbf(j)# .#chi_j = (n_j)/(n_i + n_j)# , the#\mathbf("mol")# **fraction of solvent**#\mathbf(j)# .#R# , the**universal gas constant**. To get the right units for#G# , use#"8.314472 J/mol"cdot"K"# .#T# , the**temperature**of the solution.