# What is chemical potential?

##### 2 Answers

#### Explanation:

Physically it is the change in the Gibbs free energy of the whole system if we added 1 mol of chemical

**Chemical potential** is the differential change in the molar Gibbs' free energy of the system at constant

#dG = -SdT + VdP + sum_i mu_idn_i#

In simple terms, it is analogous to the potential energy of a system: chemical potential runs downhill.

Now, if one divides the equation by

#color(blue)(mu_i = ((delG_i)/(deln_i))_(T,P,n_(j ne i)))#

*The chemical potential is where many of the thermodynamic relationships come from for ideal, ideally-dilute, and nonideal solutions.*

The following discussion will give an overview of the first two types of solutions. By no means is this exhaustive of the type of information you can learn about these systems, but I will take a look at

**IDEAL SOLUTIONS**

In an **ideal solution**, the interactions between solute (*identical*. For these solutions, we have the following major quantities describing ideal mixtures:

**Change in volume due to mixing**

#bb(DeltaV_"mix"^"id" = sum_i n_i(barV_i - barV_i^"*") = 0)# where

#((delmu_i)/(delP))_(T,n_(j ne i)) = barV_i# is the molar volume of component#i# insolution, and#((delmu_i^"*")/(delP))_(T,n_(j ne i)) = barV_i^"*"# is the molar volume of component#i# by itself(i.e. not in solution).

Since the components don't interact with each other in solution, the volume of the solution is perfectly additive. Hence,

**Change in Gibbs' Free Energy, Entropy, and Enthalpy due to mixing**

These are NOT zero. They do, however, cancel out for ideal solutions to give

#DeltaG_"mix"^"id" = sum_i n_i(barG_i - barG_i^"*")#

#= sum_i n_i(mu_i - mu_i^"*")#

But if we note that

#bb(DeltaG_"mix"^"id" = RTsum_i n_ilnchi_i)#

Similarly, the entropy of mixing also comes from the chemical potential.

#DeltaS_"mix"^"id" = sum_i n_i(barS_i - barS_i^"*")#

#= sum_i n_i(-((delmu_i)/(delT))_(P,n_(j ne i)) - ((delmu_i^"*")/(delT))_(P,n_(j ne i)))#

#= -sum_i n_i((del[mu_i - mu_i^"*"])/(delT))_(P,n_(j ne i))#

Plugging in

#bb(DeltaS_"mix"^"id") = -sum_i n_i((del[RTlnchi_i])/(delT))_(P,n_(j ne i))#

#= bb(-Rsum_i n_ilnchi_i)#

which if you notice, means that the enthalpy of mixing also follows:

#DeltaG_"mix"^"id" + TDeltaS_"mix"^"id" = bb(DeltaH_"mix"^"id")#

#= RTsum_i n_ilnchi_i - RTsum_i n_ilnchi_i = bb(0)#

where

**IDEALLY-DILUTE SOLUTIONS**

In this scenario, there is **very little solute and a lot of solvent**. In this case, we differentiate between the solute and solvent and explicitly write solute as

For the **solutes**:

#bb(DeltaV_"mix"^"id") = sum_(A) n_Acancel((barV_A - barV_A^"*"))^0 + bb(sum_(i ne A) n_i(barV_i - barV_i^"*"))#

#bb(DeltaG_"mix"^"id") = sum_(A) n_Acancel((mu_A - mu_A^"*"))^0 + bb(RTsum_(i ne A) n_ilnchi_i)#

#bb(DeltaS_"mix"^"id") = sum_(A) n_Acancel((-((delmu_A)/(delT))_(P, n_(A ne B)) - -((delmu_A^"*")/(delT))_(P, n_(A ne B))))^0 + bb( -Rsum_(i ne A) n_ilnchi_i)#

#bb(DeltaH_"mix"^"id") = sum_(A) n_Acancel((barH_A - barH_A^"*"))^0 + bb(sum_(i ne A) n_i(barH_i - barH_i^"*"))#

For the **solvents**:

They behave like the pure