Partial Molar Quantities and the Chemical Potential - PowerPoint Presentation

Slide1

Partial Molar Quantities and the Chemical Potential

Lecture 6Slide2

Partial Molar Quantities

Now that we have introduced the mole fraction,

X

, and variable composition, we want to know how the variables of our system, e.g., V, S, change as we change composition.These are partial molar quantities, usually indicated by the lower case letter.For example:Such thatThis is the partial molar volume of component i. For example, the partial molar volume of O2 dissolved in seawater.This tell us how the volume of water changes for an addition of dissolved O2 holding T, P, and the amounts of everything else constant.Slide3

Partial Molar Volumes of Ethanol and Water

If you add a shot (3

oz

) of rum to 12 oz of Coca Cola, what will be the volume of your ‘rum ‘n coke’?Less than 15 oz! Blame chemistry, not the bartender.Slide4

Other Partial Molar Quantities

We can also define partial molar quantities of other thermodynamic variables, such as entropy, and enthalpy.

One

partial molar quantity is particularly useful, that of the Gibbs Free Energy.Slide5

Chemical Potential

The chemical potential is defined as

partial molar Gibbs Free Energy

:such thator, dividing each side by the total number of moles:The chemical potential tells us how the Gibbs Free Energy will vary with the number of moles, ni, of component i holding temperature, pressure, and the number of moles of all other components constant. For a pure substance, the chemical potential is equal to its molar Gibbs Free Energy (also the molar Helmholtz Free Energy):Slide6

Gibbs Free Energy Again

In Chapter 2, we found that the Gibbs Free Energy change of a system (and we are only interested in changes, not absolute amounts) was given by:

dG

= VdP – SdT(you need to memorize this equation – think about units and how free energy is minimized when entropy is maximized.)We also said that for a reaction at constant temperature and pressure:∆Gr = ∆Hr –

T∆SrThese equation hold for a system of fixed composition. Where composition can vary, we need to modify them to account for that variance.Slide7

The total Gibbs free energy of a system will depend upon composition as well as on temperature and pressure.

The

Gibbs free energy change of a phase of variable composition is fully expressed as:

Now consider exchange of component i between two phases, α and βholding all else constant; then:For a closed system:At equilibrium (dG = 0), then:

In

a system at equilibrium, the chemical potential of every component in a phase is equal to the chemical potential of that component in every other phase in which that component is present.Slide8

Gibbs-Duhem Equation

The Free Energy of a system (or phase) is the sum of chemical potentials of its components:

Differentiating:

Equating with the earlier equation:We can rearrange this as the Gibbs-Duhem Equation:Slide9

Interpreting Gibbs-

Duhem

In a closed system at equilibrium, net changes in chemical potential will occur only as a result of changes in temperature or pressure.

At constant temperature and pressure, there can be no net change in chemical potential at equilibrium:This equation further tells us that the chemical potentials do not vary independently, but change in a related way. Slide10

In spontaneous processes, components or species are distributed between phases so as to minimize the chemical potential of all components.

Final point about chemical potential:Slide11

Ideal SolutionsSlide12

Chemical Potential in Ideal Solutions

In terms of partial molar quantities

For an ideal gas:

Integrating from P˚ to P:Where P˚ is the pressure of pure substance in its ‘standard state’ and µ˚ is the chemical potential of i in that state. In that case, P/P˚ = Xi and:Slide13

This equation tells us that in an ideal solution:

the chemical potential of component

i

is always less than that of pure i, since X is by definition less than 1 and hence ln X is negative. the chemical potential of component i increases linearly with the log of its concentration in the solution.Slide14

Volume and enthalpy changes of solutions

Water–alcohol is an example of a non-ideal solution. The volume is expressed as:

∆

Vmixing term may be negative, as in rum ‘n coke.Similarly, mix nitric acid and water and the solution gets hot – that thermal energy is the enthapy of mixing. Enthalpy of solutions is expressed as:The ∆H term is positive in the nitric acid case.For ideal solutions, however, ∆Vmixing = ∆H

mixing = 0Slide15

Entropy changes of solution

What about entropy and free energy changes of ideal solutions?

Even in ideal solutions, there is an entropy change (increase) because we have increased the randomness of the system.

The entropy change of ideal solution is:Note similarity to configurational entropy.Note negative sign. How will entropy change?The total (molar) entropy of an ideal solution is then:Slide16

What does this tell us about how Gibbs Free Energy will change in a ideal solution?

∆G

r

= ∆Hr – T∆SrSlide17

Free Energy Change of Solution

∆

G

mixing = ∆Hmixing - T∆SmixingFor an ideal solution, ∆Hmixing = 0AndBecause the log term is always negative, ideal solutions have lower free energy than a mixture

of their pure constituents and ∆G decreases with increasing T. This is why things are usually more soluble at higher T. Total Free Energy of an ideal solution is:Slide18

Total Free Energy of an Ideal Solution

Recalling that and

We can substitute

Into the above and obtain:Slide19

Total Free Energy of an Ideal Solution

The Free Energy of a solution is simply the sum of the chemical potentials of the components times their mole fractions

. (Note that I have shortened the subscript from

ideal solution to simply ideal).Since This equation is entirely equivalent to the one we previously derived:The first term on the right is the sum of the contribution of the chemical potentials of the pure components. The second term on the right is the decrease in free energy that comes from the increase in entropy.The second term on the right will always be negative because the mole fractions are by definition less than 1, hence their logs are always negative.

Equivalently:Slide20

Let’s now see what happens to Free Energy when we dissolve component 2 in component 1 to create an ideal solution at various temperatures

Recall:

The change due to solution is simply the second term on the right:Slide21

Free Energy of Mixing in an Ideal SolutionSlide22

Mixing and Mixture

We are calling the reduction in free energy due only to dissolving one component in another

∆

Gideal mixing.Once again, mixtures are not solutions.In our equation: The first term on the right is the free energy of the mixture: it is simply the sum of the molar free energies (or equivalently the chemical potentials) of the pure components weighted by their mole fractions.Again, the second term arises from the increase in entropy that occurs when we create a true solution.Slide23

Method of Intercepts

Consider a two- component ideal solution so that

X

1 = (1-X2). Since:Then:G = µ1[(1-X2) +µ2

X2 = µ1 + (µ2 - µ

1)X2This is the equation of a straight line (dashed on the figure) on a

G-bar–X2 plot.We can use it to extrapolate to chemical potentials of our two components in the solution.