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Chemical Thermodynamics 2013/2014. 7 th Lecture: Gibbs Energy and Fundamental Equations Valentim M B Nunes, UD de Engenharia. What have we learned (from the 2 nd Law!). We have seen that, in isolated systems, the entropy change give us the direction of an spontaneous transformation:.

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Chemical thermodynamics 2013 2014

ChemicalThermodynamics2013/2014

7th Lecture: Gibbs Energy and Fundamental Equations

Valentim M B Nunes, UD de Engenharia


What have we learned (from the 2nd Law!)

We have seen that, in isolated systems, the entropy change give us the direction of an spontaneous transformation:

ΔStotal > 0 – spontaneous transformation

ΔStotal = 0 – equilibrium (reversible process)

But, systems of interest generally are not isolated. It will be convenient to have a criteria for spontaneous change depending only on the system.

Gibbs (1839-1903) introduced a new state function, that plays a key role in chemistry and in the study of chemical equilibrium:


Josiah Willard Gibbs

Thermodynamicist!


Criteria for Spontaneous Change

Consider a system in thermal equilibrium with its surroundings:

By the second Law,

For any transformation in a closed system at constant p (p=pext)


Criteria for Spontaneous Change

Under constant T= Tsurr and constant p = pext, the criterion for spontaneity is:

This means that at constant p and constant T, equilibrium is achieved when the Gibbs energy is minimized.

Consider the process A  B (keeping p and T constant)

ΔG < O – A  B is spontaneous

ΔG = 0 – A and B are in equilibrium

ΔG > 0 – then B  A is spontaneous


What tell us G? (some mistakes!)

Wrong!

maximum

minimum

The second law is still valid. In some transformations the maximization of entropy can be obtained by “transferring” enthalpy to the surroundings

If Text is low (cold air masses) then |ΔSext| > | ΔSsyst|

Snowing: gas  liquid  solid


The Helmholtz energy

If we define now the Helmholtz energy, A, as:

The criterion for spontaneity under constant T = Tsurr and constant volume, V it will be:

Gibbs energy is, by far, more important that Helmholtz energy.


Physical meaning of Gibbs energy

At constant temperature:

Since H = U + pV

Now, considering that the total work is the sum of expansion work and other types of work

As result, at constant p we finally obtain

The Gibbs energy change represents the maximum available work, or the useful work that we can obtain from a given process.


Standard Gibbs energy

As for enthalpy we can also obtain standard Gibbs energy of formation, ΔGºf (at pº = 1 bar). For any element in their reference state, ΔGºf=0.

From the definition of G = H – TS we obtain:

and:

These values are tabled, and are obtained from calorimetric or spectroscopic data!


The Gibbs Energy and Fundamental Equations

We can now combine the first an second law; as we have introduced all the state functions for closed systems

From the 1st Law:

then:

U= U(S,V)

FUNDAMENTAL EQUATIONS: valid for reversible or irreversible changes for closed systems and only pV work.

Since H = U + pV

H = H(S,p)

G = H-TS

G = G(p,T)

A=U-TS

A =A(T,V)


How G changes with p and T

Since G is a state function, and as we saw before G = G(T,p) then we can write:

And easily obtain:

and



The Maxwell Relations

Recalling that df = gdx + hdy is an exact differential if

and using the fundamental equations we obtain

Maxwell Relations: these equations allow us to establish useful thermodynamic relations between state properties!


U and H from equations of state

We can now relate U and H to p-V-T data. For instance for U, we can calculate the ΠT value:

Generic equation of state!

For an ideal gas:

This proves that for an ideal gas U = U(T), that is function of T only !

Then:

U=U(T,V)

For a van der Waals gas:


The variation of Gibbs Energy with temperature

As we saw previously:

then:

If we derivate now G/T we obtain:

or:

And finally:

Gibbs-Helmholtz equation


Gibbs Helmholtz equation and Chemical Reactions

Applying the Gibbs Helmholtz equation to a chemical reaction we obtain:

Where ΔH is the reaction enthalpy change.


The variation of Gibbs Energy with pressure

Recalling that

And upon integration we obtain, at constant T:

For liquids and solids the molar volume, Vm, is small and approximately constant, so:

~0


Ideal Gases

For a perfect gas (pV = nRT) we obtain:

or:

Considering now that pi = pº (the standard pressure of 1 bar) and calculating the molar Gibbs energy, Gm (n=1), we obtain:

The molar Gibbs energy is also called the chemical potential, μ:


Final remark

We have now all the “thermodynamic tools” that allow us to apply the thermodynamic principles to all kind of physical and chemical processes, like chemical equilibrium or phase equilibrium. That will be the syllabus for the next lectures…


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