Spontaneity and Equilibrium in Chemical Systems

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Spontaneity and Equilibrium in Chemical Systems . Gibbs Energy and Chemical Potentials. The Use of  univ S to Determine Spontaneity. Calculation of T univ S  two system parameters  r S  r H Define system parameters that determine if a given process will be spontaneous?.

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### Spontaneity and Equilibrium in Chemical Systems

Gibbs Energy and Chemical Potentials

The Use of univS to Determine Spontaneity
• Calculation of TunivS  two system parameters
• rS
• rH
• Define systemparameters that determine if a given process will be spontaneous?
Entropy and Heat Flow

Distinguish between a reversible and an irreversible transformation.

Pressure Volume and Other Types of Work
• Our definition of work can be extended to include other types of work.
• Electrical work.
• Surface expansion.
• Stress-strain work.

dw=-Pext dV+dwa

where dwa includes all other types of work

Spontaneity under Various Conditions
• In an isolated system where

dq=0; dw=0; dU=0

dS  0

• Now allow the system to make thermal contact with the surroundings. For an isentropic process (dS = 0)

dU  0

Isothermal Processes

For a systems where the temperature is constant and equal to Tsurr

The Helmholtz Energy

Define the Helmholtz energy A

A(T,V) =U – TS

Note that for an isothermal process

dA  dw

A  w

For an isochoric, isothermal process

A  0

The Properties of A

The Helmholtz energy is a function of the temperature and volume

Isothermal Volume Changes

For an ideal gas undergoing an isothermal volume change

Isothermal Processes at Constant Pressure

For an isothermal, isobaric transformation

The Gibbs Energy

Define the Gibbs energy G

G(T,P) =U – TS+PV

Note that for an isothermal process

dG  dwa

G wa

For an isothermal, isobaric process

G  0

The Properties of G

The Gibbs energy is a function of temperature and pressure

Isothermal Pressure Changes

For an ideal gas undergoing an isothermal pressure change

The Chemical Potential

Define the chemical potential  = G/n

The Standard Chemical Potential

For P1 = P = 1 bar, we define the standard state chemical potential

°= (T, 1bar)

Gibbs Energy Changes for Solids and Liquids

Solids and liquids are essentially incompressible

Temperature Dependence of A

Under isochoric conditions

Helmholtz Energy Changes As a Function of Temperature

Consider the calculation of Helmholtz energy changes at various temperatures

Dependence of G on Temperature

Under isobaric conditions

Gibbs Energy Changes As a Function of Temperature

The Gibbs energy changes can be calculated at various temperatures

The Gibbs-Helmholtz relationship

Chemical Potentials of the Ideal Gas

Differentiating the chemical potential with temperature

Fundamental Relationships for a Closed, Simple System

For a reversible process

dU = TdS – PdV

The Fundamental Equation of Thermodynamics!!

Internal energy is a function of entropy and volume

The Mathematical Consequences

The total differential

The Maxwell Relationships
• The systems is described by
• Mechanical properties (P,V)
• Three thermodynamic properties (S, T, U)
• Three convenience variables (H, A, G)
An Example Maxwell Relationship
• The Maxwell relationships are simply consequences of the properties of exact differentials
• The equality of mixed partials
Other Thermodynamic Identities

The Thermodynamic Equation of State!!

Obtain relationships between the internal energy and the enthalpy

The Enthalpy Relationship

A simple relationship between (H/P)T and other parameters.

The Fundamental Equation

For a system at fixed composition

• If the composition of the system varies
The Chemical Potential

Using the chemical potential definition

Gibbs Energy of an Ideal Gas

Chemical potential is an intensive property

For an ideal gas

Note - J (T) is the Standard State Chemical Potential

of substance J

Chemical Potential in an Ideal Gas Mixture

The chemical potential of any gas in a mixture is related to its mole fraction in the mixture

Non-Reacting Mixtures

In a non-reacting mixture, the chemical potentials are calculated as above.

The total Gibbs energy of the mixture

Ideal Gas Mixtures

In an ideal gas mixture

Consider a closed system at constant pressure

The system consists of several reacting species governed by

The Gibbs Energy Change

At constant T and P, the Gibbs energy change results from the composition change in the reacting system

The Extent of Reaction

Suppose we start the reaction with an initial amount of substance J

nJ0

Allow the reaction to advance by  moles

 - the extent of reaction

The Non-standard Gibbs Energy Change

Examine the derivative of the Gibbs energy with the reaction extent

G – the non-standard Gibbs

energy change

The Equilibrium Condition

The equilibrium condition for any chemical reaction or phase change

The Gibbs Energy Profile of a Reaction

GA*

Pure components

GB*

0

min

max

Extent of Reaction, 

A (g) ⇌ B (g)

For the simple reaction

The Gibbs Energy Profile of a Reaction

GA*

Pure components

GB*

0

Mixing Contribution

min

max

Extent of Reaction, 

Adding in the contribution from mixG.

The Gibbs Energy Profile of a Reaction

GA*

Pure components

GB*

rG

0

eq

min

max

Extent of Reaction, 

The Gibbs energy of reaction.

Chemical Equilibrium in an Ideal Gas Mixture

For the reaction

aA (g) + bB (g) ⇌ pP (g) + qQ (g)

The Gibbs Energy Change

The Gibbs energy change can be written as follows

Standard Gibbs Energy Changes

fG = Jø = the molar formation Gibbs energy (chemical potential) of the substance

The Gibbs energy change for a chemical reaction?

The Reaction Quotient and G

Define the reaction quotient

The Equilibrium Point

At equilibrium, rG = 0

Equilibrium Constants and rG

At equilibrium, the non-standard Gibbs energy change is 0.

Standard State Chemical Potentials

Examine the following reaction

CO2 (g) – C (s) – ½ O2 (g) = 0

The standard state chemical potentials for the elements in their stable state of aggregation

Temperature Dependence of K

We can write the equilibrium constant as

Differentiating

The Gibbs-Helmholtz Equation

For a chemical reaction, with a standard Gibbs

energy change, rG

The van’t Hoff Equation

The van’t Hoff equation relates the temperature dependence of Kp to the reaction enthalpy change

The Integrated van’t Hoff Equation

Assuming the reaction enthalpy change is constant with temperature

The Result

If the enthalpy change for the reaction is know, we can estimate the Kp value at any temperature

The Integrated van’t Hoff Equation

H2O (l) ⇌H+ (aq) + OH- (aq)

reaction

Le Chatelier’s Principle

GA*

GB*

0

rG = 0

min

max

Extent of Reaction, 

Revisit the Gibbs energy profile!

At equilibrium, the Gibbs energy is at a minimum

The second derivative of the Gibbs energy with the extent of reaction,

= G’’ is positive!!

Le Chatelier’s Principle

rHo < 0

max

rHo >0

min

T / K

The change in the extent of reaction with temperature.

Le Chatelier’s Principle

rVo> 0

max

min

rVo < 0

P / bar

The change in the extent of reaction with pressure.