• Quantum mechanic

• Statistical physics

CHEMICAL THERMODYNAMICS

At the microscopic scale (molecular scale) , valence electrons determine the chemical reactivity of compounds .

At the macroscopic scale we are interested to know if a chemical reaction is effectively possible. Beside electronic affinity, other factors will determine the chemical reaction and its efficiency (can you cite some ? )

Thermodynamics provide a theoretical and practical framework to do that. This approach is fundamental in the study of natural systems.

* Historically TD was developed before and from QM and SP. SP shows the remarkable consistency between microscopic and thermodynamical approach.

• Stoichiometry

• Limiting Reagent

• Mole, molar mass

Chemical reactions (macroscopic scale) : basic notions

The mole is a unit of measurement used in chemistry to express amounts of a chemical substance, defined as an amount of a substance that contains as many elementary entities (e.g., atoms, molecules, ions, electrons) as there are atoms in 12 grams of pure carbon-12 (12C), the isotope of carbon with atomic weight 12. This corresponds to a value of 6.02214179(30)×1023 elementary entities of the substance.

Stoichiometry is the method by which we calculate how many grams of these reactants are needed to form a given amount of product in a chemical reaction. g.g. How many moles of O2 are required to burn 0.5 moles of CH4 ?

CH4 + 2 O2 CO2 + 2 H2O

Q: How many grams of CO2 will be produced when 0.5 moles of CH4 is burned

If there is less of one reactant than required by stoechiometric proportions then reactant is called the limiting reagent. When the limiting reagent is used up, the reaction will stop.

• Stoichiometry in Solution

e.g. : The term 2M NaOH is the abbreviation for 2 molar sodium hydroxide solution. This expression means that 2 moles (or 2 x 40 g = 80 g) of NaOH are dissolved in enough water to make one liter of solution

Q : How many mL of a 0.5 M HCl solution would be required to react with 20 g of NaOH? The balanced chemical equation is...

• Stoichiometry in the gas phase:

• Pi = Ptotal.xi

Mole fraction

Partial pressure

Mass fraction

Concentration : C [kg.m-3]. For a compound x, coenctrations are often noted [x]

Mass mixing ratio : mr [kg.kg-1]

part per part volume (ppp), ppmv, ppbv

part per part mass (pppm), ppmm, ppbm

molarity: C [kg.l-1]

molality : M [mol.l-1]

Activity of a species : {X}

Define an effective concentration, depending on the phase of the compound.

The activity of a gas corresponds to its partial pressure.

For a species dissolved in water ( aqueous phase species) under ideal conditions , the activity is equal to its concentrations. {X} ~ [X]

As the concentration of ions in solution increases (i.e. the solution becomes non ideal), solvation effects ( interactions between water molecules and the ions) take place and activities differs from concentrations. Practically activities will be expressed as “corrected concentrations” using activity coefficients.

For a species in solid phase, the activity is equal to 1.

Thermodynamical systems

• Isolated system: have boundaries that are rigid, do not permit transfer of mechanical energy, heat or matter. They have a constant energy and mass content. Universe as a whole is an isolated system.

• Closed systems: have boundaries that allow transfer of energy in or out of the system but no transfer of matter is allowed. They contain a fixed mass, but variable energy.

• Open Systems: have boundaries that allow transfer of both energy and matter to and from the system.

• Adiabatic systems: Do not allow heat transfer.

• Sate function or state variable : property of a system that depends only on the current state of the system, not on the way in which the system acquired that state (independent of the path).

• System evolves from a state 1 to a state 2 but there is no kinetics consideration.

• Extensive vs intensive variables.

First Law of Thermodynamics(cf also the atmospheric thermodynamics class)

The total energy of the universe is a constant.

Energy can, however, be converted from one form to another or transferred from a system to the surroundings or vice versa.

• For a closed system dU = δq + δw U : internal energy (nature ? ), state variabe no absolute value for U only variations are meaningful )\

∆U = Q + W = Q - P ∆V (if work is exchanged only through volume variations : cf piston model)

∆Uuniverse = ∆Usystem + ∆Usurroundings = 0

Transformation at constant volume :

Transformation at constant pressure :

∆U = Qv (no work exchanged)

Qp = ∆(U + p.V) = ∆H

H = U + p.V is the enthalpy of the system.

Joule’s law : For a prefect gas, the internal energy only depends on temperature.

Thermochemistry

Application the first principle to study chemical reactions occurring in a system. If the reaction happens at constant pressure the ‘heat of reaction’ (measurable) is equal to the system enthalpy variation. Heat of reaction can also be measured at constant volume.

Enthalpy of formation :

Heat corresponding to reaction of the formation of one mole of a complex compounds from simple compounds considered in standard conditions:P=1atm, T=298K, pure reactants in their more stable states

Practically standard enthalpy of formation are kept in thermodynamical tables (given apart)

The standard enthalpy of formation is used to find the standard enthalpy change of a reaction.

This is done by subtracting the sum of the standard enthalpies of formation of the reactants (each being multiplied by its respective stoichiometric coefficient, ν) from the sum of the standard enthalpies of formation of the products (each also multiplied by its respective stoichiometric coefficient), as shown in the equation below:

ΔHr° = Σ(ν × ΔHf°) (products) - Σ(ν × ΔHf°) (reactants)

Later we will use a more compact notation :

ΔHr° = Σ νi × ΔHf°i with

For example, for the reaction CH4 + 2 O2 → CO2 + 2 H2O:ΔHr° = [(1 × ΔHf°(CO2)) + (2 × ΔHf°(H2O))] (products) - [(1 × ΔHf°(CH4)) + (2 × ΔHf°(O2))] (reactants)

If the standard enthalpy of the products is less than the standard enthalpy of the reactants, the standard enthalpy of reaction will be negative. This implies that the reaction is exothermic. The converse is also true; the standard enthalpy of reaction will be positive for an endothermic reaction.

Enthalpy of reaction

νi > 0 for products νi < 0 for reactants

Spontaneous Processes

• Spontaneous processes are those that can proceed without any ‘outside intervention’.

• Q: Can the first principle account for the spontaneous character of a process ?

Reversible Processes

• In a reversible process the system changes in such a way that the system and surroundings can be put back in their original states by exactly reversing the process.

• Changes are infinitesimally small in a reversible process.

Irreversible Processes

• Irreversible processes cannot be undone by exactly reversing the change to the system.

• All Spontaneous processes are irreversible.• All Real processes are irreversible.

Entropy

For a process occurring at constant temperature (an isothermal process) we can define a corresponding variation of entropy as :

qrev = the heat that is transferred when the process is carried out reversibly at a constant temperature T = temperature in Kelvin.

T

dqdS rev

2

1 T

dqS revFor a finite change between two states 1 and 2

dqrev is the quantity of heat added reversibly to a system at temperature T.

A reversible (or equilibrium) transformation is one in which a system moves by infinitesimal amounts and infinitesimally slowly between equilibrium states, so that the direction of the process can be reversed at any time just by making an infinitesimal change in the surroundings. Entropy is a function of state.

Second Law of Thermodynamics

The second law of thermodnamics: The entropy of the universe does not change for reversible processes and increases for spontaneous processes.

Reversible (ideal):

Irreversible (real, spontaneous):

For an isolated system

Practically we are interested in entropy variations occurring in the studied system

For an non isolated system

ΔSsystem ≥ 0

ΔSsurroundings ≥ - ΔSsystem

Transformation that induce a diminution of the system entropy are only possible if the system can exchange energy with the surroundings (or another system).

You can create order in a system but it has to be ‘paid’ in energy to the surrounding (examples ?).

Q: Use the second law to prove that an isolated perfect gas can spontaneously expand but not contract

Third law of thermodynamics

• The number of micro-states and, therefore, the entropy tends to increase with increases in– Temperature.– Volume (gases).– The number of independently moving molecules (phase) .

• In general, entropy increases when– Gases are formed from liquids and solids.– Liquids or solutions are formed from solids.– The number of gas molecules increases.– The number of moles increases.

Entropy can be thought of as a measure of the randomness of a system.It is related to the various modes of motion in molecules.

We can define a absolute value of entropy in function of a reference state: The entropy of a pure crystalline substance at absolute zero is 0.

W : number of microscopic state (ri,vi,ei) corresponding to one macroscopic state (T,P,C)

…

Standard Entropies

These are molar entropy values of substances in their standard states.Standard entropies tend to increase with increasing molar mass.

Absolute value of entropy in standard conditions can be calculated and are referenced in tables

• S.E. are used to calculate variation of entropy relative to a reaction in standard conditions

ΔSr° = Σ(ν × S°) (products) - Σ(ν × S°) (reactants)

Open systems of variable chemical composition

More generally for an open system of variable composition (or a closed system that undergoes composition changes) the internal energy will also depend of the composition of the system (i.e. the number of mole of different component of the mixture) :

The variation of internal energy will thus also depends of the variations of the amount (in mole) of the different constituents, which we can express using the differential form :

with

* The chemical potential, μi has an important function in system’s thermodynamic behavior. A temperature difference between two bodies determines the tendency of heat to pass from one body to the other, while a pressure tendency determine tendency for bodily movement. A difference of chemical potential can be viewed as the cause of a chemical reaction or mass transfer from one phase to the other.

We introduce at this stage an important notion : the chemical potential * of a species defined as :

Combining first and second principle for a closed system we have :

Gibbs free energyCalculation of dU of a system requires the estimation of change of entropy , volume and number of moles of different constituents ni as independent variables. It is not convenient to work with these variables for natural systems: temperature and pressure are much more useful.

The thermodynamical study of chemical processes can be facilitated by introducing other thermodynamic variables . One of the most useful is the Gibbs free energy :

G = H – TS or

1 : If T and P are constant, what is the physical interpretation of G ?A variation of the internal energy of the system can be written in this case as :

dU = – P.dV + T.dS + dG

Variation of energy linked to the micro-state change of the system

Mechanical work produced

• Maximum non mechanical work that the system can produce …

• (e.g. chem reaction in a reactive system, electrical work , osmotic phenomena …).

By analogy between the previous expressions of dU, dG relates to chemical potentials and mole variations as :

• It is possible to show using the Euler theorem that G (extensive quantity) can be written in general as :

• The chemical potential of a substance can be viewed as the value of Gibbs free energy per mole of the substance (partial molar Gibbs free energy). It is an intensive quantity relative to a specific component of the system.

It explains why the general expression of G from the chemical potentials does not explicitly depends on pressure and temperature.

2 : In the more general case (T and P not anymore constant), the variation of free energy includes variation of energy produced by dT and dp and dni :

If we consider the general definition of G :

If the temperature and the pressure of the system change, there is a corresponding change of the chemical potential of each species in the system (Gibbs-Duhem relationship).

If T and P are constant, we also have :

dG = dH – TdS , in other words :

dG = dq – dqrev

If a system is in thermodynamical equilibrium, every possible infinitesimal transformation is reversible. This implies that if the transformation occurs under equilibrium conditions.

Evolution of a system

dq = dqrev and dG = 0

If a system is not in equilibrium : dq < dqrev and dG < 0

According to first and second principle, thermodynamics tends to move the system to a state that minimize its internal energy and maximizes its entropy.

The condition for spontaneous evolutionof the system is expressed as :

Example : Consider the general reaction

The evolution (‘toward the left’ or ‘toward the right’) is determined by the sign of ΔGr.In standard conditions (p=1atm, T ), we can calculate the Gibbs free energy variation

corresponding to the reaction.

ΔG0f : Is the standard Gibbs molar free energy of formation of a compound is the change of Gibbs free energy that accompanies the formation of 1 mole of a substance in its standard state from its constituent elements in their standard states (the most stable form of the element at 1 atm of pressure and the specified temperature, usually 298.15 K or 25 °C).

This data is available from tables or calculated from standard enthalpy of formation and entropy . What is the unit of DG0r ? It is equivalent to the standard chemical potential defined further.

At constant pressure and temperature, for a chemical reaction we can express the corresponding Gibbs free energy change of the reaction in term of molar quantities as :

rrr STHG

Application to chemical reactions

According to the sign of ΔG, we will know if the reaction is spontaneous or not

r

For a general reaction :

At equilibrium (P,T)

Chemical equilibrium

Chemical equilibrium is reached when : dG = 0

• νi > 0 for products

• νi < 0 for reactants

Which means that for a given p and T , the composition of the system and the chemical potentials are not evolving anymore

Where Mi are the concentrations and vi the stoechiometic coefficients

This implies the following condition on chemical potentials:

Practically, the determination of the equilibrium composition of a multiphase system requires the determination of the chemical potentials for all species as a function of the corresponding conditions of temperature, pressure and concentrations in reference to standard state :

• For a gas mixture:

Standard chemical potential (at p=1atm)

Partial pressure

• For an ideal solution :

The chemical potential of a species i in solution follows:

xi is the mole fraction of i in the solution

standard chemical potential of i in a pure state in standard conditions

Correction for pressure and composition

Q: For an ideal gas at constant T ?

• Non-Ideal solutions :

γi are called activity coefficients (determined experimentally) accounting for non ideal behavior of solutions. γi = 1 for ideal solutions, and < 1 for real solutions. Activity coefficient will be given or considered equal 1

• Solids The activity is equal to 1 so the chemical potential always equal to standard chemical potential

We have to introduce activities

The equilibrium constant of a reaction

For a general reaction:

And we can write at equilibrium :

And Equilibrium constant of the reaction.

• K is known (and referenced in tables) for a reaction since it is calculated from known standard potentials. Note that it depends on the temperature

• This fixes a condition on ratio of reactants to products concentrations (activities) at thermodynamical equilibrium.

• Q: write K in function of standard molar Gibbs free energy variation associated to the reaction

• What information can the value of K give us ?

• If K >> 1, the reaction is product-favored; product predominates at equilibrium.

• If K << 1, the reaction is reactant-favored; reactant predominates at equilibrium.

• Remember that K depends on temperature and pressure conditions !

• According to the type of reaction (or process) studied, the thermodynamical equilibrium constant K can take different names (this will be detailed in the following chapters) :

• e.g.:

• Dissolution of a gas : K=H , Henry consant• Dissolution of a solid ( NaCl <=> Na+ + Cl-) : K=Ks , solubility constant• Acid / base equilibrium : K=Ka, dissociation constants• …

•

Evolution towards equilibrium

Qi Reaction Quotient (not in equilibrium) calculated from activities of product and reactants at any time.

: Equilibrium constant (equilibrium) : αi = αi(eq) , found in tables

• if Q < Keq then rxn will shift to R (more products)• if Q > Keq then rxn will shift to L (more reactants)

Le Châtelier’s Principle: Perturbation of a system at equilibrium system will shift to minimize perturbation.

• We can also express the variation of molar Gibbs free energy at any stage of the reaction in function of Q and K at constant pressure and temperature.

(Q: Can you try to show it ?)

• ΔGr = RT ln(Q/K)

Equilibrium between gas phase and liquid phaseRaoult and Henry’s law

dG=0, which in term of chemical potential can be written as :

• If xi -->1 : Pi --> Ki ( T,p) = P0i = vapor pressure of the pure component

= Raoult’s law

The ideal gas vapor pressure of an ideal solution component is equal to the vapor pressure of the pure liquid component multiplied by the mole fraction of the component in the solution

For real solutions, Raoult's law behavior is approached as Xi ---> 1 : which means that i is the solvent

We consider a non reactive species which can exist in gas and liquid phase state.Once the system reach equilibrium, we have :

The ratio of partial pressure (in the gas phase) to the molar fraction (in liquid phase) of the chemical species is equal to the the thermodynamical equilibrium constant:

and

• If xi --> 0 : Ki(T,P) tends toward Hi the slope of the plot of Pi vs xi for sufficiently dilute solutions (i is the solute)

pi = Hi . xi = Henry’s law

For a dilute solution the ideal gas vapor pressure of a volatile solute is proportional to its mole fraction in the solution; that is, escaping tendency of the solute molecule is proportional to their mole fraction.

e.g solution of two components A and B

Non ideal solution approach ideality when concentrations of the different species are small except for one (the solvent). In this case, the solute follows Henry’s law and the solvent Raoult’s law.

In most geophysical situations, the solvent will be water.