Geochemistry on the Earth’s surface
for analytical geochemists
1a.
Mineral stability and structure
Tento učební materiál vznikl v rámci projektu Rozvoj doktorského studia chemie
č. CZ.02.2.69/0.0/0.0/16_018/0002593
EQUILIBRIUM
Dissolving limestone
• What will happen if 1 kg of limestone (calcite)
is put into a container with 10 L of clean
water?
• When does the reaction stop?
CaCO3(𝑠) → Ca2+ 𝑎𝑞 + CO3
2−
𝑎𝑞
Precipitation of limestone
• What will happen if tap water is boiled in a
kettle and let to cool again?
• When does the reaction stop?
• What determines the direction of the
reaction?
CaCO3(𝑠) ← Ca2+ 𝑎𝑞 + CO3
2−
𝑎𝑞
Chemical equilibrium
• A chemical reaction does not proceed until the
reactants are completely consumed.
• The product molecules react with each other and
change back into reactants.
• At some point, the rates of the reverse and
forward reactions become equal, and the
established steady state is called chemical
equilibrium.
• Chemical equilibrium is a state in which the
rates of opposite chemical reactions are equal.
Equilibrium constant
• It indicates the extent of the reaction before
equilibrium is reached.
• It is given by the ratio of products and
reactants of a reaction at equilibrium:
𝑎A 𝑔 + 𝑏B 𝑔 ↔ 𝑐C 𝑔 + 𝑑D(𝑔)
𝐾𝑒𝑞 =
C 𝑐
D 𝑑
A 𝑎 B 𝑏
Equilibrium constant
• From the definition:
– Concentrations for expressing the equilibrium
constant are always in mol.L−1
– The products are always in the numenator.
– The reactants are always in the denominator.
– Exponents are always coefficients in an equation.
Equilibrium constant
• Determines the ratio between reactants and
products:
• K > 103 products significantly dominating
• K < 10−3 reactants significantly dominating
• 103 > K > 10−3 the resulting concentrations are
similar
Note
• Conventionally, only the concentrations of gases and
liquids (solutions) are included in the equation,
because solids have a constant concentration that is
already included in K.
• The equilibrium constant is dependent on temperature
(and to a much lesser extent on pressure).
CaCO3(𝑠) ↔ Ca2+ 𝑎𝑞 + CO3
2−
𝑎𝑞
𝐾𝑒𝑞 = Ca2+
CO3
2−
𝐾𝑒𝑞 = 10−8.4
3H2 𝑔 + N2 𝑔 ↔ 2NH3 (𝑔)
𝐾𝑒𝑞 = ?
Example 3.1
Combining hydrogen and nitrogen to ammonia
At a temperature of 350 °C, the equilibrium molar concentrations
are:
N 2 = 0.885
H 2 = 0.665
NH 3 = 1.230
What is the value of the equilibrium constant?
Will reactants or products predominate?
Example 3.2
• Calculate the value of the equilibrium
constant , Kc , for the system shown if a 2-liter
experimental vessel contains 0.1908 mol CO2 ,
0.0908 mol H2 , 0.0092 mol CO, and 0.0092
mol water vapor at equilibrium.
STABILITY
Assessment
• Which processes will run on their own?
Assessment
• Which processes will occur spontaneously?
– Total entropy increases ( ΔS > 0)
– Energy is released ( ΔH < 0)
• We can therefore base our assessment on ΔH,
ΔS and T (particle energy).
GIBBS FUNCTION
G = H − TS
Josiah Willard Gibbs
( 1839–1903 )
Freework,https://commons.wikimedia.org/w/index.php?curid=4935118
Gibbs function
entropy change of the system
and surroundings expressed by
thermodynamic quantities,
relating to the system
Gibbs function
Gibbs function
Thermodynamic equilibrium
A natural process
Evaluation of the Gibbs function
• From the value of ΔG, we can assess whether the
reaction will take place under the considered
conditions.
• ΔG < 0 means that the products are more stable
(they have a smaller G value) and the reaction
will proceed forward.
• ΔG > 0 means that the reactants are more stable
and the reaction will proceed in the reverse
direction.
• In any case, the reaction will eventually reach
equilibrium and ΔG = 0!
Example 3.3
• Determine Δ G r
o for calcite dissolution.
CaCO 3 => Ca 2+ + CO 3
2 −
• G f
o (Ca 2+ ) = −553.54 kJ /mol
• G f
o (CO 3
2− ) = −527.89 kJ /mol
• G f
o (calcite) = −1129.26 kJ /mol
• What does this say about the nature of the
reaction? And about the nature of equilibrium (if
we know that the activity of ions in a solution
under standard conditions is equal to 1)?
Example 3.4
• What will be the heat of reaction of calcite
dissolution?
Example 3.5
• What will be the entropy change when calcite
dissolves under standard conditions?
• S f
o (Ca 2+ ) = −53.2 J/K
• S f
o (CO 3
2− ) = −56.9 J/K
• S f
o (calcite) = 92.9 J/K
Example 3.6 (final)
• What value will Δ G r
o have for the dissolution of
calcite if we calculate it from the value of H
and S for the standard state (i.e. the previous
two calculations).
Example 3.7
• Find (a) the change in Gibbs function and (b)
the total entropy change during the
polymorphic transformation of 1 mole of
calcite to aragonite at 25 °C and 1 bar
pressure.
• Gibbs function of calcite is –1128.84 kJ /mol
and of aragonite is –1127.79 kJ /mol.
Gibbs function
Gibbs function
Dependence of Gibbs function on
composition
general dendence of G on terms
chemical potential
Dependence of G on p, T and composition for substance A
Dependence of chemical potential on composition
expression
Gas solutions
0 < X A , X B < 1; ln X A , ln X B < 0
is always negative, mixing gases always increases the total
entropy
The formation of a solution will always result in a reduction of chemical potential
of individual gases and thus also to a decrease in the value of the Gibbs function
of the mixture of gases (there will be an increase in total entropy).
Gas solutions
Gas solutions
The formation of a solution will always result in a reduction of chemical potential
of individual gases and thus also to a decrease in the value of the Gibbs function
of the mixture of gases (there will be an increase in total entropy).
Summary
• The first law tells what processes take place in nature:
Processes that conserve energy.
• The second law specifies the direction of the processes :
Processes take place in the direction in which the total entropy increases.
The relationship between the change in the Gibbs function and the total entropy:
Entropy is defined by the relation
The measure of total entropy change is the Gibbs function
Summary
The value of the Gibbs function depends on temperature, pressure and composition
whereas the chemical potential μ of substance A depends on the composition
where μ° is equal to the value of the Gibbs function of one mole of pure substance.
In real systems whose behavior differs from the ideal, concentration is replaced by
activity, and deviations from ideality are concentrated into the activity coefficient:
Another view
• We can also express the relationship as:
• H = G + TS
• The enthalpy of the substance then contains
two components:
– G – the energy component that can be released by
the reaction and therefore indicates the instability
of the substance (similar to potential energy).
– TS – energy component that is inseparably bound
in the internal disorder of the substance and is
unavailable through chemical reactions.
STABILITY
Adapted from Ryan (201
Stability
• The system tries to achieve the minimum
value of the Gibbs function for the given
conditions.
Adapted from Ryan (201
Stability
• If under the given conditions (T , p and composition)
there is a state with a lower value of the Gibbs
function, Nature tries to convert the system to this
state, because this will increase the total entropy. The
system will be unstable under the given conditions .
• If the system reached the state with the lowest value of
the Gibbs function (the highest total entropy) under
the given conditions, a change in the state of the
system would lead to an increase in the value of the
Gibbs function and thus to a decrease in the total
entropy. The system will be stable under the given
conditions.
Comparison of total entropies
Entropy components of the system with H2O in different states.
Total entropy of the system with H 2 O
For the sake of illustration, the temperature dependence of the entropies of ice, water, steam and the heats of melting and boiling on the temperature was
neglected. These changes in entropies depending on the temperature would slightly modify the individual values, but the basic relationships between the total
entropies of the system in individual states would remain the same.
Comparison of total entropies of the system with H2O in different states at different
temperatures.
Comparison of system entropy during crystallization
Dependence of phase stability on temperature and pressure
Temperature can be changed, pressure and composition remain constant.
Stability is determined by the value of the Gibbs function, which is an indicator of the total entropy of the system. Under certain conditions T and p, the system
chooses the state with the lowest value of the Gibbs function (highest total entropy).
As the temperature increases, the Gibbs function decreases in direct proportion to the negative value of the system's own entropy. For crystalline phases it
decreases the slowest (low entropy value), for gaseous phases it decreases the fastest (highest entropy value). Therefore, at high temperatures, the least ordered
phase is preferred.
T
Dependence of phase stability on temperature and pressure
Pressure can be varied, temperature and composition remain constant.
Stability is determined by the value of the Gibbs function, which is an indicator of the total entropy of the system. Under certain conditions T and p, the system
chooses the state with the lowest value of the Gibbs function (highest total entropy).
With increasing pressure, the Gibbs function grows directly proportional to the volume of substances. It grows fastest for gases, slowest for solids. Therefore, at high
pressures, the state with the smallest volume (the lowest value of the Gibbs function) is preferred.
P
Systems with variable phase composition
• chemical reactions
• melting in a multi-component system
• dissolution in water
Changes in the value of the chemical potentials of individual components and the values of the Gibbs
function of individual phases occur not only as a result of changes in temperature and pressure, but
also as a result of changes in the concentration of components in some or all phases.
Temperature and pressure are usually determined from outside the system, the concentration of
components in individual phases can change not only by supplying components from outside the
system and removing components from the system, but also by transitioning components between
individual phases.
Derivation of basic relations
For a model reaction that takes place at a certain temperature and pressure, it is possible to follow the
evolution of the value of the Gibbs function depending on how "far" the reaction proceeds. The
pressure and temperature do not change during the reaction and therefore the change in the value of
the Gibbs function depending on the temperature and pressure will be zero. In the model course, only
pure starting substances A and B are present in the system at the beginning, only pure products C and
D are present in the system at the end of the reaction.
The course of the values of the Gibbs function, depending on how much of the starting substances
have been transformed into products, also shows how the total entropy of the system changes.
According to the second law of thermodynamics, changes in the system will tend to achieve the lowest
value of the Gibbs function and thus to achieve the maximum total entropy.
To express the degree of transformation of starting substances into products, it is possible to use
the so-called progress of the reaction (extent of reaction), which indicates how much of the path
the system has traveled from starting substances to products.
The change in the representation of the components in the system depending on the progress of
the reaction ξ is then given by the relations
The value of the Gibbs function of the system at any moment is given by the sum of the values of
the Gibbs function of the individual components
Dependence of the Gibbs function on the progress of the reaction
about
For the dependence of the Gibbs function on the progress of the reaction , we obtain
The value d G /d ξ represents the direction of the tangent to the dependence of the value of the
Gibbs function on the progress of the reaction. If this direction is negative, then the further
continuation of the reaction from left to right will lead to a decrease in the Gibbs function and
the reaction will tend to continue. If the slope is positive, then the further continuation of the
reaction would increase the value of the Gibbs function of the system. On the contrary, the
reaction will tend to proceed from right to left.
The minimum of the Gibbs function and thus the equilibrium will be reached at the point where
this slope has a zero value.
The direction of the dependence of the value of the Gibbs
function on the progress of the reaction is referred to as the
reaction Gibbs function Δ G r
By substituting for chemical potentials and further modification
Δ G° r is the standard reaction Gibbs function (or the standard value of the reaction Gibbs
function) and is equal to the difference of the standard values of the Gibbs function of the
substances on the right side and the substances on the left side of the chemical reaction. The
coefficient Q is referred to as the reaction quotient and is equal to the product of the
instantaneous activities (concentrations) of the products multiplied by the respective coefficients,
divided by the product of the activities (concentrations) of the reactants multiplied by the
respective coefficients.
For balance
At equilibrium, the reaction Gibbs function is equal to zero, the system tends to remain in this state with the lowest value of the
Gibbs function. The reaction quotient Q under these conditions becomes the equilibrium constant K . The equilibrium constant is
uniquely determined by the difference in the standard values of the Gibbs function of the substances that participate in the
reaction.
This key relationship applies to any transformations that result in a change in the concentrations of some or all components of
the system. In essence, the change in the value of the Gibbs function of the system is not the "connecting" and "disconnecting" of
the components of the system (atoms, ions, molecules), but the change in their concentrations as a result of this "connecting"
and "disconnecting", i.e. the change in dilution and the associated change in the Gibbs function relevant components in the form
of a RT member ln and .
If the model reaction did not take place in an environment where at least some components form a solution - there would be no
change in concentration, the values of the Gibbs function would remain constant. Under the given conditions of temperature and
pressure, the combination of substances that have a lower value of the Gibbs function (A and B or C and D) at the given
temperature and pressure would clearly be preferred . Only by changing the external conditions (temperature, pressure or both)
can a state be reached where the mentioned substances can be in equilibrium with each other. If the given substances form a
solution, then the system can reach the minimum of the Gibbs function also by internal changes in the system by changing the
concentration of the components due to chemical reactions, dissolution or melting and crystallization.
The given relationship shows how the equilibrium constant K (more precisely , ln K ) when the temperature changes by
d T . The value Δ H ° r is the standard reaction enthalpy – the difference in the enthalpies of the substances on the right
and left side of the reaction.
Dependence of the equilibrium constant on temperature
Van't Hoff's equation
We can integrate on
To express the change in K with temperature change
Example 3.8
• What is the value of the solubility product (=
equilibrium dissolution constant) of calcite at
25 °C and 1 atm ?
Example 3.9
• In underground karst systems, the average
temperature is 10 °C. What will be the
corresponding dissociation constant for
calcite?
Summary
Basic principles applicable to the assessment of system stability and transformations:
1. in any process, energy is conserved
2. in any process the total entropy increases
It then follows from these two principles
It then holds for the Gibbs function and related quantities
lnRT X 
= +
G H TS= −
ln rG
K
RT

−
=
celk
dG
dS
T
= −
dG Vdp SdT dn= − +
> 0 a tedy 0celkdS dG 
1 1rH
R T T
T T
K K e



−  
− 
 
=
With the help of these relationships, any problem of balance and transformations can be solved.
Tento učební materiál vznikl v rámci projektu Rozvoj doktorského studia
chemie
č. CZ.02.2.69/0.0/0.0/16_018/0002593
Resources
• Images without a specified source are
copyrighted, have licenses that do not require
the source or author to be indicated, or are
used with the consent of doc. Zemana.
• RYAN, Peter Crowley. Environmental and low
temperature geochemistry. Chichester, West
Sussex, UK: Wiley Blackwell, 2014. ISBN 978-1-
118-86735-8.