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Physical Chemistry
for Energy Engineering
(7th: 2013/10/8)
Takuji Oda
Assistant Professor, Department of Nuclear Engineering
Seoul National University
II-2. Phase equilibria- $23-3: The chemical potentials of a pure substance in two phases in equilibrium
are equal -
We consider a system consisting of two phases of a pure substance
in equilibrium each other. (liquid and gas, for example here)
The Gibbs energy of this system is given by
𝐺 = 𝐺𝑙 + 𝐺𝑔
where 𝐺𝑙 and 𝐺𝑔 are the Gibbs energies of the liquid and the gas phase.
Now, suppose 𝑑𝑛 mole are transferred from the liquid to the solid phase,
where T and P are kept constant. The infinitesimal change in Gibbs energy
for this process is:
𝑑𝐺 =𝜕𝐺𝑔
𝜕𝑛𝑔 𝑃,𝑇𝑑𝑛𝑔 +
𝜕𝐺𝑙
𝜕𝑛𝑙 𝑃,𝑇𝑑𝑛𝑙
As 𝑑𝑛𝑙 = −𝑑𝑛𝑔, then
𝑑𝐺 =𝜕𝐺𝑔
𝜕𝑛𝑔 𝑃,𝑇−
𝜕𝐺𝑙
𝜕𝑛𝑙 𝑃,𝑇𝑑𝑛𝑔
Here, we define chemical potentials, 𝜇𝑔 =𝜕𝐺𝑔
𝜕𝑛𝑔 𝑃,𝑇and 𝜇𝑙 =
𝜕𝐺𝑙
𝜕𝑛𝑙 𝑃,𝑇, then
𝑑𝐺 = 𝜇𝑔 − 𝜇𝑙 𝑑𝑛𝑔 (constant T and P)
II-2. Phase equilibria- $23-3: The chemical potentials of a pure substance in two phases in equilibrium
are equal -
We consider a system consisting of two phases of a pure substance
in equilibrium each other. (liquid and gas, for example here)
𝑑𝐺 = 𝜇𝑔 − 𝜇𝑙 𝑑𝑛𝑔 (constant T and P)
where chemical potentials are 𝜇𝑔 =𝜕𝐺𝑔
𝜕𝑛𝑔 𝑃,𝑇and 𝜇𝑙 =
𝜕𝐺𝑙
𝜕𝑛𝑙 𝑃,𝑇.
If the two phases are in equilibrium with each other, then 𝑑𝐺 = 0.
And because we suppose some amount of the liquid phase is
transferred to the gas phase (𝑑𝑛𝑔) here, to make 𝑑𝐺 = 0 in this
condition, 𝜇𝑔 = 𝜇𝑙 is needed.
If 𝜇𝑔 < 𝜇𝑙, [𝜇𝑔 − 𝜇𝑙] < 0. Thus, the process of 𝑑𝑛𝑔 > 0 (transfer
from the liquid phase to the gas phase) spontaneously takes place
as it holds 𝑑𝐺 < 0.
Likewise, if 𝜇𝑔 > 𝜇𝑙, 𝑑𝑛𝑔 < 0.
Hence, in general, the transfer occurs from the phase with higher
chemical potential to the phase with lower chemical potential.
II-2. Phase equilibria- $23-3: The chemical potentials of a pure substance in two phases in equilibrium
are equal -
In general, the chemical potential is defined as:
μ =𝜕𝐺
𝜕𝑛 𝑃,𝑇
Because 𝐺 is proportional to the size/amount of a system 𝐺 ∝ 𝑛,
namely an extensive thermodynamic function, we can express it as:
𝐺 = 𝑛𝜇 𝑇, 𝑃 Here, this equation is consistent with the definition of chemical
potential:
μ =𝜕𝐺
𝜕𝑛 𝑃,𝑇=
𝜕 𝑛𝜇 𝑇,𝑃
𝜕𝑛 𝑃,𝑇= 𝜇 𝑇, 𝑃
which means 𝜇 𝑇, 𝑃 , the chemical potential, is the same quantity as
the molar Gibbs energy and it is an intensive quantity.
II-2. Phase equilibria- $23-3: The chemical potentials of a pure substance in two phases in equilibrium
are equal -
As they are in equilibrium,
μ𝛼(𝑇, 𝑃) = μ𝛽(𝑇, 𝑃)Now take the total derivatives of both sides
𝜕μ𝛼
𝜕𝑃 𝑇𝑑𝑃 +
𝜕μ𝛼
𝜕𝑇 𝑃𝑑𝑇 =
𝜕μ𝛽
𝜕𝑃 𝑇𝑑𝑃 +
𝜕μ𝛽
𝜕𝑇 𝑃𝑑𝑇
Since 𝜇 is simply the molar Gibbs energy for a single substance, utilizing 𝜕𝐺
𝜕𝑃 𝑇= 𝑉 and
𝜕𝐺
𝜕𝑇 𝑃= −𝑆 (*these were previously derived along Maxwell
relations)𝜕μ
𝜕𝑃 𝑇=
𝜕 𝐺
𝜕𝑃 𝑇= 𝑉 and
𝜕μ
𝜕𝑇 𝑃=
𝜕 𝐺
𝜕𝑇 𝑃= − 𝑆
where 𝑉 and 𝑆 are the molar volume and the molar entropy. Then, 𝑉𝛼𝑑𝑃 − 𝑆𝛼𝑑𝑇 = 𝑉𝛽𝑑𝑃 − 𝑆𝛽𝑑𝑇
Since we consider the two phases are in equilibrium each other𝑑𝑃
𝑑𝑇=
𝑆𝛽− 𝑆𝛼
𝑉𝛽− 𝑉𝛼=∆𝑡𝑟𝑠 𝑆
∆𝑡𝑟𝑠 𝑉=
∆𝑡𝑟𝑠 𝐻 𝑇
∆𝑡𝑟𝑠 𝑉=
∆𝑡𝑟𝑠 𝐻
𝑇∆𝑡𝑟𝑠 𝑉
We consider two phases (α and β) are in equilibrium each other.
II-2. Phase equilibria- $23-3: The chemical potentials of a pure substance in two phases in equilibrium
are equal -
In the previous slide, we obtain for two phases in equilibrium each other:
𝑑𝑃
𝑑𝑇=∆𝑡𝑟𝑠 𝐻
𝑇∆𝑡𝑟𝑠 𝑉
This equation is called the Clapeyron equation, and relates “the slope
of the two-phase boundary line in a phase diagram with the values of
∆𝑡𝑟𝑠 𝐻 and ∆𝑡𝑟𝑠 𝑉 for a transition between these two phases”.
II-2. Phase equilibria- $23-3: The chemical potentials of a pure substance in two phases in equilibrium
are equal -
Exampe-1) Solid-liquid coexistence curve at around 1 atm for Benzene
∆𝑓𝑢𝑠 𝐻 = 9.95 kJ mol-1 and ∆𝑓𝑢𝑠 𝑉 = 10.3 cm3 mol-1 at the normal
melting point (278.7 K).
Thus, 𝑑𝑃 𝑑𝑇 at the normal melting point of benzene is:𝑑𝑃
𝑑𝑇=
∆𝑡𝑟𝑠 𝐻
𝑇∆𝑡𝑟𝑠 𝑉=
9.95 𝑘𝐽 𝑚𝑜𝑙−1
278.7 𝐾 (10.3 𝑐𝑚3𝑚𝑜𝑙−1)= 34.2 𝑎𝑡𝑚 𝐾−1
Here, by taking the reciprocal of this result:𝑑𝑇
𝑑𝑃= 0.0292 𝐾 𝑎𝑡𝑚−1
Using the above result (0.0292 K atm-1) and
assuming ∆𝑓𝑢𝑠 𝐻 and ∆𝑓𝑢𝑠 𝑉 are independent
of pressure, we predict the melting point as:
308 K at 1000 atom (experimental value
is 306 K)
570 K at 10000 atom (exp. value ~ 460 K)
*as clearly in the left figure, ∆𝑓𝑢𝑠 𝐻 and
∆𝑓𝑢𝑠 𝑉 are not independent of pressure at
high-pressure region
II-2. Phase equilibria- $23-3: The chemical potentials of a pure substance in two phases in equilibrium
are equal -
Exampe-2) Solid-liquid coexistence curve at around 1 atm for water
∆𝑓𝑢𝑠 𝐻 = 6.01 kJ mol-1 and ∆𝑓𝑢𝑠 𝑉 = -1.63 cm3 mol-1 at the normal
melting point (273.15 K).
Thus, 𝑑𝑃 𝑑𝑇 at the normal melting point of water is:𝑑𝑇
𝑑𝑃=𝑇∆𝑡𝑟𝑠 𝑉
∆𝑡𝑟𝑠 𝐻=
273.15 𝐾 (−1.63 𝑐𝑚3𝑚𝑜𝑙−1)
6.01 𝑘𝐽 𝑚𝑜𝑙−1= −0.00751 𝐾 𝑎𝑡𝑚−1
As already mentioned, the melting point of
ice decreases with increasing pressure.
Hence, the solid-liquid coexistence curve in
P-T phase diagram has a negative slope.
This clearly comes from the fact that
∆𝑓𝑢𝑠 𝑉 < 0 (the molar volume is larger in
solid than that in liquid at around melting
point, 1atm), because ∆𝑓𝑢𝑠 𝐻 = 𝑇∆𝑓𝑢𝑠 𝑆 must
be larger than 0 as ∆𝑓𝑢𝑠 𝑆 must be positive
(liquid is more disordered than solid)
II-2. Phase equilibria- $23-4: The Clausius-Clapeyron equation gives the vapor pressure of a substance
as a function of temperature -
The assumption we made for benzene two slides ago that “∆𝑓𝑢𝑠 𝐻 and
∆𝑓𝑢𝑠 𝑉 are independent of pressure” is indeed satisfactory for solid-liquid
and solid-solid transitions over a small ∆ 𝑇.
However, it is not satisfactory for liquid-gas and solid-gas transitions
because the molar volume of a gas varies strongly with pressure.
But, if the temperature is not too near the critical point, 𝑑𝑃
𝑑𝑇=
∆𝑡𝑟𝑠 𝐻
𝑇∆𝑡𝑟𝑠 𝑉can
be cast into a useful form for condensed phase-gas phase transitions.
II-2. Phase equilibria- $23-4: The Clausius-Clapeyron equation gives the vapor pressure of a substance
as a function of temperature -
We consider the Clapeyron equation in liquid-gas equilibrium.
First, we rewrite the equation:𝑑𝑃
𝑑𝑇=
∆𝑡𝑟𝑠 𝐻
𝑇(∆𝑡𝑟𝑠 𝑉)=
∆𝑣𝑎𝑝 𝐻
𝑇( 𝑉𝑔− 𝑉𝑙)
Here, if (1) the system is not too near the critical point (thus, 𝑉𝑔 ≫ 𝑉𝑙 and 𝑉𝑙
in the denominator is negligible) and (2) the pressure is not too high (thus, we
can assume the vapor is an ideal gas and 𝑉𝑔 = 𝑅𝑇 𝑃) 𝑑𝑃
𝑃𝑑𝑇=𝑑 ln 𝑃
𝑑𝑇=
∆𝑣𝑎𝑝 𝐻
𝑃𝑇( 𝑉𝑔− 𝑉𝑙)~∆𝑣𝑎𝑝 𝐻
𝑃𝑇( 𝑉𝑔)~
∆𝑣𝑎𝑝 𝐻
𝑃𝑇( 𝑅𝑇 𝑃)=∆𝑣𝑎𝑝 𝐻
𝑅𝑇2
which is know as Clausius-Clapeyron equation (derived by Clausius in 1850).
If ∆𝑣𝑎𝑝 𝐻 does not vary with temperature over integration limits of T, then
ln𝑃2
𝑃1= −
∆𝑣𝑎𝑝 𝐻
𝑅
1
𝑇2−1
𝑇1=∆𝑣𝑎𝑝 𝐻
𝑅
𝑇2−𝑇1
𝑇1𝑇2
which can be used to calculate the vapor pressure at some temperature
given the molar enthalpy of vaporization and the vapor pressure at some
other temperature.
II-2. Phase equilibria- $23-4: The Clausius-Clapeyron equation gives the vapor pressure of a substance
as a function of temperature -
Example-1) Vapor pressure of benzene at 373.2 K
We obtain an integral form of Clausius-Clapeyron equation as
ln𝑃2
𝑃1=∆𝑣𝑎𝑝 𝐻
𝑅
𝑇2−𝑇1
𝑇1𝑇2
Note that we assume ∆𝑣𝑎𝑝 𝐻 does not (largely) vary with temperature.
∆𝑣𝑎𝑝 𝐻 = 30.8 kJ mol-1 for benzene at the normal boiling point (353.2 K),
and assume it is not so dependent on temperature.
The vapor pressure of benzene is 760 torr = 1 atm at 353.2 K (as the
definition of boiling point).
Then, using the above equation:
ln𝑃
760=30.8× 103
8.31
373.2−353.2
373.2×353.2
This gives us P=1330 torr, close to the experimental value,1360 torr.
II-2. Phase equilibria- $23-4: The Clausius-Clapeyron equation gives the vapor pressure of a substance
as a function of temperature -
If we take an integral of Clausius-Clapeyron equation indefinitely rather
than between definite limits, we obtain (still assuming ∆𝑣𝑎𝑝 𝐻 is constant) 𝑑𝑃
𝑃𝑑𝑇=𝑑 ln 𝑃
𝑑𝑇=∆𝑣𝑎𝑝 𝐻
𝑅𝑇2→ ln 𝑃 = −
∆𝑣𝑎𝑝 𝐻
𝑅𝑇+ constant
This means that a plot of the logarithm of the vapor pressure against the
reciprocal of temperature is a straight line with a slope of −∆𝑣𝑎𝑝 𝐻 𝑅.
The slope of line for
benzene (left figure) over
313-353 K gives us
∆𝑣𝑎𝑝 𝐻 = 32.3 𝑘𝐽 𝑚𝑜𝑙−1,
close to 30.8 kJ mol-1
which is the experimental
value at the normal boiling
point (353 K).
<Benzene case>
II-2. Phase equilibria- $23-4: The Clausius-Clapeyron equation gives the vapor pressure of a substance
as a function of temperature -
If we consider the temperature dependence of ∆𝑣𝑎𝑝 𝐻, for example, as
∆𝑣𝑎𝑝 𝐻 = 𝐴 + 𝐵𝑇 + 𝐶𝑇2 +⋯
where A, B, C are constants.
Then, Clausius-Clapeyron equation, 𝑑𝑃
𝑃𝑑𝑇=𝑑 ln 𝑃
𝑑𝑇=∆𝑣𝑎𝑝 𝐻
𝑅𝑇2, gives us
ln 𝑃 = −𝐴
𝑅𝑇+𝐵
𝑅ln 𝑇 +
𝐶
𝑅𝑇 + 𝑘 + 𝑂 𝑇2
This equation works a wider range accurately, more than the equation that
we obtain assuming ∆𝑣𝑎𝑝 𝐻 is constant.
For example, for the vapor pressure of solid ammonia over 146-195 K is:
ln 𝑃 𝑡𝑜𝑟𝑟 = −4124.4 𝐾
𝑇+ 1.8163 ln
𝑇
𝐾+ 34.4834
II-2. Phase equilibria- $23-4: The Clausius-Clapeyron equation gives the vapor pressure of a substance
as a function of temperature -
The Clausius-Clapeyron equation also proves that the slope of the solid-gas
coexistence curve must be greater than the slope of the liquid-gas coexistence
curve near the triple pint (where these curves meet), as follows.
The Clausius-Clapeyron eq. gives the solid-gas and liquid-gas curve slopes as𝑑𝑃𝑠
𝑑𝑇= 𝑃𝑠
∆𝑠𝑢𝑏 𝐻
𝑅𝑇2and
𝑑𝑃𝑙
𝑑𝑇= 𝑃𝑙
∆𝑣𝑎𝑝 𝐻
𝑅𝑇2
here 𝑃𝑠 and 𝑃𝑙 are the vapor pressure of the solid and the liquid, respectively.
Because 𝑃𝑠 = 𝑃𝑠 at the triple point, 𝑑𝑃𝑠 𝑑𝑇
𝑑𝑃𝑙 𝑑𝑇=∆𝑠𝑢𝑏 𝐻
∆𝑣𝑎𝑝 𝐻
Since enthalpy is a state function, the transition between “s→g” and ““s→l→g”
having the same initial and final states should hold the same ∆𝑡𝑟𝑎 𝐻. Thus,
∆𝑠𝑢𝑏 𝐻 = ∆𝑓𝑢𝑠 𝐻 + ∆𝑣𝑎𝑝 𝐻 𝑑𝑃𝑠 𝑑𝑇
𝑑𝑃𝑙 𝑑𝑇=∆𝑠𝑢𝑏 𝐻
∆𝑣𝑎𝑝 𝐻= 1 +
∆𝑓𝑢𝑠 𝐻
∆𝑣𝑎𝑝 𝐻
which indicates the slope of the solid-gas curve is greater than that of the
liquid-gas curve at the triple point.
*the subscription “sub”
stand for sublimation.
II-3. Chemical Equilibrium- $26: Introduction -
Thermodynamics enables us to predict the equilibrium pressures or
concentrations of reaction mixtures.
In this chapter, we will derive a relation between the standard Gibbs
energy change and the equilibrium constant for a chemical reaction.
We will also learn how to predict the direction in which a chemical reaction
will proceed if we start with arbitrary concentrations (thus, not equilibrium)
of reactants and products.
II-3. Chemical Equilibrium- $26-1: Chemical equilibrium results when the Gibbs energy is a minimum with
respect to the extent of reaction -
Consider a general gas phase reaction, described by a balanced equation.
𝜈𝐴A(g) + 𝜈𝐵B(g) ⇌ 𝜈𝑌Y(g) + 𝜈𝑍Z(g)
The amount of species 𝑖 is 𝑛𝑖 [mol]. The Gibbs energy for this multi-component
system is a function of 𝑇, 𝑃, 𝑛𝐴, 𝑛𝐵 , 𝑛𝑌 𝑎𝑛𝑑 𝑛𝑍, then the total derivative is:
dG =𝜕𝐺
𝜕𝑇𝑃,𝑛𝐴,𝑛𝐵,𝑛𝑌,𝑛𝑍
𝑑𝑇 +𝜕𝐺
𝜕𝑃𝑇,𝑛𝐴,𝑛𝐵,𝑛𝑌,𝑛𝑍
𝑑𝑃 +𝜕𝐺
𝜕𝑛𝐴 𝑇,𝑃,𝑛𝐵,𝑛𝑌,𝑛𝑍
𝑑𝑛𝐴
+𝜕𝐺
𝜕𝑛𝐵 𝑇,𝑃,𝑛𝐴,𝑛𝑌,𝑛𝑍
𝑑𝑛𝐵 +𝜕𝐺
𝜕𝑛𝑌 𝑇,𝑃,𝑛𝐴,𝑛𝐵,𝑛𝑍
𝑑𝑛𝑌 +𝜕𝐺
𝜕𝑛𝑍 𝑇,𝑃,𝑛𝐴,𝑛𝐵,𝑛𝑌
𝑑𝑛𝑍
Using some equations derived around the Maxwell relation derivation (of 𝐺):
𝑑𝐺 = −𝑆𝑑𝑇 + 𝑉𝑑𝑃 + 𝜇𝐴𝑑𝑛𝐴 + 𝜇𝐵𝑑𝑛𝐵 + 𝜇𝑌𝑑𝑛𝑌 + 𝜇𝑍𝑑𝑛𝑍
𝜇𝐴 =𝜕𝐺
𝜕𝑛𝐴 𝑇,𝑃,𝑛𝐵,𝑛𝑌,𝑛𝑍
, 𝑒𝑡𝑐
If the reaction takes place in constant T and P,
dG = 𝜇𝐴𝑑𝑛𝐴 + 𝜇𝐵𝑑𝑛𝐵 + 𝜇𝑌𝑑𝑛𝑌 + 𝜇𝑍𝑑𝑛𝑍 (constant T and P)
II-3. Chemical Equilibrium- $26-1: Chemical equilibrium results when the Gibbs energy is a minimum with
respect to the extent of reaction -
Consider a general gas phase reaction, described by a balanced equation.
𝜈𝐴A(g) + 𝜈𝐵B(g) ⇌ 𝜈𝑌Y(g) + 𝜈𝑍Z(g)
We define a quantity 𝜉, called as the “extent of reaction”. Here 𝑛𝑖0 is the
initial number of moles for species 𝑖, then :
𝑛𝐴 = 𝑛𝐴0 − 𝜈𝐴𝜉 𝑛𝐵 = 𝑛𝐵0 − 𝜈𝐵𝜉
𝑛𝑌 = 𝑛𝑌0 − 𝜈𝑌𝜉 𝑛𝑍 = 𝑛𝑍0 − 𝜈𝑍𝜉
(reactants)
(products)
In this case, 𝜉 has units of moles. Then, the variations of 𝑛𝑖 is:
𝑑𝑛𝐴 = −𝜈𝐴𝑑𝜉 𝑑𝑛𝐵 = 𝜈𝐵𝑑𝜉
𝑑𝑛𝑌 = −𝜈𝑌𝑑𝜉 𝑑𝑛𝑍 = 𝜈𝑍𝑑𝜉
(reactants)
(products)
which means that as the reaction (left to right) proceeds, the reactants
decrease and the products increase according to the stoichiometry.
Using these equations:
dG = 𝜇𝐴𝑑𝑛𝐴 + 𝜇𝐵𝑑𝑛𝐵 + 𝜇𝑌𝑑𝑛𝑌 + 𝜇𝑍𝑑𝑛𝑍= −𝜈𝐴𝜇𝐴 − 𝜈𝐵𝜇𝐵 + 𝜈𝑌𝜇𝑌 + 𝜈𝑍𝜇𝑍 𝑑𝜉 (constant T and P)
II-3. Chemical Equilibrium- $26-1: Chemical equilibrium results when the Gibbs energy is a minimum with
respect to the extent of reaction -
Consider a general gas phase reaction, described by a balanced equation.
𝜈𝐴A(g) + 𝜈𝐵B(g) ⇌ 𝜈𝑌Y(g) + 𝜈𝑍Z(g)
dG = −𝜈𝐴𝜇𝐴 − 𝜈𝐵𝜇𝐵 + 𝜈𝑌𝜇𝑌 + 𝜈𝑍𝜇𝑍 𝑑𝜉 (constant T and P)
𝜕𝐺
𝜕𝜉𝑇,𝑃
= 𝜈𝑌𝜇𝑌 + 𝜈𝑍𝜇𝑍 − 𝜈𝐴𝜇𝐴 − 𝜈𝐵𝜇𝐵
Here, we define 𝜕𝐺
𝜕𝜉 𝑇,𝑃= ∆𝑟𝐺, which is the change in Gibbs energy
when the extent of reaction changes by one mole, and its unit is 𝐽 𝑚𝑜𝑙−1.
Assuming each species behaves as ideal gas, as the pressure dependence
of chemical potential is written as 𝜇𝑗 𝑇, 𝑃 = 𝜇°𝑗 𝑇 + 𝑅𝑇 ln 𝑃𝑗 𝑃° , then:
∆𝑟𝐺 = ∆𝑟𝐺° + 𝑅𝑇 ln𝑄
∆𝑟𝐺° = 𝜈𝑌𝜇°𝑌(𝑇) + 𝜈𝑍𝜇°𝑍(𝑇) − 𝜈𝐴𝜇°𝐴(𝑇) − 𝜈𝐵𝜇°𝐵(𝑇)
𝑄 = 𝑃𝑌 𝑃°
𝜈𝑌 𝑃𝑍 𝑃°𝜈𝑍
𝑃𝐴 𝑃°𝜈𝐴 𝑃𝐵 𝑃°
𝜈𝐵
𝑃° is the pressure of standard
state (1 bar) and 𝑃𝐴 is the
partial pressure of species A.
II-3. Chemical Equilibrium- $26-1: Chemical equilibrium results when the Gibbs energy is a minimum with
respect to the extent of reaction -
Consider a general gas phase reaction, described by a balanced equation.
𝜈𝐴A(g) + 𝜈𝐵B(g) ⇌ 𝜈𝑌Y(g) + 𝜈𝑍Z(g)
(ideal gas, constant T and P)∆𝑟𝐺 = ∆𝑟𝐺° + 𝑅𝑇 ln𝑄
∆𝑟𝐺° = 𝜈𝑌𝜇°𝑌 𝑇 + 𝜈𝑍𝜇°𝑍 𝑇−𝜈𝐴𝜇°𝐴(𝑇) − 𝜈𝐵𝜇°𝐵(𝑇)
𝑄 = 𝑃𝑌 𝑃°
𝜈𝑌 𝑃𝑍 𝑃°𝜈𝑍
𝑃𝐴 𝑃°𝜈𝐴 𝑃𝐵 𝑃°
𝜈𝐵
Here, the quantity ∆𝑟𝐺° is the change in standard Gibbs energy for the
reaction between unmixed reactants to form unmixed products. All
species in their standard states at 𝑇 and 𝑃°. Note that 𝑃° = 1 bar.
When the reaction system is equilibrium, the Gibbs energy must
minimum with respect to any change from the equilibrium state, thus 𝜕𝐺
𝜕𝜉 𝑇,𝑃= ∆𝑟𝐺 = ∆𝑟𝐺° + 𝑅𝑇 ln𝑄𝑒𝑞 = 0 at an equilibrium state. Thus:
∆𝑟𝐺° = −𝑅𝑇 ln 𝑃𝑌𝜈𝑌𝑃𝑍
𝜈𝑍
𝑃𝐴𝜈𝐴𝑃𝐵
𝜈𝐵𝑒𝑞
= −𝑅𝑇 ln𝐾𝑃(𝑇)
II-3. Chemical Equilibrium- $26-1: Chemical equilibrium results when the Gibbs energy is a minimum with
respect to the extent of reaction -
Consider a general gas phase reaction, described by a balanced equation.
𝜈𝐴A(g) + 𝜈𝐵B(g) ⇌ 𝜈𝑌Y(g) + 𝜈𝑍Z(g)
∆𝑟𝐺° = −𝑅𝑇 ln𝐾𝑃(𝑇)
𝐾𝑃 𝑇 = 𝑄𝑒𝑞 = 𝑃𝑌 𝑃°
𝜈𝑌 𝑃𝑍 𝑃°𝜈𝑍
𝑃𝐴 𝑃°𝜈𝐴 𝑃𝐵 𝑃°
𝜈𝐵𝑒𝑞
= 𝑃𝑌𝜈𝑌𝑃𝑍
𝜈𝑍
𝑃𝐴𝜈𝐴𝑃𝐵
𝜈𝐵𝑒𝑞
*the subscript eq emphasizes that the
partial pressures are in an equilibrium.
𝐾𝑃 𝑇 is called as equilibrium constant. Be sure that 𝐾𝑃 𝑇 has no unit.
As seen in the definition, this constant is defined after the target equation
is given. For example, if the 𝜈𝐴 in the equation is changed (even keeping
the same meaning of reaction, like 2𝜈𝐴A(g) + 2𝜈𝐵B(g) ⇌ 2𝜈𝑌Y(g) +
2𝜈𝑍Z(g) ), 𝐾𝑃 𝑇 value is changed.
𝑃° = 1 𝑏𝑎𝑟
II-3. Chemical Equilibrium- $26-1: Chemical equilibrium results when the Gibbs energy is a minimum with
respect to the extent of reaction -
(Example-1a) For reaction “3 H2(g) + N2(g) ⇌ 2 NH3(g)”, the
equilibrium pressures are given as 𝑃𝐻2, 𝑃𝑁2, and 𝑃𝑁𝐻3 .
𝐾𝑃, 𝑇 = 𝑃𝑁𝐻32
𝑃𝐻23 𝑃𝑁2 𝑒𝑞
*Be sure that these pressures are
pressures at equilibrium, as in the
definition of equilibrium constant.
(Example-1b) For reaction “3/2 H2(g) + ½ N2(g) ⇌ NH3(g)”, the
equilibrium pressures are given as 𝑃𝐻2, 𝑃𝑁2, and 𝑃𝑁𝐻3 .
𝐾𝑃 𝑇 = 𝑃𝑁𝐻3
𝑃𝐻23/2𝑃𝑁21/2
𝑒𝑞
≠ 𝐾𝑃 𝑇 𝑜𝑓 𝑒𝑥𝑎𝑚𝑝𝑙𝑒 − 1𝑎
Although the reactions themselves are identical, the equilibrium
constants are not the same, because the equilibrium constant
depends on the expression of chemical reaction equation.
II-3. Chemical Equilibrium- $26-2: An equilibrium constant is a function of temperature only -
Consider a general gas phase reaction, described by a balanced equation.
PCl5(g) ⇌ PCl3(g) + Cl2(g)
The equilibrium-constant expression for this reaction is:
𝐾𝑃, 𝑇 = 𝑃𝑃𝐶𝑙3𝑃𝐶𝑙2
𝑃𝑃𝐶𝑙5 𝑒𝑞
Suppose we have 1 mol of PCl5 (g) and no PCl3 or Cl2 at the beginning.
When the reaction occurs to an extent 𝜉, PCl5: 1 mol → (1- 𝜉) mol
PCl3: 0 mol → 𝜉 mol, Cl2: 0 mol → 𝜉 mol
Total: 1 mol → (1+ 𝜉) mol
If 𝜉𝑒𝑞 is the extent of reaction at equilibrium, then the partial pressures are:
𝑃𝑃𝐶𝑙3 = 𝑃𝐶𝑙2 = 𝜉𝑒𝑞𝑃 1 + 𝜉𝑒𝑞 , 𝑃𝑃𝐶𝑙5 = 1 − 𝜉𝑒𝑞 𝑃 1 + 𝜉𝑒𝑞
where 𝑃 is the total pressure. Then, the equilibrium constant is:
𝐾𝑃, 𝑇 = 𝜉𝑒𝑞
2
1 − 𝜉𝑒𝑞2 𝑃
II-3. Chemical Equilibrium- $26-2: An equilibrium constant is a function of temperature only -
Consider a general gas phase reaction, described by a balanced equation.
PCl5(g) ⇌ PCl3(g) + Cl2(g)
𝐾𝑃 𝑇 = 𝜉𝑒𝑞
2
1 − 𝜉𝑒𝑞2 𝑃
PCl5: 1 mol → (1- 𝜉) mol
PCl3 , Cl2: 0 mol → 𝜉 mol,
Total: 1 mol → (1+ 𝜉) mol
𝐾𝑃 𝑇 only depends on 𝑇, but
not 𝑃. So, if 𝑃 (total pressure)
is changed, 𝜉𝑒𝑞 must be
changed so that 𝐾𝑃 𝑇 is kept
constant. For example, 𝐾𝑃 𝑇of this reaction is 5.4 (no unit)
at 200ºC.
II-3. Chemical Equilibrium- $26-2: An equilibrium constant is a function of temperature only -
Consider a general gas phase reaction, described by a balanced equation.
𝜈𝐴A(g) + 𝜈𝐵B(g) ⇌ 𝜈𝑌Y(g) + 𝜈𝑍Z(g)
So far, we express the equilibrium constant regarding pressure. We can
also express the equilibrium constant in terms of concentrations, etc, by
using the ideal-gas relation “𝑃 = 𝑐𝑅𝑇”, where 𝑐 = 𝑛 𝑉 is the concentration:
∆𝑟𝐺° = −𝑅𝑇 ln𝐾𝑃(𝑇)
𝐾𝑃 𝑇 = 𝑄𝑒𝑞 = 𝑃𝑌 𝑃°
𝜈𝑌 𝑃𝑍 𝑃°𝜈𝑍
𝑃𝐴 𝑃°𝜈𝐴 𝑃𝐵 𝑃°
𝜈𝐵𝑒𝑞
=𝐶𝑌𝜈𝑌𝐶𝑍
𝜈𝑍
𝐶𝐴𝜈𝐴𝐶𝐵
𝜈𝐵𝑒𝑞
𝑅𝑇
𝑃°
𝜈𝑌+𝜈𝑍−𝜈𝐴−𝜈𝐵
*As the same with 𝐾𝑃,
𝐾𝑐 has also no unit.
Here, we consider some standard concentration 𝑐° (like 𝑃° ), often taken to
be “1 mol L-1”. Then:
𝐾𝑃 𝑇 = 𝐾𝐶 𝑇𝑐°𝑅𝑇
𝑃°
𝜈𝑌+𝜈𝑍−𝜈𝐴−𝜈𝐵
𝐾𝑐 𝑇 = 𝑐𝑌 𝑐°
𝜈𝑌 𝑐𝑍 𝑐°𝜈𝑍
𝑐𝐴 𝑐°𝜈𝐴 𝑐𝐵 𝑐°
𝜈𝐵𝑒𝑞
II-3. Chemical Equilibrium- $26-2: An equilibrium constant is a function of temperature only -
(Example-2) For reaction “NH3(g) ⇌ 3/2 H2(g) +1/2 N2(g)”, 𝐾𝑃 𝑇 =1.36 × 10−3 at 298.15 K. Determine the corresponding 𝐾𝐶 𝑇 .
𝐾𝑃 𝑇 = 𝐾𝐶 𝑇𝑐°𝑅𝑇
𝑃°
3/2+1/2−1
= 𝐾𝐶 𝑇𝑐°𝑅𝑇
𝑃°
1
𝐾𝐶 𝑇 = 𝐾𝑃 𝑇𝑐°𝑅𝑇
𝑃°
−1
= 1.36 × 10−3 ×1 𝑚𝑜𝑙 𝐿−1 × 0.0831 𝐿 𝑏𝑎𝑟 𝑚𝑜𝑙−1𝐾−1 × 298.15 𝐾
1 𝑏𝑎𝑟
−1
= 5.49× 10−5
II-3. Chemical Equilibrium- $26-3: Standard Gibbs energies of formation can be used to calculate
equilibrium constants -
∆𝑟𝐺° = 𝜈𝑌𝜇°𝑌 𝑇 + 𝜈𝑍𝜇°𝑍 𝑇 − 𝜈𝐴𝜇°𝐴(𝑇) − 𝜈𝐵𝜇°𝐵(𝑇)
∆𝑟𝐺° = −𝑅𝑇 ln𝐾𝑃(𝑇)
As already derived, 𝐾𝑃 is related to the difference between the standard
chemical potentials of the products and the reactants.
Because a chemical potential is an energy (it is the molar Gibbs energy of
a pure substance), we need to define a “zero” value.
As in the same manner with “standard molar enthalpy of formation”, we
can define “the standard molar Gibbs energy of formation” according to
∆𝑟𝐺° = ∆𝑟𝐻° − 𝑇∆𝑟𝑆°referring to standard molar entropies.
So, for “𝜈𝐴A(g) + 𝜈𝐵B(g) → 𝜈𝑌Y(g) + 𝜈𝑍Z(g)”, for example, we have
∆𝑟𝐺° = 𝜈𝑌∆𝑓𝐺° 𝑌 + 𝜈𝑍∆𝑓𝐺° 𝑍 − 𝜈𝐴∆𝑓𝐺° 𝐴 − 𝜈𝐵∆𝑓𝐺°[𝐵]
where ∆𝑓𝐺° 𝑌 is the standard molar Gibbs energy of formation for
substance 𝑌.
II-3. Chemical Equilibrium- $26-3: Standard Gibbs energies of formation can be used to calculate
equilibrium constants -
∆𝑟𝐺° = 𝜈𝑌𝜇°𝑌 𝑇 + 𝜈𝑍𝜇°𝑍 𝑇 − 𝜈𝐴𝜇°𝐴(𝑇) − 𝜈𝐵𝜇°𝐵(𝑇)
∆𝑟𝐺° = −𝑅𝑇 ln𝐾𝑃(𝑇)
As already derived, 𝐾𝑃 is related to the difference between the standard
chemical potentials of the products and the reactants.
Because a chemical potential is an energy (it is the molar Gibbs energy of
a pure substance), we need to define a “zero” value.
As in the same manner with “standard molar enthalpy of formation”, we
can define “the standard molar Gibbs energy of formation” according to
∆𝑟𝐺° = ∆𝑟𝐻° − 𝑇∆𝑟𝑆°referring to standard molar entropies.
So, for “𝜈𝐴A(g) + 𝜈𝐵B(g) → 𝜈𝑌Y(g) + 𝜈𝑍Z(g)”, for example, we have
∆𝑟𝐺° = 𝜈𝑌∆𝑓𝐺° 𝑌 + 𝜈𝑍∆𝑓𝐺° 𝑍 − 𝜈𝐴∆𝑓𝐺° 𝐴 − 𝜈𝐵∆𝑓𝐺°[𝐵]
where ∆𝑓𝐺° 𝑌 is the standard molar Gibbs energy of formation for
substance 𝑌.
II-3. Chemical Equilibrium- Appendix) how to calculate ∆𝒇𝑮° -
We can find thermodynamic database, usually the one at 298.15 K (but not
necessarily and not limited to):
∆𝑓𝐻°: Standard molar enthalpy (heat) of formation (kJ mol-1)
∆𝑓𝐺°: Standard molar Gibbs energy of formation (kJ mol-1)
𝑆°: Standard molar entropy at 298.15 K in (J mol-1 K-1)
𝐶𝑃: Molar heat capacity at constant pressure (J mol-1 K-1)
*As definition, the value is for 1 bar.
If the Gibbs energy is not available, you can calculate from enthalpy
and entropy. For example, for H2O(g) formation: "𝐻2 +1
2𝑂2 → 𝐻2𝑂“
∆𝑓𝐺° T K 𝐻2𝑂 g
= ∆𝑓𝐻° T K 𝐻2𝑂 g − ∆𝑓𝐻° T K 𝐻2 g −1
2∆𝑓𝐻° T K 𝑂2 g
− T 𝑆° T K 𝐻2𝑂 g − 𝑆° T K 𝐻2 g −1
2𝑆° T K 𝑂2 g
By this way, as the same with the standard molar enthalpy of formation,
pure elemental substances that appear as the equilibrium phase at the
temperature have ∆𝑓𝐺° T K = 0, and the standard molar Gibbs energies
of other chemicals are aligned to them.
II-3. Chemical Equilibrium- Appendix) how to calculate ∆𝒇𝑮° -
In addition, temperature dependence of ∆𝑓𝐺° T K can be evaluated
using 𝐶𝑃 (for constant pressure process) as:
𝐻° 𝑇2𝐾 = 𝐻° 𝑇1𝐾 + 𝑇1
𝑇2
𝐶𝑃𝑑𝑇
𝑆° 𝑇2𝐾 = 𝑆° 𝑇1𝐾 + 𝑇1
𝑇2 𝐶𝑃𝑇𝑑𝑇
𝐺° 𝑇2𝐾 = 𝐻° 𝑇2𝐾 − 𝑇2𝑆° 𝑇2𝐾
where 𝐻°, 𝑆° and 𝐺° are molar enthalpy, molar entropy, and molar Gibbs
energy. For evaluation of the standard Gibbs energy of formation, they
can be replaced with ∆𝑓𝐻°, 𝑆° and ∆𝑓𝐺°.
𝐶𝑃 is molar heat capacity at constant pressure. If 𝑇2 is enough close to
𝑇1, 𝐶𝑃 can be regarded as a constant in most case.
II-3. Chemical Equilibrium- Appendix) how to calculate ∆𝒇𝑮° -
y = -0.1323x + 10.192
(liquid)
y = -0.2483x + 49.356
(gas)
-50
-45
-40
-35
-30
-25
-20
300 320 340 360 380
Mo
lar
Gib
bs
ener
gy /
kJ
mo
l-1
Temperature / K
y = 0.2681x + 48.961
(liquid)
y = 0.149x + 193.83
(gas)
0
50
100
150
200
250
300
300 320 340 360 380
Mo
lar
entr
op
y /
J m
ol-1
Temperature / K
y = 0.0851x - 16.241
(liquid)
y = 0.0544x + 29.463
(gas)
0
10
20
30
40
50
60
300 320 340 360 380Mo
lar
enth
alp
y /
kJ
mo
l-1
Temperature / K
Boiling pointComparison among 𝐻° (a molar
enthalpy of formation), 𝑆° (a molar
entropy) and 𝐺° (a molar Gibbs
energy of formation) for methanol
at 1 atm, around the boiling point.
Note that 𝐺° is a continuous
function, but 𝐻° and 𝑆° are not.
𝐻°𝑆°
𝐺°
II-3. Chemical Equilibrium- $26-3: Standard Gibbs energies of formation can be used to calculate
equilibrium constants -
∆𝑟𝐺° = 3 2 ∆𝑓𝐺° 𝐻2(𝑔) + 1 2 ∆𝑓𝐺° 𝑁2(𝑔) − 1 ∆𝑓𝐺° 𝑁𝐻3 𝑔
= 3 2 0 + 1 2 0 − 1 −16.637 𝑘𝐽 𝑚𝑜𝑙−1 = 16.637 𝑘𝐽 𝑚𝑜𝑙−1
(Example-3) Using the standard molar Gibbs energies of formation, calculate
∆𝑟𝐺° and 𝐾𝑃 at 298.15 K for
NH3(g) ⇌ 3/2 H2(g) + 1/2 N2(g)
∆𝑓𝐺° 𝑁𝐻3(𝑔) = −16.637 𝑘𝐽 𝑚𝑜𝑙−1, ∆𝑓𝐺° 𝐻2(𝑔) = ∆𝑓𝐺° 𝑁2(𝑔) = 0 𝑘𝐽 𝑚𝑜𝑙
−1
ln𝐾𝑃(𝑇) = −∆𝑟𝐺°
𝑅𝑇= −
16.637 × 103 𝐽 𝑚𝑜𝑙−1
8.31 𝐽 𝐾−1 𝑚𝑜𝑙−1 298.15 𝐾= −6.60
Hence, 𝐾𝑃 𝑇 = 1.36 × 10−3 at 298.15 K.
II-3. Chemical Equilibrium- $26-4: A plot of the Gibbs energy of a reaction mixture against the extent of
reaction is a minimum at equilibrium -
Consider the thermal decomposition of N2O4 (g) at 298.15 K as an example
to treat the Gibbs energy of a reaction mixture:
N2O4 (g) ⇌ 2 NO2 (g)
𝐺 𝜉 = 1 − 𝜉 𝐺𝑁2𝑂4 + 2𝜉 𝐺𝑁𝑂2
= 1 − 𝜉 𝐺°𝑁2𝑂4 + 𝑅𝑇 ln𝑃𝑁2𝑂4 + 2𝜉 𝐺°𝑁𝑂2 + 𝑅𝑇 ln𝑃𝑁𝑂2
Suppose the initial state is “N2O4 (g) = 1 mol and NO2 (g) = 0 mol” and the
reaction brings the system to “N2O4 (g) = 1- 𝜉 mol and NO2 (g) = 2 𝜉 mol”.
Here we assume the reaction occurs at a constant pressure (𝑃𝑡𝑜𝑡𝑎𝑙 = 1 𝑏𝑎𝑟).
Then, as the molar fraction is 𝑥𝑁2𝑂4 =1−𝜉
1+𝜉for N2O4 and 𝑥𝑁𝑂2 =
2𝜉
1+𝜉for NO2:
𝑃𝑁2𝑂4 = 𝑥𝑁2𝑂4𝑃𝑡𝑜𝑡𝑎𝑙 = 𝑥𝑁2𝑂4, 𝑃𝑁𝑂2 = 𝑥𝑁𝑂2𝑃𝑡𝑜𝑡𝑎𝑙 = 𝑥𝑁𝑂2
𝐺 𝜉 = 1 − 𝜉 𝐺°𝑁2𝑂4 + 𝑅𝑇 ln1 − 𝜉
1 + 𝜉+ 2𝜉 𝐺°𝑁𝑂2 + 𝑅𝑇 ln
2𝜉
1 + 𝜉
= 1 − 𝜉 ∆𝑓𝐺°𝑁2𝑂4 + 𝑅𝑇 ln1 − 𝜉
1 + 𝜉+ 2𝜉 ∆𝑓𝐺°𝑁𝑂2 + 𝑅𝑇 ln
2𝜉
1 + 𝜉
II-3. Chemical Equilibrium- $26-4: A plot of the Gibbs energy of a reaction mixture against the extent of
reaction is a minimum at equilibrium -
Consider the thermal decomposition of N2O4 (g) at 298.15 K as an example
to treat the Gibbs energy of a reaction mixture:
N2O4 (g) ⇌ 2 NO2 (g)
𝐺 𝜉 = 1 − 𝜉 ∆𝑓𝐺°𝑁2𝑂4 + 𝑅𝑇 ln1 − 𝜉
1 + 𝜉+ 2𝜉 ∆𝑓𝐺°𝑁𝑂2 + 𝑅𝑇 ln
2𝜉
1 + 𝜉
This equation gives the Gibbs energy of the reaction mixture, 𝐺, as a
function of the extent of the reaction, 𝜉. Substituting the values for ∆𝑓𝐺°:
𝐺 𝜉 = 1 − 𝜉 97.8 𝑘𝐽 𝑚𝑜𝑙−1 + 𝑅𝑇 ln1 − 𝜉
1 + 𝜉+ 2𝜉 51.3 𝑘𝐽 𝑚𝑜𝑙−1 + 𝑅𝑇 ln
2𝜉
1 + 𝜉
II-3. Chemical Equilibrium- $26-4: A plot of the Gibbs energy of a reaction mixture against the extent of
reaction is a minimum at equilibrium -
Consider the thermal decomposition of N2O4 (g) at 298.15 K as an example
to treat the Gibbs energy of a reaction mixture:
N2O4 (g) ⇌ 2 NO2 (g)
𝐺 𝜉 = 1 − 𝜉 97.8 𝑘𝐽 𝑚𝑜𝑙−1 + 𝑅𝑇 ln1 − 𝜉
1 + 𝜉+ 2𝜉 51.3 𝑘𝐽 𝑚𝑜𝑙−1 + 𝑅𝑇 ln
2𝜉
1 + 𝜉
II-3. Chemical Equilibrium- $26-4: A plot of the Gibbs energy of a reaction mixture against the extent of
reaction is a minimum at equilibrium -
Consider the thermal decomposition of N2O4 (g) at 298.15 K as an example
to treat the Gibbs energy of a reaction mixture:
N2O4 (g) ⇌ 2 NO2 (g)
𝐺 𝜉 = 1 − 𝜉 97.8 𝑘𝐽 𝑚𝑜𝑙−1 + 𝑅𝑇 ln1 − 𝜉
1 + 𝜉+ 2𝜉 51.3 𝑘𝐽 𝑚𝑜𝑙−1 + 𝑅𝑇 ln
2𝜉
1 + 𝜉
𝐺 𝜉 is minimized at 𝜉 = 0.1892 𝑚𝑜𝑙, corresponding to the equilibrium state,
thus 𝜉 = 𝜉𝑒𝑞 = 0.1892 𝑚𝑜𝑙.
Then, the equilibrium constant is:
𝐾𝑃 =𝑃𝑁𝑂2
2
𝑃𝑁2𝑂4=
2𝜉𝑒𝑞 1 + 𝜉𝑒𝑞2
1 − 𝜉𝑒𝑞 1 + 𝜉𝑒𝑞= 0.148
We can also calculate it from ∆𝑟𝐺°:
ln𝐾𝑃 = −∆𝑟𝐺°
𝑅𝑇= −
2 ∆𝑓𝐺° 𝑁𝑂2 𝑔 − 1 ∆𝑓𝐺° 𝑁2𝑂4 𝑔
8.31 𝐽 𝐾−1 𝑚𝑜𝑙−1 298.15 𝐾= −1.908
Hence, 𝐾𝑃 = 0.148
II-3. Chemical Equilibrium- $26-4: A plot of the Gibbs energy of a reaction mixture against the extent of
reaction is a minimum at equilibrium -
Consider the thermal decomposition of N2O4 (g) at 298.15 K as an example
to treat the Gibbs energy of a reaction mixture:
N2O4 (g) ⇌ 2 NO2 (g)
𝐺 𝜉 = 1 − 𝜉 97.8 𝑘𝐽 𝑚𝑜𝑙−1 + 𝑅𝑇 ln1 − 𝜉
1 + 𝜉+ 2𝜉 51.3 𝑘𝐽 𝑚𝑜𝑙−1 + 𝑅𝑇 ln
2𝜉
1 + 𝜉
We may also differentiate this equation with respect to 𝜉, then finally obtain:
𝜕𝐺
𝜕𝜉𝑇,𝑃
= ∆𝑟𝐺° + 𝑅𝑇 ln𝑃𝑁𝑂2
2
𝑃𝑁2𝑂4
Since 𝜕𝐺
𝜕𝜉 𝑇,𝑃= 0 at equilibrium,
∆𝑟𝐺° = −𝑅𝑇 ln𝑃𝑁𝑂2
2
𝑃𝑁2𝑂4 𝑒𝑞
= −𝑅𝑇 ln𝐾𝑃
which is the same equation that we have derived several slides ago.
In addition, solving 𝜕𝐺
𝜕𝜉 𝑇,𝑃= 0 explicitly, 𝜉𝑒𝑞 = 0.1892 𝑚𝑜𝑙 is obtained as well.
II-3. Chemical Equilibrium- $26-5: The ratio of the reaction quotient to the equilibrium constant determines
the direction in which a reaction will proceed -
Consider a general reaction: 𝜈𝐴A(g) + 𝜈𝐵B(g) ⇌ 𝜈𝑌Y(g) + 𝜈𝑍Z(g)
∆𝑟𝐺 𝑇 = ∆𝑟𝐺° 𝑇 + 𝑅𝑇 ln𝑃𝑌𝜈𝑌𝑃𝑍
𝜈𝑍
𝑃𝐴𝜈𝐴𝑃𝐵
𝜈𝐵
Realize that this equation is a general equation, so pressures are not
necessarily the equilibrium pressures, but are arbitrary.
Generally, this equation gives the value of ∆𝑟𝐺 when
“A(g) of [𝜈𝐴 mol at 𝑃𝐴 bar] react with B(g) of [𝜈𝐵 mol at 𝑃𝐵 bar] to produce
Y(g) of [𝜈𝐴 mol at 𝑃𝐴 bar] and Z(g) of [𝜈𝑍 mol at 𝑃𝑍 bar] ”.
1) If all the partial pressures are equl to 1 bar, ∆𝑟𝐺 𝑇 = ∆𝑟𝐺° 𝑇 . In
other words, the Gibbs energy change will be equal to the standard
Gibbs energy change.
2) If the pressures are the equilibrium pressures, ∆𝑟𝐺 𝑇 = 0 and then
we obtain results that were derived in previous slides.
II-3. Chemical Equilibrium- $26-5: The ratio of the reaction quotient to the equilibrium constant determines
the direction in which a reaction will proceed -
Consider a general reaction: 𝜈𝐴A(g) + 𝜈𝐵B(g) ⇌ 𝜈𝑌Y(g) + 𝜈𝑍Z(g)
∆𝑟𝐺 𝑇 = ∆𝑟𝐺° 𝑇 + 𝑅𝑇 ln𝑃𝑌𝜈𝑌𝑃𝑍
𝜈𝑍
𝑃𝐴𝜈𝐴𝑃𝐵
𝜈𝐵
By introducing a quantity called the “reaction quotient” 𝑄𝑃 =𝑃𝑌𝜈𝑌𝑃𝑍
𝜈𝑍
𝑃𝐴𝜈𝐴𝑃𝐵
𝜈𝐵, and
using an equation to correlate ∆𝑟𝐺° with 𝐾𝑃, ∆𝑟𝐺° = −𝑅𝑇 ln𝐾𝑃, then
∆𝑟𝐺 𝑇 = −𝑅𝑇 ln𝐾𝑃 + 𝑅𝑇 ln𝑄𝑃 = 𝑅𝑇 ln 𝑄𝑃 𝐾𝑃
Note that this is not for the equilibrium; ∆𝑟𝐺 𝑇 = 0 at equilibrium.
With this equation,
At equilibrium, ∆𝑟𝐺 𝑇 = 0 and thus 𝑄𝑃 = 𝐾𝑃.
If 𝑄𝑃 < 𝐾𝑃, then 𝑄𝑃 must increase (because 𝐾𝑃 is a constant) as the
system proceeds toward equilibrium.
This is achieved by increasing the partial pressures of the
products and decreasing those of the reactants. Hence, the
reaction spontaneously proceeds from left to right.
If 𝑄𝑃 > 𝐾𝑃, 𝑄𝑃 decrease as the system proceeds toward equilibrium.
In the same though, the reaction is spontaneous from right to left.
II-3. Chemical Equilibrium- $26-5: The ratio of the reaction quotient to the equilibrium constant determines
the direction in which a reaction will proceed -
(Example-4: #26-5) Consider “ 2SO2(g) + O2(g) ⇌ 2SO3(g) ”. The
equilibrium constant 𝐾𝑃 = 10 at 960 K. Calculate ∆𝑟𝐺 and check in which
direction the reaction will proceed spontaneously for
2 SO2 (1.0 × 10−3 𝑏𝑎𝑟) + O2 (0.20 𝑏𝑎𝑟) ⇌ 2 SO3(1.0 × 10
−4 𝑏𝑎𝑟)
𝑄𝑃 =𝑃𝑆𝑂32
𝑃𝑆𝑂22 𝑃𝑂2
=1.0 × 10−4 2
1.0 × 10−3 2 0.20= 5.0 × 10−2
Note that the value for 𝑄𝑃 is unitless because the definition of 𝑄 is, for
example in a general form, 𝑄 = 𝑃𝑌 𝑃°
𝜈𝑌 𝑃𝑍 𝑃°𝜈𝑍
𝑃𝐴 𝑃° 𝜈𝐴 𝑃𝐵 𝑃° 𝜈𝐵where 𝑃° = 1 𝑏𝑎𝑟.
∆𝑟𝐺 𝑇 = 𝑅𝑇 ln 𝑄𝑃 𝐾𝑃 = 𝑅𝑇 ln 5.0 × 10−2 10 < 0
Hence, due to ∆𝑟𝐺 𝑇 < 0 (or due to 𝑄𝑃 < 𝐾𝑃), the reaction proceed
from left to right.
II-3. Chemical Equilibrium- $26-6: The sign of ∆𝑟𝐺 and not that of ∆𝑟𝐺° 𝑇 determines the direction of reaction
spontaneity -
It should be clearly understood that ∆𝑟𝐺° is the value of ∆𝑟𝐺 when all the
reactants and products are unmixed at partial pressures equal to 1 bar.
Namely, ∆𝑟𝐺° is the standard Gibbs energy changes. (recall that “standard”
assumes 1 bar) Hence,
If ∆𝑟𝐺° < 0, then 𝐾𝑃 > 1, meaning that the reaction will proceed from
reactants to products if all species are mixed at 1 bar (for each partial
pressure; not total pressure).
If ∆𝑟𝐺° > 0, then 𝐾𝑃 < 1, thus reaction proceeds from products to
reactants in the same condition.
So, the sing of ∆𝑟𝐺° just indicates the spontaneous reaction direction
for the condition of 𝑃𝑝𝑎𝑟𝑡𝑖𝑎𝑙 = 1 bar (for all species of both reactants
and products); not necessarily for all conditions.
II-3. Chemical Equilibrium- $26-6: The sign of ∆𝑟𝐺 and not that of ∆𝑟𝐺° 𝑇 determines the direction of reaction
spontaneity -
(Example-5) Consider “ N2O4 (g) ⇌ 2 NO2 (g) ”,
for which ∆𝑟𝐺° = 4.729 kJ 𝑚𝑜𝑙−1 and 𝐾𝑃 = 0.148 at 298.15 K.
First, make sure that this ∆𝑟𝐺° > 0 does NOT mean “no N2O4 dissociate at
298.15 K when we place some of N2O4 in a reaction vessel”.
To correctly consider, we have to calculate ∆𝑟𝐺 as:
∆𝑟𝐺 = ∆𝑟𝐺° + 𝑅𝑇 ln𝑄𝑃 = 4.729 𝑘𝐽 𝑚𝑜𝑙−1 + 2.479 𝑘𝐽 𝑚𝑜𝑙−1 ln
𝑃𝑁𝑂22
𝑃𝑁2𝑂4
If we just ”place some of N2O4 in a reaction vessel”, ln 𝑃𝑁𝑂22 𝑃𝑁2𝑂4 has a
large negative value, thus ∆𝑟𝐺 < 0. Accordingly, some N2O4 dissociate.
The equilibrium state is achieved by the condition ∆𝑟𝐺 = 0 (note ∆𝑟𝐺° is a
constant while ∆𝑟𝐺 changes depending on 𝑃𝑁𝑂2 and 𝑃𝑁2𝑂4 ), at which point
𝑄𝑃 = 𝐾𝑃. Until this point is achieved, 𝑃𝑁𝑂2 increases and 𝑃𝑁2𝑂4 decreases.
II-3. Chemical Equilibrium- $26-6: The sign of ∆𝑟𝐺 and not that of ∆𝑟𝐺° 𝑇 determines the direction of reaction
spontaneity -
(Example-6) Consider “ H2 (g) + ½ O2 (g) ⇌ H2O (l) ”,
for which ∆𝑟𝐺° = −237 kJ 𝑚𝑜𝑙−1 at 298.15 K.
In this case, ∆𝑟𝐺° has a large negative value, thus basically H2O (l) is
much more stable than the reactants at 298.15 K. However, a mixture of
H2 (g) and O2 (g) remains unchanged.
If a spark or a catalyst is introduced, then the reaction occurs explosively.
The “no” of the thermodynamics is emphatic: If thermodynamics
insists that a certain process will not occur spontaneously, then it will
not occur.
On the other hand, the “yes” is actually “maybe”. The fact that a
process will occur spontaneously does not imply that it will occur at a
detectable rate.
Diamond remains its form, although a graphite is more favorable
energetically, is another example.
The speed of reaction can be analyzed in the framework
of rate theory. (next topic)
II-3. Chemical Equilibrium- $26-7: The variation of an equilibrium constant with temperature is given by the
Van’t Hoff Equation -
We utilize the Gibbs-Helmohltz equation:
Substitute ∆𝑟𝐺° = −𝑅𝑇 ln𝐾𝑃 (note this is a definition of ∆𝑟𝐺°, not a condition
achieved at equilibrium state) for this equation:
𝜕∆𝐺° 𝑇
𝜕𝑇𝑃
=∆𝐻°
𝑇2
𝜕 ln𝐾𝑃(𝑇)
𝜕𝑇𝑃
=𝑑 ln𝐾𝑃(𝑇)
𝑑𝑇=∆𝑟𝐻°
𝑅𝑇2
This means that
If ∆𝑟𝐻° > 0 (endothermic reaction), 𝐾𝑃(𝑇) increases with temperature.
If ∆𝑟𝐻° < 0 (exothermic reaction), 𝐾𝑃(𝑇) decreases with temperature.
Integrate the equation:ln𝐾𝑃(𝑇2)
𝐾𝑃(𝑇1)=
𝑇1
𝑇2 ∆𝑟𝐻°(𝑇)
𝑅𝑇2𝑑𝑇
If the temperature range is small enough to consider ∆𝑟𝐻° constant:
ln𝐾𝑃(𝑇2)
𝐾𝑃(𝑇1)= −
∆𝑟𝐻°
𝑅
1
𝑇2−1
𝑇1
II-3. Chemical Equilibrium- $26-7: The variation of an equilibrium constant with temperature is given by the
Van’t Hoff Equation -
(Example-7) Consider “ PCl3 (g) + Cl2 (g) ⇌ PCl2 (g) ”.
Given that ∆𝑟𝐻° has an average value of -69.8 kJ mol-1 over 500-700 K and
𝐾𝑃 is 0.0408 at 500K, evaluate 𝐾𝑃 at 700K.
As ∆𝑟𝐻° can be assumed as a constant over the concerned temperatures,
ln𝐾𝑃(𝑇2)
𝐾𝑃(𝑇1)= −
∆𝑟𝐻°
𝑅
1
𝑇2−1
𝑇1
Substituting provided values gives:
ln𝐾𝑃(700 𝐾)
𝐾𝑃(500 𝐾)= ln
𝐾𝑃(700 𝐾)
0.0408= −
−69.8 × 103
𝑅
1
700−1
500
𝐾𝑃 700 𝐾 = 3.36 × 10−4
Note that since the reaction is exothermic, 𝐾𝑃 700 𝐾 is less than 𝐾𝑃 700 𝐾 ;
namely less product (PCl2) at higher temperatures as 𝐾𝑃 = 𝑃𝑃𝐶𝑙2
𝑃𝑃𝐶𝑙3𝑃𝐶𝑙2.
II-3. Chemical Equilibrium- $26-7: The variation of an equilibrium constant with temperature is given by the
Van’t Hoff Equation -
ln𝐾𝑃(𝑇2)
𝐾𝑃(𝑇1)=
𝑇1
𝑇2 ∆𝑟𝐻°(𝑇)
𝑅𝑇2𝑑𝑇
Finally, we consider the temperature dependence of ∆𝑟𝐻° constant in:
This equation can write down as:
ln𝐾𝑃(𝑇) = ln𝐾𝑃(𝑇1) + 𝑇1
𝑇 ∆𝑟𝐻°(𝑇′)
𝑅𝑇′2𝑑𝑇′
For example, as we learned, the temperature dependence of ∆𝑟𝐻° may
be written as:
∆𝑟𝐻° 𝑇2 = ∆𝑟𝐻° 𝑇1 + 𝑇1
𝑇2
∆𝐶°𝑃 𝑇 𝑑𝑇
or expanded in respect to temperature as:
∆𝑟𝐻° 𝑇 = 𝛼 + 𝛽𝑇 + 𝛾𝑇2 + 𝛿𝑇3 +⋯
In this latter case, ln𝐾𝑃(𝑇) becomes:
ln𝐾𝑃(𝑇) = −𝛼
𝑅𝑇+𝛽
𝑅ln 𝑇 +
𝛾
𝑅𝑇 +
𝛿
2𝑅𝑇2 + (𝑖𝑛𝑡𝑒𝑔𝑟𝑎𝑡𝑖𝑜𝑛_𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡)…