Thermodynamic properties of pure phases
B. Mishra
PART – 1 Thermodynamic properties of pure phases
Introduction
This introductory section deals with some of the integral properties of chemical thermodynamics along with the most pertinent expressions and finally derivation of equation of state (EOS) for pure phases involving G, H, S, V and Cp at equilibrium P-T. System:
The study of any special branch of physics starts with separation of a restricted
region of space or finite portion of matter separated from the rest of the universe by a distinct boundary. The portion that is set aside (in the imagination) and on which the attention is focused is called the system, and everything outside the system, which has a direct bearing on its behaviour, is known as the surroundings. Hence, system is a restricted region of space or a finite portion of matter separated from the rest of the universe by a distinct boundary. Furthermore, there are open-, closed- and isolated systems, depending on transfer of energy (heat) and matter (mass). While open systems transfer both, the closed ones only transfer energy not mass. Isolated systems on the other hand do not transfer either energy or mass.
Phase:
A phase is defined as a system or a part of the system composed of any
number of chemical constituents satisfying the requirements that, (a) it is chemically homogenous, and (b) it has a definite boundary.
Components: The minimum number of chemical species necessary to explain the chemical variability of a system (system component) or phase (phase component). Thermodynamic coordinates of a system
In thermodynamics, the attention is directed to the interior of a system, and the need to evolve a set of parameters that have bearing on the internal state of a system. These parameters (macroscopic or microscopic) are then related through equation of state that,
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Thermodynamic properties of pure phases
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describes the equilibrium condition of a system. These parameters are called thermodynamic coordinates. Say for e.g., a system consisting of one mole of gas and at a low pressure has the equation of state PV = RT
(1a)
where P, V and T are thermodynamic coordinates.
Again, at high pressures, the equation of state is represented as
a ⎞ ⎛ ⎜ P + 2 ⎟ (V − b ) = RT V ⎠ ⎝
(1b)
Eqns. (1a) and (1b) represent equations of state of one mole of gas respectively at low and high pressure but with the same thermodynamic coordinates (P, V, T). Geochemical thermodynamics commonly involve macroscopic coordinates, which in contrast to microscopic coordinates have the following common characteristics. (a) they involve no special assumption concerning crystal structure. (b) only a few coordinates are needed. (c) Macro-coordinates are suggested more or less directly by our sense of perception. (d) Macro-coordinates can in general be directly measured.
Equation of state: At equilibrium, there exists an equation of equilibrium that connects the thermodynamic coordinates and makes one of its independent. Such an equation is called as an equation of state (EOS).
a ⎞ ⎛ For e.g., PV = RT or ⎜ P + 2 ⎟ (V − b ) = RT (for one mole) are EOS in P, V, T. Once V ⎠ ⎝ any two coordinates (say T, V) are chosen, the value of the other (P) is determined by nature. State variables are those that describe the microscopic state of the system. State variables may be extensive or intensive. While the extensive ones are those that depend on the amount of material e.g., V, H, S and G the intensive variables include those, which do not depend on the amount of the material. E.g. P, T, μ, X and R.I.
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For a state function z and its differential dz, the following statements are valid: (i) the differential dz is mathematically exact. Any expression that may be written as M(x,y)dx + N(x,y)dy is an exact differential if
there exists a function z(x,y) that is written as
⎛ ∂z ⎞ ⎛ ∂z ⎞ dz = ⎜ ⎟ dx + ⎜⎜ ⎟⎟ dy = Mdx + Ndy ⎝ ∂x ⎠ y ⎝ ∂y ⎠ x
⇒ if dz is an exact differential, then ∂M ∂N = , which is equivalent to ∂x ∂y
⎛ ∂M ⎜⎜ ⎝ ∂y
⎞ ⎛ ∂N ⎞ ⎟⎟ = ⎜ ⎟ ⎠ x ⎝ ∂x ⎠ y
The rule of reciprocity relation or cross differentiation is a necessary and sufficient test of exactness. i.e., if dz is exact then z is a continuous function of x and y and z is uniquely defined for every value of x and y, then
⎛ ⎛ ∂z ⎞ ⎜ ∂⎜ ⎟ ⎜ ⎝ ∂x ⎠ y ⇒⎜ ∂y ⎜ ⎜ ⎝
⎛ ⎛ ∂z ⎞ ⎞ ⎜ ∂⎜⎜ ⎟⎟ ⎟ ⎜ ⎝ ∂y ⎠ x ⎟ = ⎜ ⎟ ⎜ ∂x ⎟ ⎟ ⎜ ⎠x ⎝
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠y
For example, if a function V = V (T, P) is V = RT/P, where R is a constant term. Then
⎛ ∂V ⎞ ⎛ ∂V ⎞ ⎛R⎞ ⎛ − RT ⎞ dV = ⎜ ⎟ dT + ⎜ ⎟ dP = ⎜ ⎟dT + ⎜ 2 ⎟dP ⎝ ∂T ⎠ p ⎝ ∂P ⎠ t ⎝P⎠ ⎝ P ⎠
We can check the exactness by cross differentiation in the following manner
⎛ ⎛ R⎞⎞ ⎛ ⎛ − RT ⎞ ⎞ ⎜ ∂⎜ ⎟ ⎟ ⎜ ∂⎜ 2 ⎟ ⎟ ⎜ ⎝ P ⎠⎟ = ⎜ ⎝ P⎠⎟ = − R ⎜ ∂P ⎟ ⎜ ∂T ⎟ P2 ⎜ ⎟ ⎜ ⎟ ⎠T ⎝ ⎠P ⎝ 3
Thermodynamic properties of pure phases
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(ii) For any process Z 1 → Z 2 , the change ΔZ is independent of the path chosen to effect that change. Thus, ΔZ can be calculated from the initial and final states of the system 2
only, i.e., ∫ dZ = Z 2 − Z 1 = ΔZ 1
(iii) In a close cycle, the net change in Z is always zero. Hence, ∫ dZ = 0
The state of a system
Based on the changes in state variables, we can classify systems as follows. The equilibrium state is the one in which state variables do not change with space and time, whereas in the steady state, state variables change in space, but not in time. Additionally, we come across the transient states, where the state variables change both in space and time. The Equilibrium state can be of the following types. •
Stable equilibrium: is one characterized by the lowest free energy
•
Metastable equilibrium: is an equilibrium state that is time-space invariant, but not in the lowest free energy configuration, and which may convert to stable equilibrium if kinetic barriers are removed.
•
Partial equilibrium: is the state of the system in which some variables are timespace invariant, but other variables are not.
From the first law of thermodynamics, the internal energy (U) of a system is related to work done (W) and heat involved (Q) through the relation
dU = dW + dQ (dQ = dU – dW)
(1)
Mathematically, the first law is related two important properties of the system: heat and work. From classical mechanics, the work done by applying a force to a body is simply defined as,
W=
∫ F.dx , where F is the force vector and dx is the differential displacement 4
Thermodynamic properties of pure phases
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Work done on the system = Force × dis tan ce = Force × dx = P. A × dx = P( Adx) = Pdv ∴ dW = − PdV
(the negative sign stems from the opposing directions of work done and pressure exerted by the gas to oppose compression)
We can explain the first law in a different way with the help of a special kind of work relating compression or “PV work”, which is especially important in thermodynamics. Let us consider a gas held under pressure by a piston in a vertical cylinder. If the piston is released, the gas will expand a distance dx against the external pressure, which is equivalent to the atmospheric force on the piston divided by the cross sectional area ‘A’ of the cylinder. For a differential amount of work done, we can write dW = –Fdx = − Pex dAdx = − Pex dV
If the change is reversible, then the external pressure ( Pex ) = pressure (P) of the system so that dWrev = − PdV . If any gas leaves the system, as in the performance of work by an expanding gas, then the sign of the energy term is negative.
From the second law of thermodynamics, entropy (S) is related to heat through the relation dS ≥
dQ (For reversible and irreversible processes) T
For reversible processes, dS = dQ/T or dQ = TdS 5
(2)
Thermodynamic properties of pure phases
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Now, dU = dW + dQ
⇒ dU = dQ − PdV
(1)
Combining equations (1) & (2), we get (3)
dU = TdS – PdV
Eqn. (3) does not involve any transfer of matter between the system and the surroundings, i.e. equation for a closed system. For open system, eqn. (3) is modified to
dU = TdS − PdV + ∑ μ i dni
(4)
i
⎛ ∂G ⎞ ⎟⎟ where μ i is the chemical potential, expressed as μ i = ⎜⎜ ⎝ ∂ni ⎠ S ,V ,n j ≠ i
Since S, V and n are extensive properties, eqn. (4) can be written in the integral form U = TS – PV +
∑μ n i
i
and its differential form can be expressed as
dU = TdS + SdT – PdV – VdP +
∑ μ dn + ∑ n dμ i
i
i
i
Now inserting eqn. (4), dU = TdS –PdV + ∑ μ i dni , in the above eqn., we get
SdT –VdP + ∑ ni dμ i = 0
(5)
Eqn. (5) is termed as the Gibbs-Duhem equation that relates the intensive and extensive properties of a phase. It is evident that amongst n variables, (n – 1) can change independently. However, for a pure phase eqn. (5) is simplified to
SdT – VdP =0
(6a)
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Thermodynamic properties of pure phases
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For condensed pure phases, the enthalpy (H), Gibbs free energy (G), entropy (S) and internal energy (U) are related by the following relations.
H = U + PV and
G = U + PV – TS
(6b)
The differential of eqn. (6b) is dG = dU + PdV + VdP –TdS – SdT Substituting eqn. (3) (dU = TdS – PdV), the above eqn. reduces to
dG = –SdT + VdP
(7)
Again, since G = G (P, T) for a pure phase
⎛ ∂G ⎞ ⎛ ∂G ⎞ dG = ⎜ ⎟ dT + ⎜ ⎟ dP ⎝ ∂T ⎠ p ⎝ ∂P ⎠ T
(8)
Comparing eqns. (7) and (8) we get ⎛ ∂G ⎞ ⎜ ⎟ = −S ⎝ ∂T ⎠ P ⎛ ∂G ⎞ ⎜ ⎟ =V ⎝ ∂P ⎠T
(9)
Similarly since S = S (P, T) and H = H (P, T) ⎛ ∂S ⎞ ⎛ ∂S ⎞ dS = ⎜ ⎟ dT + ⎜ ⎟ dP ⎝ ∂T ⎠ P ⎝ ∂P ⎠ T ⎛ ∂H ⎞ ⎛ ∂H ⎞ dH = ⎜ ⎟ dT + ⎜ ⎟ dP ⎝ ∂P ⎠ T ⎝ ∂T ⎠ P
(10)
It is now important to express the differentials in eqn. (10) in terms of determinable parameters, namely Cp and V. 7
Thermodynamic properties of pure phases
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⎛ ∂S ⎞ ⎛ ∂Q ⎞ Since C P = ⎜ ⎟ ⎟ = T⎜ ⎝ ∂T ⎠ P ⎝ ∂T ⎠ P C ⎛ ∂S ⎞ ∴⎜ ⎟ = P T ⎝ ∂T ⎠ P
(11a)
⎛ ∂G ⎞ Further, by substituting for ⎜ ⎟ from eqn. (9), in the LHS of the eqn. below, we ⎝ ∂T ⎠ P
obtain ⎛ ∂H ⎞ ⎛ ∂S ⎞ ⎛ ∂G ⎞ ⎟ T −S ⎟ −⎜ ⎟ = −S = ⎜ ⎜ ⎝ ∂T ⎠ P ⎝ ∂T ⎠ P ⎝ ∂T ⎠ P ⎛ ∂H ⎞ Hence, ⎜ ⎟ = ⎝ ∂T ⎠ P
Cp ⎛ ∂S ⎞ = Cp ⎜ ⎟ T = T× T ⎝ ∂T ⎠ P
(11b)
Again, as G is a state variable and thus an exact differential, its crossed partial derivatives are equal
⎛ ∂ 2G ⎞ ⎛ ∂ 2G ⎞ ⎟⎟ ⎜⎜ ⎟⎟ = ⎜⎜ ⎝ ∂P∂T ⎠T , P ⎝ ∂T∂P ⎠ P ,T ⎛ ∂ G ⎞ ⎟⎟ ⇒ ⎜⎜ ⎝ ∂P∂T ⎠T , P 2
⎛ ∂ G ⎞ ⎜⎜ ⎟⎟ ⎝ ∂T∂P ⎠ P ,T 2
⎡ ⎛ ∂G ⎞ ⎢ ∂⎜ ∂P ⎟ ⎝ ⎠T =⎢ ⎢ ∂T ⎢ ⎣
⎤ ⎥ ∂V ⎞ ⎥ = ⎛⎜ ⎟ ⎥ ⎝ ∂T ⎠ P ⎥ ⎦P
(12a)
⎡ ⎛ ∂G ⎞ ⎤ ⎢ ∂⎜ ∂T ⎟ ⎥ ⎛ ∂S ⎞ ⎡ ∂ (− S ) ⎤ ⎝ ⎠P ⎥ = −⎜ ⎟ =⎢ =⎢ ⎥ ⎢ ∂P ⎥ ⎝ ∂P ⎠T ⎣ ∂P ⎦ T ⎥ ⎢ ⎦T ⎣ (12b)
From eqns. (12a) and (12b) ⎛ ∂S ⎞ ⎛ ∂V ⎞ −⎜ ⎟ = ⎜ ⎟ ⎝ ∂P ⎠ T ⎝ ∂T ⎠ P
(13)
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Thermodynamic properties of pure phases
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⎛ ∂H ⎞ Finally, ⎜ ⎟ can be derived as follows. ⎝ ∂P ⎠
Since, G = H – TS or, H = G + TS ⎛ ∂H ⎞ ⎛ ∂G ⎞ ⎛ ∂S ⎞ ⇒⎜ ⎟ =⎜ ⎟ + T⎜ ⎟ ⎝ ∂P ⎠ T ⎝ ∂P ⎠ T ⎝ ∂P ⎠ T ⎛ ∂H ⎞ ⎛ ∂V ⎞ ⇒⎜ ⎟ = V −T⎜ ⎟ ⎝ ∂P ⎠ T ⎝ ∂T ⎠ P
(14)
⎛ ∂S ⎞ ⎛ ∂S ⎞ Thus from eqn. (10a) replacing the differential ⎜ ⎟ eqn.(11a )and ⎜ ⎟ (eqn.13) we ⎝ ∂T ⎠ P ⎝ ∂P ⎠ T
get
⎛ ∂V ⎞ ⎛ Cp ⎞ dS = ⎜ ⎟ dP ⎟dT − ⎜ ⎝ ∂T ⎠ P ⎝ T ⎠
(15a)
⎛ ∂H ⎞ Similarly from eqn. (10b), replacing the differential ⎜ ⎟ (eqn.11b)and (eqn.14), we ⎝ ∂T ⎠ P
obtain, ⎡ ⎛ ∂V ⎞ ⎤ dH = CpdT + ⎢V − T ⎜ ⎟ ⎥ dP ⎝ ∂T ⎠ P ⎦ ⎣
(15b)
Integration of eqns. (15a) and (15b) between the standard state ( P0 , T0 ) to any desired state (P, T) leads to
⎡ ⎛ ∂V ⎞ ⎤ H P ,T = H P0T 0 + ∫ C P dT + ∫ ⎢V − T ⎜ ⎟ ⎥dP T ∂ ⎠P ⎦ ⎝ T0 P0 ⎣ T
P
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Thermodynamic properties of pure phases
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and
⎛C ⎞ ⎛ ∂V ⎞ + ∫ ⎜ P ⎟dT − ∫ ⎜ ⎟ dP ∂ T T ⎝ ⎠ ⎝ ⎠P T0 P0 T
S P ,T = S P0T0
P
∵ G P ,T = H P ,T − TS P ,T T P ⎡ ⎤ ⎡ ⎛ ∂V ⎞ ⎛ CP ⎞ ⎛ ∂V ⎞ ⎤ + ∫ C P dT + ∫ ⎢V − T ⎜ ⎟ dP ⎥ ⎟dT − ∫ ⎜ ⎟ ⎥dP − T ⎢ S P0T0 + ∫ ⎜ T ⎠ ∂T ⎠ P ⎥⎦ ⎝ ∂T ⎠ P ⎦ ⎢⎣ T0 P0 ⎣ T0 ⎝ P0 ⎝ T
GP ,T = H P T
0 0
G P ,T = H P0 ,T0 − TS P0 ,T0
P
T P ⎡T ⎛ Cp ⎞ ⎤ + ⎢ ∫ Cp.dT − T ∫ ⎜ ⎟dT ⎥ + ∫ V .dP T ⎠ ⎦⎥ P0 T0 ⎝ ⎣⎢T0
(16)
Fig.1 A typical Cp –T plot
But as seen from the above figures Cp is commonly a function of T and can be derived as a polynomial in T. Typically Cp is expressed in the form
Cp = a + bT + cT −2 + dT −0.5 + eT 2 + fT −3 where a, b, c, d, e and f are constants of Cp (Cp coefficients).
∫ (a + bT + cT
T
T
T0
T0
∴ ∫ Cp.dT =
−2
)
+ dT −0.5 + eT 2 + fT −3 .dT
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Thermodynamic properties of pure phases
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T
b e f 1 1 1 1 3 2 0.5 ⇒ ∫ Cp (T ).dT = a(T − T0 ) + (T 2 − T0 ) − c( − ) + 2d (T 0.5 − T0 ) + (T 3 − T0 ) − ( 2 − 2 ) T T0 2 3 2 T T0 T0 (17) Again,
a + bT + cT −2 + dT −0.5 + eT 2 + fT −3 ⎡ Cp ⎤ ( dT = ∫ ⎢ T ⎥⎦ ∫ T T0 ⎣ T0 T
T
).dT
T c 1 e f 1 1 1 1 1 ⎡ Cp ⎤ 2 ⇒ ∫ ⎢ ⎥ dT = a ln + b(T − T0 ) − ( 2 − 2 ) − 2d ( 0.5 − 0.5 ) + (T 2 − T0 ) − ( 3 − 3 ) T ⎦ T0 2 T 2 3 T T T0 T0 T0 T0 ⎣ T
(18) Combining eqns. (17) and (18) T ⎡T ⎛ Cp ⎞ ⎤ ⎢ ∫ Cp.dT − T ∫ ⎜ ⎟ dT ⎥ T ⎠ ⎥⎦ ⎢⎣T0 T0 ⎝
⎡ T ⎤ ⎡b ⎤ ⎡ ⎛ 1 1 ⎞ cT = ⎢ a (T − T0 ) − aT ln ⎥ + ⎢ (T 2 − T0 2 ) − bT (T − T0 ) ⎥ − ⎢ c ⎜ − ⎟ − T0 ⎦ ⎣ 2 ⎦ ⎣⎢ ⎝ T T0 ⎠ 2 ⎣ ⎡ ⎛ 1 1 0.5 0.5 ⎢ 2d (T − T0 ) + 2d .T ⎜ 0.5 − 0.5 T0 ⎝T ⎣⎢
⎛ 1 1 ⎞⎤ ⎜ 2 − 2 ⎟⎥ + T0 ⎠ ⎦⎥ ⎝T
⎞⎤ ⎡ e 3 1 ⎞ fT ⎛ 1 eT 2 1 ⎞⎤ ⎤ ⎡f ⎛ 1 3 T − T0 2 ) ⎥ − ⎢ ⎜ 2 − 2 ⎟ − ( ⎟ ⎥ + ⎢ (T − T0 ) − ⎜ 3 − 3 ⎟⎥ 2 T0 ⎠ 3 ⎝ T T0 ⎠ ⎦⎥ ⎦ ⎣⎢ 2 ⎝ T ⎠ ⎦⎥ ⎣ 3
(19a) Eqn. (19a) can be written as T ⎡T ⎛ Cp ⎞ ⎤ . Cp dT T − ⎢∫ ⎜ ⎟ dT ⎥ ∫ T ⎝ ⎠ ⎦⎥ T0 ⎣⎢T0 2 ⎡ ⎛ ⎤ b T ⎞ c ⎡ (T − T0 ) ⎤ 2 = a ⎢T ⎜1 − ln ⎟ − T0 ⎥ − ⎡⎣ (T − T0 ) ⎤⎦ − ⎢ ⎥ 2 ⎢⎣ 2T • T02 ⎥⎦ T0 ⎠ ⎣ ⎝ ⎦ 2
⎡ ⎤ ⎢ 0.5 ⎥ 2 0.5 2 ⎢ (T − T0 ) ⎥ e f ⎡ (T − T0 ) (T0 + 2T ) ⎤ 2 −2 d ⎢ ⎥ ⎥ − ⎡⎣(T − T0 ) (T + 2T0 ) ⎤⎦ − ⎢ T 0.5 T03T 2 6⎣ ⎦ ⎢ ⎥ 6 ⎢ ⎥ ⎢⎣ ⎥⎦
where T0 = 298.15K
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(19b)
Thermodynamic properties of pure phases
B. Mishra
P
Now, let us solve the last term
∫ V .dP in eqn. (16) in terms of differentiable parameters.
P0
Since volume is a state variable, for a pure phase we can write
V = V (P, T) ⎛ ∂V ⎞ ⎛ ∂V ⎞ Hence, dV = ⎜ ⎟ dT + ⎜ ⎟ dP ⎝ ∂T ⎠ p ⎝ ∂P ⎠ T
(20)
The two partial derivatives in eqn. (20) are expressed in terms of thermal expansivity
(α ) and isothermal compressibility ( β ), as follows.
α=
1 ⎛ ∂V ⎞ ⎛ ∂V ⎞ ⎟ = αVP0 ,T0 ⎜ ⎟ or , ⎜ VP0 ,T0 ⎝ ∂T ⎠ P ⎝ ∂T ⎠ P
β =−
1 ⎛ ∂V ⎞ ⎛ ∂V ⎞ ⎜ ⎟ or , ⎜ ⎟ = − βVP0 ,T0 VP0 ,T0 ⎝ ∂P ⎠ T ⎝ ∂P ⎠ T
where VP0 ,T0 refers to molar volume at the chosen standard state (P0, T0). By substituting values of the partial derivatives in eqn. (20), we get dV = VP0 ,T0 (α .dT − β .dP)
(21)
Now, both α and β can be expressed as linear function of T and P respectively, as follows.
α = α (T ) = α 0 + α 1T β = β ( P) = β 0 + β1 P
Substituting the above two equations in eqn. (21), we obtain dV = VP0 ,T0 [α (T ).dT − β ( P).dP ] ⎛ ∂α ⎞ ⎛ ∂β ⎞ Now on integrating the above differential for dV assuming ⎜ ⎟ = 0 and ⎜ ⎟ =0 ⎝ ∂P ⎠ T ⎝ ∂T ⎠ P
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Thermodynamic properties of pure phases PT
⇒
∫
PT
dV =
P0T0
∫
B. Mishra
VP0T0 ⎡⎣(α 0 + α1T ) dT − ( β 0 + β1 P ) dP ⎤⎦
P0T0
⇒ VP ,T = VP0T0 ⎡⎣1 + α 0 (T − T0 ) + 0.5α1 (T 2 − T0 2 ) − β 0 ( P − P0 ) − 0.5β1 ( P 2 − P0 2 ) ⎤⎦
P ⎡ = V . dP ∫P ∫P ⎢⎣VP0T0 0 0 P
α1 2 ⎤ P ⎧ 2 ⎫ + − + − 1 ( T T ) ( T T ) dP α ⎨ 0 0 0 ⎬ ⎥ − ∫ VP0T0 2 ⎩ ⎭ ⎦ P0
⎡⎧ β1 2 ⎤ 2 ⎫ ⎢ ⎨ β 0 ( P − P0 ) + 2 ( P − P0 ) ⎬ dP ⎥ ⎭ ⎦ ⎣⎩
(22)
⎡⎧ ⎤ α1 2 ⎧ β 0 β1 ⎫ 2⎫ = + − + − − + + − V dP V T T T T P P . 1 ( ) ( 2 ( 1) α ( ) ⎨ ⎬ ⎨ ⎬ P T 0 0 0 ⎢ ⎥ ( P − 1) ∫1 0 0 2 2 6 ⎩ ⎭ ⎩ ⎭ ⎣ ⎦ P
(23a)
Observing that the integral of a difference is identical to the difference is identical to the P
difference in the integrals, we may derive
∫ ΔVdP
for any reaction as follows
1
P
⎡
∫ ΔV .dP = ⎢⎣V
P0T0
1
1 1 ⎤ ⎧ ⎧1 ⎫ 2 2⎫ ⎨ +Δ (α 0VP0T0 )(T − T0 ) + Δ (α1VP0T0 )(T − T0 ⎬ − ⎨ Δ ( β 0VP0T0 ) + Δ ( β1VP0T0 ) ( P + 2 ) ⎬ ( P − 1) ⎥ ( P − 1) 2 6 ⎩ ⎭ ⎩2 ⎭ ⎦
Now, eventually substituting eqns. (19) and (23) into eqn. (16) leads to eqns. (24) which is applicable for pure phases at P0 = 1 bar and T0 = 298.15K.
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Thermodynamic properties of pure phases GP ,T = H1,298 − TS1,298
B. Mishra
2 ⎡ ⎛ ⎤ b T ⎞ c ⎡ (T − 298 ) ⎤ 2 + a ⎢T ⎜ 1 − ln ⎥ ⎟ − T0 ⎥ − ⎡⎣(T − 298 ) ⎤⎦ − ⎢ 298 ⎠ 2 ⎢ 2T • 2982 ⎥ ⎣ ⎝ ⎦ 2 ⎣ ⎦
⎡ ⎤ ⎢ 0.5 ⎥ 2 0.5 2 ⎢ (T − 298 ) ⎥ e f ⎡ (T − 298 ) ( 298 + 2T ) ⎤ 2 ⎡ ⎤ −2d ⎢ ⎥ (24) ⎥ − ⎣ (T − 298 ) (T + 2 • 298 ) ⎦ − ⎢ 6 ⎣⎢ 2980.5 2983 T 2 ⎢ ⎥ 6 ⎦⎥ ⎢ ⎥ ⎣⎢ ⎦⎥ ⎡⎧ ⎤ β α ⎫ ⎫ ⎧β +V1,298 ⎢ ⎨1 + α 0 (T − 298 ) + 1 (T 2 − 2982 ⎬ − ⎨ 0 + 1 ( P + 2) ⎬ ( P − 1) ⎥ ( P − 1) 2 6 ⎭ ⎩2 ⎭ ⎣⎩ ⎦
Eqn. (24) is the most fundamental eqn. for expressing Gibbs free energy of a pure phase at any desired equilibrium P, T. The determinable/known thermodynamic coordinates are (i) H 0,1, 298 , S 0,1, 298 , and V0,1, 298 which are standard state enthalpy, entropy and volume of the
pure phase. (ii) The Cp coefficients (a, b, c, d, e, f) that are obtained from the variation of Cp with T. (iii) α 0 , α 1 and β 0 , β1 are independently known from experiments. If the above coordinates are known, then
G0, P ,T is a function of T & P, i.e. if P, T are
known for a pure phase G is automatically defined.
For any reaction such as A + B = C + D eqn. (24) can be written as GPT
2 T ⎞ Δc ⎡⎛ (T − 298) ⎞ ⎤ ⎡ ⎛ ⎤ Δb ⎡ 2 ⎤ = ΔH1,298 − T ΔS1,298 + Δa ⎢T ⎜ 1 − ln ⎟⎥ (T − 298) ⎦ − ⎢⎜⎜ ⎟ − 298⎥ − 298 ⎠ 2 ⎢⎝ T • 2982 ⎟⎠ ⎥ ⎣ ⎝ ⎦ 2 ⎣ ⎣ ⎦
2 ⎡ ( T 0.5 − 2980.5 )2 ⎤ Δe Δf ⎡ (T − 298) (298 + 2T ) ⎤ 2 ⎡ ⎤ ⎢ ⎥ −2 Δ d − (T − 298) (T + 2 • 298) ⎦ − ⎢ ⎥ 2980.5 6 ⎣⎢ 2983 T 2 ⎢ ⎥ 6 ⎣ ⎦⎥ ⎣ ⎦ Δα1 2 Δβ ⎡⎧ ⎤ ⎫ ⎧ Δβ ⎫ (T − 2982 ⎬ − ⎨ 0 + 1 ( P + 2) ⎬ ( P − 1) ⎥ ( P − 1) +ΔV1,298 ⎢ ⎨1 + Δα 0 (T − 298) + 2 6 ⎭ ⎩ 2 ⎭ ⎣⎩ ⎦
(25) where in stead of absolute values (G, H, S, V etc), we rather calculate the changes in them (ΔG, ΔH, ΔS, ΔV etc).
14