Relations between Partial Molal and Pure Properties
5. Relations between Partial Molal and Pure Properties
We have discussed the partial molal volume and now focus on other partial molal properties.
a. Partial Molal Enthalpy and Gibbs function Since the enthalpy H(T, P, N 1 ,N 2 , ... .) = U(T, P, N 1 ,N 2 , ... .) + PV(T, P, N 1 ,N 2 , ... .), the partial molal enthalpy ˆh i =( ∂H/∂N i ) T, P, N N N 1 , 2 ,..., ji ≠ ,..., N K can be expressed as
ˆh i =( ∂U/∂N i ) T, P, N N N 1 , 2 ,..., ji ≠ ,..., N K + P( ∂V/∂N i ) T, P, N N N 1 , 2 ,..., ji ≠ ,..., N K , i.e., (26) ˆh i = ˆu i +P ˆv , where
(27) ˆu i =( ∂(N u )/ ∂N i ) T, P, N N N 1 , 2 ,..., ji ≠ ,..., N K = N( ∂ u / ∂N i ) T, P, N N N 1 , 2 ,..., ji ≠ ,..., N K + u .
Similarly, since G = H – TS, ˆg i =( ∂H/∂N i ) T, P, N N N 1 , 2 ,..., ji ≠ ,..., N K – T( ∂S/∂N i ) T, P, N N N 1 , 2 ,..., ji ≠ ,..., N K , i.e.,
(28) ˆg i = ˆh i –T ˆs i .
(29) Likewise,
ˆa i = ˆu i –T ˆs i . (30)
b. Differentials of Partial Molal Properties Applying the Gibbs–Duhem equation
(12c) in terms of the Gibbs energy, i.e., B = G, ∂G/∂T = –S, and ∂G/∂P = V, so that
( ∂B/∂T) P,N dT + ( ∂B/∂T) T,N dP – Σ k d ˆb k N k =0
–S dT + V dP – Σ k d ˆg k N k = 0.
In terms of intensive properties, this relation may be written in the form – Σ k ˆs k N k dT + Σ k ˆv k N k dP – Σ k d ˆg k N k =– Σ k N k (d ˆg k + ˆs k dT– ˆv k dP) = 0, i.e.,
For arbitrary N k >0,
d ˆg k =– ˆs k dT + ˆv k dP. (31)
Differentiating Eq. (29) and using (31) to eliminate dg ˆ k ,
d ˆh k = Td ˆs k + ˆv k dP. (32a) Dividing Eq. (32) by dT at constant pressure, the partial molal specific heat ˆc pk = ∂ ˆh k / ∂T = T ∂ ˆs k / ∂T.
(32b) Subtracting the term d(Pv k ) from Eq. (32a), we obtain the relation
d ˆu k =Td ˆs k –Pd ˆv k . (33a) Similarly
ˆc vk =( ∂ ˆu k / ∂T) = T (∂ ˆs k / ∂T). (33b) These relations for partial molal properties are similar to those for pure substances.
Maxwell’s relations can be likewise derived. Subtracting d(T ˆs k ) from Eq. (33a), we obtain the relation
d ˆa k =– ˆs k dT – P d ˆv k . (34) This implies that ˆs k =– ∂ ˆa k / ∂T, P = – ∂ ˆa k / ∂ ˆv k . These expressions are similar to those for pure
properties. Maxwell’s relations can be likewise derived. Furthermore, from Eq. (32a) ∂ ˆh k /dP = T ∂ ˆs k /dP + ˆv k .
(35) Using the Maxwell’s relations we can show that ∂ ˆh k /dP = – T ∂ ˆv k /dT + ˆv k . For the entropy, the G–D relation is
(36) Since S = Σ k ˆs k N k and ∂ ˆs k / ∂T = ˆc pk /T, we may use Maxwell’s relations to simplify the second
( ∂S/∂T) P,N dT + ( ∂S/∂P) T,N dP – Σ k d ˆs k N k =0
term, i.e., ( ∂S/∂P) T,N = –( ∂V/∂T) P , where V = Σ k Ν k ˆv k so that
Σ k N k ( ˆc p,k /T)dT – Σ k (N k ∂ ˆv k / ∂T) dP – Σ k d ˆs k N k = 0, or
Σ k N k (( ˆc p,k /T)dT – ( ∂ ˆv k / ∂T)dP – d ˆs k ) = 0. Therefore,
d ˆs k =( ˆc p,k /T)dT – ( ∂ ˆv k / ∂T) dP. (37) Using Eqs. (37) in Eq. (32a),
d ˆh k = ˆc p,k dT + ( ˆv k –T( ∂ ˆv k / ∂T)) dP, (38) which is again similar to the corresponding expression for a pure substance. Likewise,
d ˆu k = ˆc v,k dT + (T( ∂P/∂T) – P) d ˆv k (39) Equations (37) to (39) are similar to the corresponding expressions for a pure substance. Thus
if state equations are available for mixtures, ˆu k , ˆh k and ˆs k can be determined.
i. Remarks Maxwell’s relations can be obtained using Eqs. (31)–(34). These relations are similar
to those for pure components. Consider the derivative ( ∂/∂N i ( ∂V/∂T) P, N N N 1 , 2 ,..., ji ≠ ,..., N K ) T, P, N N N 1 , 2 ,..., ji ≠ ,..., N K . Switching or-
der of differentiation, the expression equals the term
( ∂/∂T(∂V/∂N i ) T, P, N N N 1 , 2 ,..., ji ≠ ,..., N K ) P, N N N 1 , 2 ,..., ji ≠ ,..., N K = ∂ ˆv i / ∂T.
Likewise, ( ∂/∂N i ( ∂S/∂T) P, N N N 1 , 2 ,..., ji ≠ ,..., N K ) T, P, N N N 1 , 2 ,..., ji ≠ ,..., N K =T ∂ ˆs i / ∂T = ˆc pi ,
(40) which is again an expression that is similar to that for a pure substance.
Parts
» COMPUTATIONAL MECHANICS and APPLIED ANALYSIS
» Explicit and Implicit Functions and Total Differentiation
» Exact (Perfect) and Inexact (Imperfect) Differentials
» Intermolecular Forces and Potential Energy
» Internal Energy, Temperature, Collision Number and Mean Free Path
» Vector or Cross Product r The area A due to a vector product
» First Law for a Closed System
» First Law For an Open System
» STATEMENTS OF THE SECOND LAW
» Cyclical Integral for a Reversible Heat Engine
» Irreversibility and Entropy of an Isolated System
» Degradation and Quality of Energy
» SINGLE–COMPONENT INCOMPRESSIBLE FLUIDS
» Evaluation of Entropy for a Control Volume
» Internally Reversible Work for an Open System
» MAXIMUM ENTROPY AND MINIMUM ENERGY
» Generalized Derivation for a Single Phase
» LaGrange Multiplier Method for Equilibrium
» Absolute and Relative Availability Under Interactions with Ambient
» Irreversibility or Lost Work
» Applications of the Availability Balance Equation
» Closed System (Non–Flow Systems)
» Heat Pumps and Refrigerators
» Work Producing and Consumption Devices
» Graphical Illustration of Lost, Isentropic, and Optimum Work
» Flow Processes or Heat Exchangers
» CLASSICAL RATIONALE FOR POSTULATORY APPROACH
» Generalized Legendre Transform
» Van der Waals (VW) Equation of State
» Other Two–Parameter Equations of State
» Compressibility Charts (Principle of Corresponding States)
» Boyle Temperature and Boyle Curves
» Three Parameter Equations of State
» Empirical Equations Of State
» State Equations for Liquids/Solids
» Internal Energy (du) Relation
» EXPERIMENTS TO MEASURE (u O – u)
» Vapor Pressure and the Clapeyron Equation
» Saturation Relations with Surface Tension Effects
» Temperature Change During Throttling
» Throttling in Closed Systems
» Procedure for Determining Thermodynamic Properties
» Euler and Gibbs–Duhem Equations
» Relationship Between Molal and Pure Properties
» Relations between Partial Molal and Pure Properties
» Mixing Rules for Equations of State
» Partial Molal Properties Using Mixture State Equations
» Ideal Solution and Raoult’s Law
» Completely Miscible Mixtures
» DEVIATIONS FROM RAOULT’S LAW
» Mathematical Criterion for Stability
» APPLICATION TO BOILING AND CONDENSATION
» Physical Processes and Stability
» Constant Temperature and Volume
» Equivalence Ratio, Stoichiometric Ratio
» Entropy, Gibbs Function, and Gibbs Function of Formation
» Entropy Generated During an Adiabatic Chemical Reaction
» MASS CONSERVATION AND MOLE BALANCE EQUATIONS
» Evaluation of Properties During an Irreversible Chemical Reaction
» Criteria in Terms of Chemical Force Potential
» Generalized Relation for the Chemical Potential
» Nonideal Mixtures and Solutions
» Gas, Liquid and Solid Mixtures
» Availability Balance Equation
Show more