Explicit and Implicit Functions and Total Differentiation
1. Explicit and Implicit Functions and Total Differentiation
If P is a known function of T and v, the explicit function for P is P = P(v,T),
(14) and its total differential may be written in the form ∂ P
dP =
(15) ∂ v T
dv
∂ T v
dT .
Consider the P, T, v relation
(16) where a and b are constants. Equation (16) is explicit with respect to P, since it is an explicit
P = RT/(v–b) – a/v 2 ,
function of T and v. On the other hand, v cannot be explicitly solved in terms of P and T, and, hence, it is an implicit function of those variables. The total differential is useful in situations that require the differential of an implicit function, as illustrated below.
b. Example 2 If state equation is expressed in the form
(A) find an expression for ( ∂v/∂T) P , and for the isobaric thermal expansion coefficient β P =
P = RT/(v–b) – a/v 2 ,
(1/v) ( ∂v/∂T) P . Solution
For given values of T and v, and the known parameters a and b, values of P are unique (P is also referred to as a point function of T and v). Using total differentiation
dP = ( ∂P/∂v) T dv + ( ∂P/∂T) v dT. (B) From Eq. (A)
( ∂P/∂v) T = –RT/(v – b) 2 + 2a/v 3 , and
(C)
( ∂P/∂T) v = R/(v – b) (D) Substituting Eqs.(C) and (D) in Eq. (B) we obtain
(E) We may use Eq. (E) to determine ( ∂v/∂T) P or ( ∂v/∂P) T . At constant pressure, Eq. (E)
dP = (–RT/(v – b) 2 + 2a/v 3 ) T dv + (R/(v – b)) v dT.
yields
(F) so that (
0 = (–RT/(v – b) 2 + 2a/v 3 ) T dv + (R/(v – b)) v dT,
2 ∂v 3 P / ∂T P )=( ∂v/∂T) P = –(R/(v – b))/(–RT/(v – b) + 2a/v ), (G) and the isobaric compressibility
= 1/v ( = –R/(v(–RT/(v–b) + 2a(v–b)/v β 3 P ∂v/∂T) P )). (H) Remarks
It is simple to obtain ( ∂P/∂T) v or ( ∂P/∂v) T from Eq. (A). It is difficult, however, to obtain values of ( ∂v/∂T) P or ( ∂v/∂P) T from that relation. Therefore, the total differentiation is em- ployed.
Note that Eqs. (C) and (D) imply that for a given state equation: ( ∂P/∂T) v = M(T,v), and
(I) ( ∂P/∂v) T = N(T,v), and
(J) Since,
dP = M(T,v) dv + N(T,v) dT, (K) Differentiating Eq. (C) with respect to T,
(L) Likewise, differentiating Eq. (D) with respect to v,
∂/∂T (∂P/∂v) = (∂M/∂T) v = –R/(v – b) 2 .
(M) From Eqs. (L) and (M) we observe that
∂/∂v (∂P/∂T) = (∂N/∂v) T = –R/(v – b) 2 .
2 ∂M/∂T = ∂N/∂v or ∂ 2 P/ ∂T∂v = ∂ P/ ∂v∂T. (N) Eq. (N) illustrates that the order of differentiation does not alter the result. The equation
applies to all state equations or, more generally, to all point functions (see next section for more details). From Eq. (B), at a specified pressure
( ∂P/∂v) T dv + ( ∂P/∂T) v dT = M(T,v) dv + N(T,v) dT = 0. Therefore, ( ∂v/∂T) P = –M(T,v)/N(T,v) = –( ∂P/∂T) v /( ∂P/∂v) T .
(O) Eq. (O) can be rewritten in the form
( ∂v/∂T) P ( ∂T/∂P) v ( ∂P/∂v) T = –1, (P) ( ∂v/∂T) P ( ∂T/∂P) v ( ∂P/∂v) T = –1, (P)
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
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