General thermodynamic properties and its relation

S.Gunabalan Associate Professor Mechanical Engineering Department Bharathiyar College of Engineering & Technology Karaikal - 609 609. e-Mail : gunabalans@yahoo.com

Part - 2

HELM HOLTZ AND GIBBS FUNCTIONS

The work done in a non-flow reversible system (per unit mass) is given by W = Q – (u0 – u1) = T.ds – (u0 – u1) = T.(s0-s1) – (u0 – u1) = (u1 – Ts1) – (u0 – Ts0) The term (u – Ts) is known as Helmholtz function

This gives maximum possible output – when the heat Q is transferred at constant temperature This can be achieved through a very large source.

work against atmosphere = p0 (v0 – v1) then the maximum work available, Wmax = W – work against atmosphere = W – p0 (v0 – v1) = (u1 – Ts1) – (u0 – Ts0) – p0 (v0 – v1) = (u1 + p0v1 – Ts1) – (u0 + p0v0 – Ts0) = (h1 – Ts1) – (h0 – Ts0) Wmax = g1 – g0 where g = h – T.s is known as Gibb’s function or free energy function

Gibbs Phase Equilibrium Condition

• A system is said to be in thermodynamic equilibrium if, when it is isolated from its surroundings, there would be no macroscopically observable changes.

• An important requirement for equilibrium is that the temperature be uniform throughout the system or each part of the system in thermal contact.

Consider the compressible system of fixed mass for which temperature and pressure are uniform with position throughout. In the absence of overall system motion and ignoring the influence of gravity, the energy balance in differential form

푑푈 = 훿푄 − 훿W 훿W = pdV

푑푈 = 훿푄 − 푝푑푉

훿푄 = 푑푈 + 푝푑푉

푇푑푆 − 푇훿휎 = 푑푈 + 푝푑푉 푇푑푆 − 푑푈 − 푝푑푉 = 푇푑휎 ------------ (eq.1) Entropy is produced (푇푑휎)in all actual processes and conserved only in the absence of irreversibilities. Hence, Eq.1 provides a constraint on the direction of processes. The only processes allowed are those for which 훿휎 >= 0. Thus

푇푑푆 − 푑푈 − 푝푑푉 ≥ 0 This equation used to study equilibrium under various conditions.

a process taking place in an insulated, constant-volume vessel, where dU = 0 and dV = 0, must be such that

푑푆 ≥ 0 this equation suggests that changes of state of a closed system at constant internal energy and volume can occur only in the direction of increasing entropy. The expression also implies that entropy approaches a maximum as a state of equilibrium is approached.

In a study of chemical and phase equilibria is one in which temperature and pressure are fixed

Gibbs Phase Equilibrium Condition

Any process taking place at a specified temperature and pressure (dT = 0 and dp = 0) must be such that

The equilibrium state is the one having the minimum value of the Gibbs function

Values of two independent

intensive properties are sufficient to fix the intensive state

of a simple compressible system

of specified mass and composition

Thermodynamic Relations

Similar to

Clapeyran Equation

During a phase change at fixed temperature, the pressure is independent of specific volume and is determined by temperature alone. Thus,

Clapeyran Equation

(1) 푣f is negligible in comparison to 푣g, (2) the pressure is low enough that 푣g can be evaluated from the ideal gas equation of state as 푣g = RT/p

Reference • Moran, M. J. 2011. Fundamentals of engineering thermodynamics. Wiley,

[Hoboken, N.J.?].