Physical Processes and Stability

1. Physical Processes and Stability

Consider the isothermal compression of water at 593 K. (cf. Figure 11 ). As the fluid is compressed from state B towards states E, C, G, etc., the volume decreases with the increase in pressure, since the intermolecular spacing decreases. The intermolecular attraction forces slowly increase as the states E, C, G, etc., are approached. At larger volumes (i.e. lower pres- sures) the first term in RK equation dominates, i.e., b « v, P ≈ RT/v so that P ∝ 1/v at a speci-

fied temperature. The Gibbs energy value is lower at larger volumes, and gradually increases as the pressure is raised (BECG in Figure 11 ), indicating that the fluid accumulates a larger potential to perform work. The rate of pressure increase with decreasing volume is lower at smaller volumes due to the larger intermolecular attraction forces and the second term in the RK equation a/(T 1/2 v(v+b)) becomes significant. Beyond a maximum pressure at state N, the intermolecular attraction forces are so large that this second term dominates and tends to lower the pressure. Consequently, the pressure starts to decrease with compression at smaller vol- umes due to the very small intermolecular spacing. (The pressure can sometimes be negative indicating that the fluid is under tension. In case of water the tension can be as high as –40 bar without evaporation occurring.) Consequently the “g” decreases along NDUHJM. Upon com- pression beyond states M, R, A, etc., the body volume effect (which reduces the space avail- able for the movement of molecules) becomes dominant and results in a higher number density so that the first term in the RK equation again dominates. Thereupon, the pressure again in- creases rapidly and P ≈ RT/(v–b), i.e., P ∝ 1/(v–b) and the Gibbs energy again increases

(MRAFL). All fluid states along the path BECGKNDHJMRFL are equilibrium states. The sta- bility criterion ∂P/∂v < 0 suggests that the path NDUHJM is an unstable branch along which

inherent disturbances exist and hence a uniform intermolecular spacing or stable state cannot

be maintained.

a. Physical Explanation At T = 593 K, g F =g G at P = 133 bars. Hence P sat = 133 bar; the fluid in the context of Example 10 typically changes its state from vapor to liquid (from state G to F along line GHF (cf. Figure

11 )). However the Gibbs

energy

changes 2500 ∆g GH = g H – g G and ∆g FH =

g H – g F represent

the adverse poten- N

1500 e

tials at the satura-

ar F H G C E k ,b mol P

B tion condition that J/

1000 g, k

the fluid has to

overcome to either

form a vapor em- 500 bryo at state G

0 from the liquid 0

mother phase at

v, m 3 /kmole

state L, or a liquid embryo at state F from the vapor

Figure 11: Plot of P (and ¯g ) vs. volume using the RK equation.

Unstable H L 1000

Stable Liq k mol

A C F,G

J/ Saturation g, k

Stable vapor

P, bars

Figure 12: Variation of ¯g with pressure. state G ( Figure 13a ). We note from Figure 11 or Figure 12 that when P < 50 bar, a single state

is possible (i.e., superheated vapor). If 50 < P < 155 bar, there are three possible states for the same Gibbs energy value, and when P > 155 bar there is a single liquid solution (i.e., com- pressed liquid).

Consider a constant T, P, and m system at an arbitrary state J. A reduction in the fluid volume from a value v J reduces the internal pressure exerted by the fluid further compressing the fluid. Assume that equilibrium is achieved at a liquid volume v A at which P A =P J ( Figure

with a volume v C . The implication is that at the pressure P J there are three plausible solutions for the Gibbs energy. The questions are as follows. Which are stable states? Which are nonstable?

12 ). Likewise a corresponding vapor state C exists at which P C =P J

Since dG T =VdP, dG embryo = (V( ∂P/∂V)) dV. An increased volume (e.g., during evaporation) implies that ∂P/∂V < 0 so that ∂G/∂V < 0. Suppose a disturbance at J causes the embryo phase to expand to a volume slightly higher than v J from state J, the volume increase

tends to increase the embryo phase pressure. Since the embryo phase pressure is higher than the mother phase pressure, which is held fixed, the embryo expands to larger and larger vol- umes, eventually to the vapor state B. The first bubble during boiling is formed through this process. The embryo phase bubble is associated with a lower Gibb’s free energy ( Figure 12 ) as compared to the mother phase that is still at state J. In Chapter 3 we discussed that a Gibbs energy gradient produces a species flow from a system at a higher Gibbs energy to that at a lower Gibbs energy. In that context, the molecules from the mother phase migrate to the vapor phase during vaporization as long as the pressure and temperature are maintained constant. If the disturbance at J results in reduction of volume, the embryo pressure decreases; since the mother phase is at higher pressure, embryo is compressed further until it forms a liquid droplet at A. In the case of flow processes, a sudden condensation or vaporization produces a severe pressure disturbance or a sudden acceleration of the flow, leading to local turbulence (e.g., in boiler tubes and in clouds).

In the context of the above discussion, we now consider the H 2 O at 140 bar and 593 K for which the specific volume v D is at an unstable state. An embryo may form, but a decrease in the embryo volume causes the pressure to decrease below 140 bar, which results in com- pression of the embryo by the mother phase that is still at 140 bar. This accelerates the forma- tion of a drop at state L. Since the Gibbs energy of the compressed liquid at state L is lower than that of the mother phase at state K, the fluid molecules will tend to migrate to the liquid state. State H is also unstable and only a microscopic disturbance is required to drive the state

to either of states F or G. At saturation, the two minima are equal, i.e., g sat

F = g G = g . This process is called a first order phase transition during which both the liquid and vapor states are probable, which is a consequence of boiling at a specified temperature and pressure. In this case, the liquid and vapor molecules can exchange phase if a disturbance is strong enough to

overcome the potential (g H –g G ), which results in a wet mixture. Note that thermodynamics do not specify the time scales (called relaxation time scale) required to effect the change from meta-stable or unstable state to stable state. Constitute equations for the transport processes are required to determine those time scales.

These examples pertain to phase equilibrium. An analogous situation exists during chemical equilibrium where, at a specified temperature and pressure, the Gibbs energy of products reaches a minimum value.