THE VARIATION OF GIBBS FREE ENERGY WITH TEMPERATURE AT CONSTANT PRESSURE

At a total pressure of 1 atm, ice and water are in equilibrium with one another at 0°C, and hence, for these values of temperature and pressure, the Gibbs free energy, G
, of the system has its minimum value. Any transfer of heat to the system causes some of the ice to melt at 0°C and 1 atm pressure, and, provided that some ice remains, the equilibrium between the ice and the water is not disturbed and the value of G
for the system is unchanged. If, by the addition of heat, 1 mole of ice is melted, then for the change of state

Thus, at the state of equilibrium between ice and water,

(7.1)

where is the molar Gibbs free energy of H₂O in the solid (ice) phase, and is the molar Gibbs free energy of H₂O in the liquid (water) phase. For the system of ice+water containing n moles of H₂O, which are in the ice phase and of which are in the water phase, the Gibbs free energy of the system, G
, is

(7.2)

and from Eq. (7.1) it is seen that, at 0°C and 1 atm pressure, the value of G
is independent of the proportions of the ice phase and the water phase present.

The equality of the molar Gibbs free energies of H₂O in the solid and liquid phase at 0°C and 1 atm corresponds with the fact that, for equilibrium to occur, the escaping tendency of H₂O from the liquid phase must equal the escaping tendency of H₂O from the solid phase. Hence it is to be expected that a relationship exists between the molar Gibbs free energy and the chemical potential of a component in a phase. Integration of Eq. (5.25) at constant T and P gives

(7.3)

Comparison of Eqs. (7.2) and (7.3) shows that or, in general, _i=G_i; i.e., the chemical potential of a species in a particular state equals the molar Gibbs free energy of the species in the particular state.

This result could also have been obtained from a consideration of Eq. (5.16)

In a one-component system, the chemical potential of the component equals the increase in the value of G
which occurs when 1 mole of the component is added to the system at constant T and P. That is, if the component is the species i,

and as the increase in the value of G
for the one-component system is simply the molar Gibbs free energy of i, then

If the ice+water system is at 1 atm pressure and some temperature greater than 0°C, then the system is not stable and the ice spontaneously melts. This process decreases the Gibbs free energy of the system, and equilibrium is attained when all of the ice has melted. That is, for the change of state H₂O_(s) → H₂O_(l) at T>273 K, and P=1 atm,

i.e.

The escaping tendency of H₂O from the solid phase is greater than the escaping tendency of H₂O from the liquid phase. Conversely, if, at P=1 atm, the temperature is less than 0°C, then

which, for the ice+water system, is written as

176 Introduction to the Thermodynamics of Materials

The variations of and with temperature at constant pressure are shown in Fig. 7.1 and the variation of G_s→l with temperature at constant pressure is shown in Fig. 7.2.

Figs. 7.1 and 7.2 show that, at 1 atm pressure and temperatures greater than 0°C, the minimum Gibbs free energy occurs when all of the H₂O is in the liquid phase,

Figure 7.1 Schematic representation of the variations of the molar Gibbs free energies of solid and liquid water with temperature at constant pressure.

and at 1 atm pressure and temperatures lower than 0°C, the minimum Gibbs free energy occurs when all of the H₂O is in the solid phase. The slopes of the lines in Fig. 7.1 are obtained from Eq. (5.25) as

Similarly, the slope of the line in Fig. 7.2 is given as

where S is the change in the molar entropy which occurs as a result of the change of state. The slope of the line in Fig. 7.2 is negative, which shows that, at all temperatures,

Figure 7.2 Schematic representation of the variation of the molar Gibbs free energy of melting of water with temperature at constant pressure.

as is to be expected in view of the fact that, at any temperature, the liquid phase is more disordered than is the solid phase.

and the curvatures of the lines are obtained from Eq. (6.12) as

178 Introduction to the Thermodynamics of Materials

equilibrium with one another can be determined from consideration of the molar enthalpy H and the molar entropy S of the system. From Eq. (5.2),

This can be written for both the solid and the liquid phases,

and

For the change of state solid → liquid, subtraction gives

where H_(s→l) and S_(s→l) are, respectively, the changes in the molar enthalpy and molar entropy which occur as a result of melting at the temperature T. From Eq. (7.1)

The state in which the solid and liquid phases of a one-component system are in

Figure 7.3 The variations, with temperature, of the molar enthalpies of solid and liquid water at 1 atm pressure.

The molar enthalpy of liquid water at 298 K is arbitrarily assigned the value of zero.

equilibrium between the solid and the liquid phases occurs at that state at which

G_(s→l)=0. This occurs at that temperature T_m at which

(7.4)

For H₂O

Fig. 7.3 shows the variations of H_(s) and H_(l) at 1 atm pressure, in which, for convenience, H_(l),298K is arbitrarily assigned the value of zero, in which case

and

The molar enthalpy of melting at the temperature T, H_(s→l),T is the vertical separation between the two lines in Fig. 7.3.

180 Introduction to the Thermodynamics of Materials

and

Figure 7.4 The variations, with temperature, of the molar entropies of solid and liquid water at 1 atm pressure.

Fig. 7.4 shows the variations of S_(s) and S_(l) with temperature at 1 atm pressure, where

Figure 7.5 The variation h, with temperature, of TS for solid and liquid water at 1 atm pressure.

The molar entropy of melting at the temperature T, _S(s→l) is the vertical separation between the two lines in Fig. 7.4. Fig. 7.5 shows the corresponding variations of TS_(S) and TS_(l) with temperature. Equilibrium between the solid and liquid phases occurs at that temperature at which the vertical separation between the two lines in Fig. 7.3 equals the vertical separation between the two lines in Fig. 7.5. This unique temperature is T_m, and, at this temperature,

In Fig. 7.6, H_(s→l), TS_(s→l), and G_(s→l) are plotted as functions of temperature using the data in Figs. 7.3 and 7.5. This figure shows that G_{(s → l)}=0 at T=T_m= 273 K, which is thus the temperature at which solid and liquid water are in equilibrium with one another at 1 atm pressure.

Equilibrium between two phases thus occurs as the result of a compromise between enthalpy considerations and entropy considerations. Equilibrium requires that G
for the system have its minimum value at the fixed values of P and T, and Eq. (5.2) shows that minimization of G
requires that H be small and S be large. Fig. 7.3 shows that, at all temperatures, H_(l)>H_(s), and thus, from consideration of the contribution of enthalpy to the Gibbs free energy, and in the absence of any other consideration, it would seem that the solid would always be stable with respect to the liquid. However Fig 7.4 shows that,

182 Introduction to the Thermodynamics of Materials

at all temperatures, S_(l)>S_(s). Thus from consideration of the contribution of entropy to the Gibbs free energy, in the absence of any other consideration, it would seem that the liquid phase is always stable with respect to the solid phase. However, as the contribution of the entropy, TS, to G is

Figure 7.6 The variations, with temperature, of the molar Gibbs free energy of melting, the molar enthalpy of melting, and T
the molar entropy of melting of water at 1 atm pressure.

dependent on temperature, a unique temperature T_m occurs above which the contribution of the entropy outweighs the contribution of the enthalpy and below which the reverse is the case. The temperature T_m is that at which H_(l)–T_mS_(l) equals H_(s)– T_mS_(s) and hence is the temperature at which the molar Gibbs free energy of the solid has the same value as the molar Gibbs free energy of the liquid. This discussion is analogous to that presented in Sec. 5.3 where, at constant T and V, the equilibrium between a solid and its vapor was examined in terms of minimization of the Helmholtz free energy, A, of the system.

equilibrium with one another at 0°C, when the pressure exerted on the system is increased to a value greater than 1 atm. Le Chatelier’s principle states that, when subjected to an external influence, the state of a system at equilibrium shifts in that direction which tends to nullify the effect of the external influence. Thus when the pressure exerted on a system is increased, the state of the system shifts in the direction which causes a decrease in its volume. As ice at 0°C has a larger molar volume than has water at 0°C, the melting of ice is the change in state caused by an increase in pressure.

The influence of an increase in pressure, at constant temperature, on the molar Gibbs free energies of the phases is given by Eq. (5.25) as

i.e., the rate of increase of G with increase in pressure at constant temperature equals the molar volume of the phase at the temperature T and the pressure, P For the change of the state solid → liquid,

and as V_(s→l) for H₂O at 0°C is negative, the ice melts when the pressure is increased to a value greater than 1 atm. Thus, corresponding to Fig. 7.1, which showed the variation of G_(s) and G_(l) with T at constant P, Fig. 7.7 shows the variation of G_(s) and G_(l) with P at constant T. Water is anomalous in that, usually, melting causes an increase in volume.

7.3 THE VARIATION OF GIBBS FREE ENERGY WITH PRESSURE AT