The variation of Gibbs energy with pressure

The thermodynamics of transition

3.2 The variation of Gibbs energy with pressure

To discuss how phase transitions depend on the pressure and to lay the

foundation for understanding the behavior of solutions of biological macromolecules, we need to know how the molar Gibbs energy varies with pressure.

Why should biologists be interested in the variation of the Gibbs energy with the pressure of a gas, since in most cases their systems are at pressures close to 1 atm?

You should recall the discussion in Section 1.3, where we pointed out that to study the thermodynamic properties of a liquid (in which biochemists do have an inter- est), we can explore the properties of a gas, which is easy to formulate, and then bring the gas into equilibrium with its condensed phase. Then the properties of the liquid mirror those of the vapor, and we can expect to find a similar dependence on the pressure. That is the strategy we adopt throughout this chapter. First we es- tablish equations that apply to gases. Then we consider equilibria between gases and liquids and adapt the gas-phase expressions to describe what really interests us, the properties of liquids.

We show in the following Derivationthat when the temperature is held con- stant and the pressure is changed by a small amount p, the molar Gibbs energy of a substance changes by

G_mV_mp (3.1)

whereV_mis the molar volume of the substance. This expression is valid when the molar volume is constant in the pressure range of interest.

DERIVATION 3.1 The variation of G with pressure

We start with the definition of Gibbs energy, GHTS, and change the tem- perature, volume, and pressure by an infinitesimal amount. As a result, Hchanges to HdH, T changes to TdT, S changes to SdS, and G changes to GdG. After the change

GdGHdH(TdT)(SdS) HdHTSTdSSdTdTdS

The Gon the left cancels the HTSon the right, the doubly infinitesimal dTdScan be neglected, and we are left with

dGdHTdSSdT

To make progress, we need to know how the enthalpy changes. From its defi- nitionHUpV, in a similar way (letting Uchange to UdU, and so on, and neglecting the doubly infinitesimal term dpdV) we can write

dHdUpdVVdp

At this point we need to know how the internal energy changes and write dUdqdw

If initially we consider only reversible changes, we can replace dqby TdS(be- cause dSdq_rev/T) and dwbypdV(because dw p_exdVandp_expfor a reversible change) and obtain

dUTdSpdV

Now we substitute this expression into the expression for dHand that expres- sion into the expression for dGand obtain

dGTdSpdVpdVVdpTdSSdT VdpSdT

Now here is a subtle but important point. To derive this result we have sup- posed that the changes in conditions have been made reversibly. However, Gis a state function, and so the change in its value is independent of path. There- fore, the expression is valid for any change, not just a reversible change.

At this point we decide to keep the temperature constant and set dT0;

this leaves dGVdp

and, for molar quantities, dGmVmdp. This expression is exact but applies only to an infinitesimal change in the pressure. For an observable change, we replace dGmand dpbyGmandp, respectively, and obtain eqn 3.1, provided the mo- lar volume is constant over the range of interest.

A note on good practice:When confronted with a proof in thermodynamics, go back to fundamental definitions (as we did three times in succession in this derivation: first of G, then of H, and finally of U).

Equation 3.1 tells us that, because all molar volumes are positive, the molar Gibbs energy increases(G_m0)when the pressure increases(p0). We also see that, for a given change in pressure, the resulting change in molar Gibbs energy is greatest for substances with large molar volumes. Therefore, because the molar vol- ume of a gas is much larger than that of a condensed phase (a liquid or a solid), the dependence of G_monpis much greater for a gas than for a condensed phase.

For most substances (water is an important exception), the molar volume of the liquid phase is greater than that of the solid phase. Therefore, for most substances, the slope of a graph of G_magainstpis greater for a liquid than for a solid. These characteristics are illustrated in Fig. 3.1.

As we see from Fig. 3.1, when we increase the pressure on a substance, the mo- lar Gibbs energy of the gas phase rises above that of the liquid, then the molar Gibbs energy of the liquid rises above that of the solid. Because the system has a tendency to convert into the state of lowest molar Gibbs energy, the graphs show that at low pressures the gas phase is the most stable, then at higher pressures the liquid phase becomes the most stable, followed by solid phase. In other words, un- der pressure the substance condenses to a liquid, and then further pressure can re- sult in the formation of a solid.

We can use eqn 3.1 to predict the actual shape of graphs like those in Fig. 3.1.

For a solid or liquid, the molar volume is almost independent of pressure, so eqn 3.1 is an excellent approximation to the change in molar Gibbs energy, and with G_mG_m(p_f)G_m(p_i) and pp_fp_iwe can write

G_m(p_f)G_m(p_i)V_m(p_fp_i) (3.2a)

This equation shows that the molar Gibbs energy of a solid or liquid increases lin- early with pressure. However, because the molar volume of a condensed phase is so small, the dependence is very weak, and for typical ranges of pressure of inter- est to us, we can ignore the pressure dependence of G. The molar Gibbs energy of a gas, however, does depend on the pressure, and because the molar volume of a gas is large, the dependence is significant. We show in the following derivation that

G_m(p_f)G_m(p_i)RTlnp p

f

i (3.2b)

This equation shows that the molar Gibbs energy increases logarithmically (as lnp) with the pressure (Fig. 3.2). The flattening of the curve at high pressures re- flects the fact that as Vmgets smaller, Gmbecomes less responsive to pressure.

DERIVATION 3.2 The pressure variation of Gibbs energy of a perfect gas

We start with the exact expression for the effect of an infinitesimal change in pressure obtained in Derivation3.1, that dG_mV_mdp. For a change in pressure

Gas

Liquid Solid

g l s

Pressure,p Molar Gibbs energy, Gm

Fig. 3.1 The variation of molar Gibbs energy with pressure. The region where the molar Gibbs energy of a particular phase is least is shown by a dark line and the corresponding region of stability of each phase is indicated in the band at the top of the illustration.

Molar Gibbs energy, {Gm(pf)−Gm(pi)}/RT

−2

−1 0 1 2

−3

Pressure ratio, p_f/p_i

0 1 2 3 4 5 Fig. 3.2 The variation of the molar Gibbs energy of a perfect gas with pressure.

fromp_itop_f, we need to add together (integrate) all these infinitesimal changes and write

G_m

冕

_p^p_i^fVmdp

To evaluate the integral, we must know how the molar volume depends on the pressure. The easiest case to consider is a perfect gas, for which VmRT/p. Then

Perfect gas, VmRT/p

G_m

冕

_p^p_i^fVmdp

冕

_p^p_i^{f dpRT}

冕

_p^p_i^f

Isothermal,Tconstant

RTln

We have used the standard integral described in Comment 1.3. Finally, with G_mG_m(p_f)G_m(p_i), we get eqn 3.2b.