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Properties of the Gibbs free energy:
Properties of the Gibbs free energy
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We have defined the Gibbs free energy G and the enthalpy H as: and hence: i.e.: But for a system doing no nonexpansion work, dU may be replaced by the fundumental equation of thermodynamics, i.e.: and hence we have: which simplifies to: which since dG is an exact differential gives us: and:
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The temperature dependence of G is governed by the following relationship: This means that because S is positive, then G decreases when the temperature increases at constant pressure (G vs T plot, see fig. 1).Then sharpness of the decrease of G decreases with temperature at constant pressure is determined by the entropy of the system. This means that the sharpness of the decrease of the G vs T (i.e. the sensitivity of G with changes in T, = gradient of G vs T plot, see fig. 1) is greatest for a gas, then liquid, then solid. Let us now derive a relationship between the Gibbs energy and enthalpy (The GibbsHelmholz equantion). From the definition of G , we have: i.e.: i.e.: which simplifies to the GibbsHelmholtz equation (see proof below): The GibbsHelmholtz equation is most useful when applied to changes, including change of phase or chemical reactions at constant pressures. Then since DG = G_{f}  G_{i}, and since the GibbsHelmholtz equation applies to both G_{f }and G_{i}, we can write:Fig. 1: The variation of the Gibbs energy with the temperature is determined by the entropy. Aside: Proof of the GibbsHelmholtz equation: [ TOP ]
The pressure dependence of G is governed by the following relationship: This means that because V is positive, then G always increases when the pressure increases at constant temperature (G vs p plot, see fig. 2). Also, since V is the gradient of the G vs p plot, then the sensitivity of G to changes in p is greatest for gases and negligible for liquids and solids (see fig. 2).Fig. 2: The variation of the Gibbs energy with the pressure We can find the change in the Gibb's energy due to a change in pressure at constant temperature through: For solids and liquids, we then treat V as constant, whilst for an ideal gas we use the pV=nRT relationship, and we get: or if we assume p_{i} = p^{0} then we get: 
