Question

Show that when the two phases of a two-component system are in equilibrium, the specific Gibbs function of each phase of each component is the same.

Short answer

Answer: When the two phases of a two-component system are in equilibrium, the specific Gibbs function of each phase of each component is the same.

Step-by-step solution

Definition of Gibbs function

The Gibbs function (G) is a thermodynamic potential that measures the maximum reversible work that can be performed by a system. For a single component in a single phase, the specific Gibbs function (g) is the Gibbs function per unit mass, which can be defined as: g = h - T * s Here, g is the specific Gibbs function, h is the specific enthalpy, T is the temperature, s is the specific entropy.

Two-component system in equilibrium

For a two-component system in equilibrium, there are two phases (α and β) and two components (1 and 2). The total Gibbs function for each phase can be written as: G_α = n1_α * g1_α + n2_α * g2_α G_β = n1_β * g1_β + n2_β * g2_β Here, n1_α and n2_α are the amounts of component 1 and 2 in phase α, n1_β and n2_β are the amounts of component 1 and 2 in phase β, g1_α and g2_α are the specific Gibbs function of component 1 and 2 in phase α, g1_β and g2_β are the specific Gibbs function of component 1 and 2 in phase β.

Equilibrium conditions

For the two phases to be in equilibrium, the chemical potential of each component in both phases should be the same. The chemical potentials of components in phase α and β can be written as: μ1_α = (∂G_α/∂n1_α)_T,V,n2_α = g1_α + n2_α * (∂g2_α/∂n1_α)_T,V μ2_α = (∂G_α/∂n2_α)_T,V,n1_α = g2_α + n1_α * (∂g1_α/∂n2_α)_T,V μ1_β = (∂G_β/∂n1_β)_T,V,n2_β = g1_β + n2_β * (∂g2_β/∂n1_β)_T,V μ2_β = (∂G_β/∂n2_β)_T,V,n1_β = g2_β + n1_β * (∂g1_β/∂n2_β)_T,V At equilibrium, we have: μ1_α = μ1_β μ2_α = μ2_β

Deriving the relation between specific Gibbs functions

From the equilibrium conditions, we can derive the relation between the specific Gibbs functions of each phase of each component. Subtracting μ2_α = μ2_β from μ1_α = μ1_β, we get: (g1_α - g2_α) = (g1_β - g2_β) Similarly, subtracting μ1_α = μ1_β from μ2_α = μ2_β, we get: (g2_α - g1_α) = (g2_β - g1_β) By solving these equations, we find that: g1_α = g1_β g2_α = g2_β So, when the two phases of a two-component system are in equilibrium, the specific Gibbs function of each phase of each component is the same.