Question

Show that two phases are in thermal equilibrium only if their temperatures are the same.

Answer 1

Show that two phases are in thermal equilibrium only if their temperatures are the same

1)

i)Phase: A phase is homogeneous region of matter in which there is no spatial variation in average density, energy, composition, or other macroscopic properties.

ii) Phase equilibrium means that there are two or more phase system present. These systems exist spontaneously on their own without the use of partition or membranes or other interventions and since they are in mutual equilibrium.

iii)Systems in equilibrium can exchange energy, volume and particles.

2)

i)The condition of equilibrium are given by maximization of the entropy subject to the constraints of constant total energy, volume and particles.

ii)For two phases involving a single-component, this becomes,

max[S₁(E₁,V₁,N₁) + S₂(E₂,V₂,N₂)]

At a maximum, we have,

From this its clear that, the condition for equilibrium between two phases must then be given as,

T₁ = T₂, P₁ = P₂, μ₁ = μ₂

3)Note the conditions for thermodynamic equilibrium:

a)For a completely isolated system, S is maximum at thermodynamic equilibrium.

b)For a system with controlled constant temperature and volume, A is minimum at thermodynamic equilibrium.

c)For a system with controlled constant temperature and pressure, G is minimum at thermodynamic equilibrium.

4)We can achieve various equilibriums as follows,

i)Thermal equilibrium when temperatures are the same (T₁ = T₂)

ii)Mechanical equilibrium when pressures are same (P₁ = P₂)

iii)Diffusive equilibrium when their chemical pressures are same (μ₁ = μ₂).

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