12.2: Thermodynamics: Basic Terms

Internal Energy, U

The internal energy of a system is the sum of the potential energy and the kinetic energy. For many applications it is necessary to consider a small change in the internal energy, dU, of a system.

dU = dq + dw = CdT - PdV

dq = the heat supplied to a system
dw = the work performed on the system
C = heat capacity
dT = change in temperature
P = pressure
dV = change in volume

At constant volume,

dU = C_VdT

Enthalpy, H

Enthalpy is the constant pressure version of the internal energy. Enthalpy,

H = U + PV.

Therefore, for small changes in enthalpy,

dH = dU + PdV + VdP.

At constant pressure,

dH = C_PdT.

Entropy, S

Entropy is a measure of the disorder of a system. In terms of molecular disorder, the entropy consists of the configurational disorder (the arrangement of different atoms over identical sites) and the thermal vibrations of the atoms about their mean positions. A change in entropy is defined as,

\[dS \geq \frac{dq}{T}\]

For reversible changes, i.e. changes under equilibrium conditions,

dq = TdS

For natural changes, i.e. under non-equilibrium conditions,

dq < TdS

Gibbs free energy, G

The Gibbs free energy can be used to define the equilibrium state of a system. It considers only the properties of the system and not the properties of its surroundings. It can be thought of as the energy which is available in the system to do useful work.

Free energy, G, is defined as,

G = H - TS = U + PV - TS

For small changes,

dG = dH - TdS - SdT = VdP - SdT + (dq - TdS)

For changes occurring at constant pressure and temperature,

dG = dq - TdS

Therefore, dG = 0 for reversible (equilibrium) changes, and dG < 0 for non-reversible changes.

From this it is clear that G tends to a minimum at equilibrium.

The Helmholtz free energy, F, is sometimes used instead of G, and is the equivalent of G for changes at constant volume. It is defined as,

F = U - TS

Thermodynamics of Solutions

Consider a mechanical mixture of two phases, A and B. If this is then transformed into a single solution phase with A and B atoms distributed randomly over the atomic sites, then there will be,

• An enthalpy change associated with interactions between the A and B atoms, ΔH_mix
• An entropy change, ΔS_mix, associated with the random mixing of the atoms
• A free energy of mixing, ΔG_mix = ΔH_mix - TΔS_mix

Assume that the system consists of N atoms: x_AN of A and x_BN of B, where,

x_A = fraction of A atoms and x_B = (1 - x_A) = fraction of B atoms

Enthalpy of mixing

In calculating ΔH_mix it is assumed that only the potential energy term undergoes any significant change during mixing. This change arises from the interactions between nearest-neighbour atoms. Consider an alloy consisting of atoms A and B. If the atoms prefer like neighbours, A atoms will tend to cluster and likewise B atoms, so a greater number of A-A and B-B bonds will form. If the atoms prefer unlike neighbours a greater number of A-B bonds will form. If there is no preference A and B atoms will be randomly distributed.

Let w_AA be the interaction energy between A - A nearest neighbours, w_BB that for B - B nearest neighbours and w_AB that for A - B nearest neighbours.

All of these energies are negative, as the zero in potential energy is for infinite separation between atoms.

Let each atom of A and B have co-ordination number z.

Therefore, the total number of nearest-neighbour pairs is Nz/2.

Probability of A - A neighbours = x_A²

Probability of B - B neighbours = x_B²

Probability of A - B neighbours = 2x_Ax_B

For a solid solution the total interaction energy is,

H_s - U_s = Nz/2 (x_A² w_AA + x_B² w_BB + 2x_Ax_B w_AB)

For pure A, H_A = (Nz/2)w_AA

For pure B, H_B = (Nz/2)w_BB

Hence the enthalpy of mixing is given by,

ΔH_mix = H_s - (x_AH_A + x_BH_B) = (Nz/2)x_Ax_B (2w_AB - w_AA - w_BB)

We can define an interaction parameter

W = (Nz/2)(2w_AB - w_AA - w_BB)

Therefore,

ΔH_mix = Wx_Ax_B

If A-A and B-B interactions are energetically more favourable than A-B interactions then W > 0. So, ΔH_mix > 0 and there is a tendency for the solution to form A-rich and B-rich regions.

If A-B interactions are energetically more favourable than A-A and B-B interactions, W < 0, ΔH_mix < 0, and there is a tendency to form ordered structures or intermediate compounds.

Finally if the solution is ideal and all interactions are energetically equivalent, then W = 0 and ΔH_mix = 0.

Entropy of mixing

Per mole of sites, this is

ΔS_mix = kN (- x_Alnx_A - x_Blnx_B)

(the derivation of this result makes use of Stirling's approximation)

where N = Avogadro's number, and kN = R, the gas constant.

Hence,

ΔS_mix = R (- x_Alnx_A - x_Blnx_B)

A graph of ΔS_mix versus x_A has a different form from ΔH_mix. The curve has an infinite gradient at x_A = 0 and x_A = 1.

The free energy of mixing is now given by,

ΔG_mix = ΔH_mix - TΔS_mix = x_Ax_BW + RT (x_A lnx_A + x_Blnx_B)

For W < 0, ΔG_mix is negative at all temperatures, and mixing is exothermic. For W > 0, ΔH_mix is positive and mixing is endothermic.