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 = CVdT
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 = CPdT.
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,
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, ΔHmix
- An entropy change, ΔSmix, associated with the random mixing of the atoms
- A free energy of mixing, ΔGmix = ΔHmix - TΔSmix
Assume that the system consists of N atoms: xAN of A and xBN of B, where,
xA = fraction of A atoms and xB = (1 - xA) = fraction of B atoms
Enthalpy of mixing
In calculating ΔHmix 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 wAA be the interaction energy between A - A nearest neighbours, wBB that for B - B nearest neighbours and wAB 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 = xA2
Probability of B - B neighbours = xB2
Probability of A - B neighbours = 2xAxB
For a solid solution the total interaction energy is,
Hs - Us = Nz/2 (xA2 wAA + xB2 wBB + 2xAxB wAB)
For pure A, HA = (Nz/2)wAA
For pure B, HB = (Nz/2)wBB
Hence the enthalpy of mixing is given by,
ΔHmix = Hs - (xAHA + xBHB) = (Nz/2)xAxB (2wAB - wAA - wBB)
We can define an interaction parameter
W = (Nz/2)(2wAB - wAA - wBB)
ΔHmix = WxAxB
If A-A and B-B interactions are energetically more favourable than A-B interactions then W > 0. So, ΔHmix > 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, ΔHmix < 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 ΔHmix = 0.
Entropy of mixing
Per mole of sites, this is
ΔSmix = kN (- xAlnxA - xBlnxB)
(the derivation of this result makes use of Stirling's approximation)
where N = Avogadro's number, and kN = R, the gas constant.
ΔSmix = R (- xAlnxA - xBlnxB)
A graph of ΔSmix versus xA has a different form from ΔHmix. The curve has an infinite gradient at xA = 0 and xA = 1.
The free energy of mixing is now given by,
ΔGmix = ΔHmix - TΔSmix = xAxBW + RT (xA lnxA + xBlnxB)
For W < 0, ΔGmix is negative at all temperatures, and mixing is exothermic. For W > 0, ΔHmix is positive and mixing is endothermic.
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