Professor Claude Lupis
July 1, 2002
3.025 Thermodynamics and Kinetics of Materials
Example of calculations: H, S and G versus T
Magnesium melts at 922oK with a heat of fusion equal to 8.95 kJ/mol and boils at 1363oK
with a heat of vaporization equal to 145.25 kJ/mol. The heat capacity Cp of the liquid and
of the gas can be assumed to be constant at 32.6 J/oK.mol and 20.8 J/oK.mol,
respectively, but is temperature dependent for the solid. For the latter, Cp is measured to
be 24.9 J/oK.mol at room temperature and 30.8 J/oK.mol at 800oK; assume that a linear
variation of Cp with T is justified.
a/ Derive analytical representations of the enthalpy, entropy and Gibbs free energy for all
three phases of Mg.
b/ Plot carefully and accurately the enthalpy, entropy and Gibbs free energy versus
temperature in the interval 298 - 1800oK. Indicate on your plots all relevant features (e.g.,
entropies of fusion and vaporization).
(The pressure is fixed at 1 atm)
2
a/
Calculation of the Enthalpy
In the temperature interval: 298 < T < 922 where Mg is solid;
in J/oK.mol, Cp is equal to 24.9 at 298oK and 30.8 at 800oK, a change of 5.9 for
502o. A linear interpolation yields:
Csp = 24.9 + 5.9 (T - 298)/502 = 24.9 + 1.18x10-2(T - 298)
or:
(J/oK.mol)
Csp = 21.4 + 1.18x10-2T
Hence:
HsT  H o298 = [21.4 + 1.18x10-2T] dT = 21.4(T-298) + 0.59x10-2 (T2 - 2982)
o
HsT -H298
= -6900 + 21.4 T + 0.59 10-2 T2
(J/mol)
At 922oK, Hs922  H o298 = 17,835 J and there is a discontinuity in the enthalpy
curve corresponding to the fusion of the solid: Hf = 8950 J/mol.
Hence, at 922oK: H922  H o298 = 17835 + 8950 = 26785 J.
In the temperature interval: 922 < T < 1363 where the liquid is stable, Cp is
equal to 32.6 J/oK.mol and:
H T  H o298 = 26785 + Cp dT = 26785 + 32.6 ( T - 922)
or:
o
H T -H298
= -3270 + 32.6 T
(J/mol)
At 1363oK, H1363  H o298 = 41160 J/mol and there is a discontinuity which
corresponds to the vaporization of the Mg: Hvap = 145,250 J/mol.
g
 H o298 = 41160 + 145250 = 186410 J/mol.
At T = 1363, H1363
At T > 1363 , where the gas is the stable phase, Cgp = 20.8 J/oK.mol and:
H gT  H o298 = 186410 + Cgp dT = 186410 + 20.8 (T - 1363)
or:
o
HgT -H298
= 158,060 + 20.8 T
(J/mol)
3
Calculation of the Entropy
For the analytical representation of the entropy S, we note that:
SsT  So298 =
( Csp /T) dT
and for 298 < T < 922 , replacing Csp by its analytical representation:
SsT  So298 =
[(21.4/ T) + 1.18 10-2] dT
or:
SsT  So298 =
21.4 ln (T/298) + 1.18 10-2 (T - 298)
For So298 = 32.7 J/oK.mol, it can be rewritten:
S sT = -92.7 + 21.4 ln T + 1.18 10-2 T
(J/oK.mol)
At T = 922 oK, Ss922 = 64.27 J/oK.mol. At that temperature there is a
discontinuity arising from the S of fusion. It is calculated by:
Sf = Hf / Tf = 8950 / 922 = 9.71 J/oK.mol
Hence: S922 = 64.27 + 9.71 = 73.98 J/oK.mol.
Proceeding as before, in the temperature interval 922 < T < 1363 , we have:
ST  S922 =
32.6 ln (T/922)
and:
S T = -148.6 + 32.6 ln T
(J/oK.mol)
At the boiling point, T = 1363, S1363 = 86.69 J/oK.mol. The Svap is equal to
145,250/1363 or 106.57 J/oK.mol.
Hence:
g
S1363
= 86.69 + 106.57 = 193.26 J/oK.mol.
Above the boiling point, T > 1363 , for Cgp equal to 20.8 J/oK.mol:
g
SgT  S1363
= SgT - 193.26 = 20.8 ln (T/1363)
or:
S gT =
43.1 + 20.8 ln T
(J/oK.mol)
4
Calculation of the Gibbs Free Energy
To calculate the Gibbs free energy, we note that: G = H - TS, and that:
G - H o298 = (H  H o298 ) - TS
In each temperature interval, we have the functional dependence of (H  H o298 )
and S on T. Hence, we immediately deduce the analytical representations of G.
For 298 < T < 922:
Gs - H o298 = - 6900 + 11.41 T - 0.59 10-2 T2 -21.4 T ln T
(J/mol)
For 922 < T < 1363:
G
- H o298 = -3270 + 181.2 T -32.6 T ln T
(J/mol)
For T > 1363:
Gg - H o298 = 158060 - 22.3 T - 20.8 T ln T
(J/mol)
One should verify that at 922oK, Gs - H o298 is equal to G - H o298 , and similarly
that at T = 1363oK, G - H o298 = Gg - H o298 . Of course, it is possible to
extrapolate the expressions of G beyond the temperature intervals where they
have been derived (for example for a supercooled liquid).
b/
See attached figures.