HW #11 Problem 4: Graphite-diamond equilibrium
Since the molar Gibbs free energy of diamond is 2.87 kJ/mol higher than
that of graphite, graphite is the stable phase under these conditions. Recall
that the stable phase always has the lower molar Gibbs free energy g. To find
the pressure at which these two carbon phases are in equilibrium, we first recall
that the pressure dependence of g is given by
µ ¶
∂g
= v,
(1)
∂p T
where v is the molar volume. Applying Eq.(1) to diamond and graphite and
subtracting the two equations gives us
µ
¶
∂∆g
= ∆v,
(2)
∂p T
where
∆g = gd − gg ,
(3)
∆v = vd − vg .
(4)
and
The molar volume is easily related to the density ρ by the following equation
v = M/ρ,
(5)
where, in this case, M is the atomic weight of carbon (be careful of units).
All that remains is to integrate Eq.(2) from ambient pressure (1 atm), at
which ∆g = ∆g0 = 2.87 kJ/mol, to the very large equilibrium pressure, peq , at
which ∆g = 0. The integration is done assuming that the two solid phases are
incompressible, i.e., that ∆v is constant. This gives us
−∆g0 = ∆v∆p.
(6)
Solve for ∆p and substitute numerical values to find
∆p = (−2.87 kJ/mol)/(−1.92 × 10−6 m3 /mol),
(7)
∆p = 1.49 × 109 P a.
(8)
or
The initial pressure of 1 atm is completely negligible in comparison to this large
pressure difference, so we have peq = 1.49 GP a. This is reasonable agreement
with the value on a theoretical phase diagram found at the website
http://en.wikipedia.org/wiki/File:Carbon_basic_phase_diagram.png See
below for a copy of the phase diagram.
1
File:Carbon basic phase diagram.png - Wikipedia, the free encyclopedia
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http://en.wikipedia.org/wiki/File:Carbon_basic_phase_diagram.png
File:Carbon basic phase diagram.png
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Theoretical phase diagram of carbon, which shows the state of matter for varying temperatures and pressures.
The hatched regions indicate conditions under which one phase is metastable, so that two phases can coexist.
Accurate illustration but not a pixel-exact plot. Also note that a considerable disagreement exists between
theory and experiment, e.g., for the triple point doi:10.1016/0008-6223(76)90010-5 (http://dx.doi.org
/10.1016%2F0008-6223%2876%2990010-5)
4/1/2010 8:23 PM
File:Carbon basic phase diagram.png - Wikipedia, the free encyclopedia
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http://en.wikipedia.org/wiki/File:Carbon_basic_phase_diagram.png
Adapted from a diagram at J.M. Zazula, On Graphite Transformations at High Temperature and Pressure
Induced by Absorption of the LHC Beam, 1997 (http://lbruno.home.cern.ch/lbruno/documents/Bibliography
/LHC_Note_78.pdf) as based on:
F.P. Bundy, Pressure-Temperature Phase Diagram of Elemental Carbons, Physica A 156, 169 (1989).
J. Steinback et al., A Model for Pulsed Laser Melting of Graphite, J. Appl. Phys. 58 (11), 4374 (1985).
Source: English Wikipedia, original upload 16 April 2005 by Femto
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(Basic phase diagram of
carbon, which shows the state
of matter for varying
temperatures and pressures.
The hatched regions indicate
Saperaud
conditions under which one
phase is metastable, so that
two phases can coexist.
Accurate illustration but not a
pixel-exac)
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Carbon
Diamond
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炭素
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