# [Pw_forum] help:free energy under pressure

Stefano Baroni baroni at sissa.it
Sun Sep 3 18:32:47 CEST 2006

On Sep 3, 2006, at 1:45 PM, hailin yu wrote:

> Dear all:
>       I want to reserach the free energy of alpha-Sn under
> different external pressure. I can calculate the free energy at
> P=0, but I don't sure it is same under non-zero pressure.

Sure, it is not, even though your statement is a bit confusing (or,
at least, it denotes some degree of confusion)

What you probably calculate (what is easiest to calculate from the
ingredients you have) is the Helmholtz free energy, which is the
thermodinamical potential function of the volume and temperature, F=E-
TS. The volume derivative of F wrt V is the pressure: P = {\partial F
over \partial V}. The thermodynamical potential whose independent
variables are the pressure (instead of the volume) and the
temperature is the Gibbs free energy, G=E+PV-TS=F+PV. In order to
calculate G as a function of pressure and temperature, you usually
have to express V as a function of P by inverting the relation: P =
{\partial F over \partial V}, which gives P as a function of V and T.
In practice, this is done numerically.

I would recommend that you take a few hours studying the relevant
chapter of any textbook of elementary statistical mechanics, such as
Landau's, Chandler's, or your favorite textbook. If you do not have a
text in statistical mechanics handy, you may find the relevant
definitions and formulas by googling "helmholtz gibbs free energy"

hope this helps

Stefano B.

> Can you tell me? Thanks in advance!
>
> Thanks a lot!
> yuhailin
>
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---
Stefano Baroni - SISSA  &  DEMOCRITOS National Simulation Center -
Trieste
[+39] 040 3787 406 (tel) -528 (fax) / stefanobaroni (skype)

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