- #1
- 757
- 355
For any system where the thermodynamic limit exists, we know that the internal energy U, the entropy σ, the total particle number N and the total volume V are all extensive. Because of this, we know that the Euler relation holds true
U = -PV + [itex]\tau[/itex]σ + [itex]\mu[/itex]N
and that the chemical potential is just the Gibbs free energy per particle.
G = [itex]\mu[/itex] N
We can use the differential relation of the Gibbs free energy
dG = V dP -σ d[itex]\tau[/itex] + [itex]\mu[/itex] dN
to find expressions for the derivatives of the chemical potential.
d[itex]\mu[/itex]/d[itex]\tau[/itex] = 1/N * dG/d[itex]\tau[/itex] =-σ/N
d[itex]\mu[/itex]/dP = 1/N * dG/dP = V/N
Since N, [itex]\sigma[/itex], and V are all positive, it appears that the chemical potential must increase with pressure at constant temperature, and decrease with temperature at constant pressure.
My question is this:
the 1D fermi gas is a case where the chemical potential actually increases with temperature for small temperatures, though we would think in all cases the chemical potential should be decreasing with temperature.
How do we reconcile this result with the previous assumptions about the chemical potential?
-James
U = -PV + [itex]\tau[/itex]σ + [itex]\mu[/itex]N
and that the chemical potential is just the Gibbs free energy per particle.
G = [itex]\mu[/itex] N
We can use the differential relation of the Gibbs free energy
dG = V dP -σ d[itex]\tau[/itex] + [itex]\mu[/itex] dN
to find expressions for the derivatives of the chemical potential.
d[itex]\mu[/itex]/d[itex]\tau[/itex] = 1/N * dG/d[itex]\tau[/itex] =-σ/N
d[itex]\mu[/itex]/dP = 1/N * dG/dP = V/N
Since N, [itex]\sigma[/itex], and V are all positive, it appears that the chemical potential must increase with pressure at constant temperature, and decrease with temperature at constant pressure.
My question is this:
the 1D fermi gas is a case where the chemical potential actually increases with temperature for small temperatures, though we would think in all cases the chemical potential should be decreasing with temperature.
How do we reconcile this result with the previous assumptions about the chemical potential?
-James