Gibbs Energy: Understanding A=G+hM

[KFC]

In a book about stat. mech., I read the following relaiton for magnetic system

$$A = G + hM$$

where A is the Helmholtz energy, G is the Gibbs energy and h is the external magnetic field, M is the magetization. I know from thermodynamic, we have

$$A = U - TS$$

or

$$A = G - PV$$

so, $$hM = -PV$$ ?

I don't understand what is $$hM$$. If I know magnetization, external field and Gibbs energy, how to get Helmholtz energy?

 

[Mapes]

This goes back to that issue I believe you mentioned earlier: magnetization is usually ignored in thermodynamics, so authors can be inconsistent when adding the relevant terms. When we're including magnetization in an open system, energy is

$$U=TS-PV+\mu N+hM$$

The relevant potential when temperature, pressure, and field are kept constant is acquired by the Legendre transform

$$\Phi=U-TS+PV-hM$$

Some authors may call $\Phi$ the Gibbs energy, which risks great confusion. (I'm using the symbol $\Phi$ as a dummy variable here.)

To figure out the potentials, just remember that you need to remove (by Legendre transform) any conjugate pairs associated with constant variables. Don't rely on the consistency of names like Helmholtz, Gibbs, etc. If someone's working with a system at constant temperature and field, for example, you know you need to consider the potential

$$\Lambda=U-TS-mH$$

whatever it might be called. (Again, $\Lambda$ is just a dummy variable.) I hope it's helpful to see the method to constructing these potentials. Does this make sense?

(To reiterate, it's definitely not true that $hM=-PV$.)

 

[sir-pinski]

The relationship A = G + hM, where A is the Helmholtz energy, G is the Gibbs energy, h is the external magnetic field, and M is the magnetization, is known as the Gibbs energy equation for magnetic systems. This equation is derived from statistical mechanics and is a useful tool for understanding the thermodynamic behavior of magnetic systems.

In thermodynamics, we have two main equations for calculating the Helmholtz energy: A = U - TS and A = G - PV. The first equation relates the Helmholtz energy to the internal energy (U), temperature (T), and entropy (S) of a system. The second equation relates the Helmholtz energy to the Gibbs energy (G), pressure (P), and volume (V) of a system.

In the context of magnetic systems, the external magnetic field (h) is an important factor that affects the thermodynamic behavior. The term hM in the Gibbs energy equation represents the work done by the external magnetic field on the system, which is equivalent to -PV in the A = G - PV equation. This is because in a magnetic system, the magnetic field can do work by aligning the magnetic dipoles of the system, which is similar to the work done by pressure in a non-magnetic system.

If you know the magnetization, external field, and Gibbs energy of a magnetic system, you can use the Gibbs energy equation to calculate the Helmholtz energy. Simply rearrange the equation to isolate A, and then plug in the known values for G, h, and M. This will give you the Helmholtz energy, which is a measure of the total energy available to do work in the system at constant temperature and volume.

In summary, the Gibbs energy equation for magnetic systems, A = G + hM, is a valuable tool in understanding the thermodynamic behavior of magnetic materials. It takes into account the effects of external magnetic fields and allows for the calculation of the Helmholtz energy, which is an important thermodynamic quantity.