Gibbs Free Energy and Equation of State

[electricspit]

I'm wondering why the Gibbs function is related to the equation of state as follows (supposedly):

[tex]V=(\frac{∂G}{∂P})_T[/tex]

I found a thread on here that mentions this relationship, but doesn't explain it at all. Any help understanding this would be appreciated, this is my first introduction to the Gibbs and Helmholtz functions and I'm trying to understand as fully as I can.

 

[Jano L.]

It is a consequence of the definition G:

$$
G = U - TS + PV.
$$

Since

$$
dU = TdS - PdV,
$$

it follows that

$$
dG = -SdT + VdP
$$

and thus the volume V is given by

$$
V = \left(\frac{\partial G}{\partial P}\right)_T.
$$

 

[electricspit]

Awesome thank you, just what I was looking for!

So then a state function itself is just any relation that relates 2 or more extensive variables?

 

[hilbert2]

So then a state function itself is just any relation that relates 2 or more extensive variables?

If you have a simple system ("simple" here means that the system can do work only by volume expansion) that consists of one pure substance, giving the values of three independent thermodynamic variables fixes the values of all other variables. An equation of state is an equation that can be used to solve a fourth variable when three are known. A familiar example is the ideal gas equation PV=nRT, but other variables like entropy or free energy can also appear in an equation of state.

 

[PJC01]

The Gibbs free energy and the equation of state are closely related because they both describe the thermodynamic properties of a system. The equation of state relates the macroscopic variables of temperature, pressure, and volume, while the Gibbs free energy takes into account the internal energy and entropy of the system.

The relationship between the two can be understood by looking at the definition of the Gibbs free energy:

G = U + PV - TS

where U is the internal energy, P is the pressure, V is the volume, T is the temperature, and S is the entropy. We can rewrite this equation as:

G = U + PV - T(S-PV/T)

Since PV/T is equal to the Gibbs-Helmholtz equation (Gibbs free energy minus the product of pressure and volume divided by temperature), we can substitute this into the equation above to get:

G = U + PV - T(S-G/T)

Now, if we take the derivative of this equation with respect to pressure at constant temperature (dT = 0), we get:

(∂G/∂P)_T = (∂U/∂P)_T + V - T(∂S/∂P)_T

But (∂U/∂P)_T is equal to -V, so we can rewrite this as:

(∂G/∂P)_T = V - T(∂S/∂P)_T

This is the same as the equation you mentioned in your post:

V=(\frac{∂G}{∂P})_T

Therefore, the Gibbs free energy and the equation of state are related through the derivative of the Gibbs free energy with respect to pressure at constant temperature. This relationship is important in understanding the thermodynamic behavior of a system and can be used to calculate various properties of a system.

 

[PJC01]

The Gibbs free energy (G) and the equation of state are both important concepts in thermodynamics. The equation of state describes the relationship between the state variables of a system, such as pressure (P), volume (V), and temperature (T). On the other hand, the Gibbs free energy is a thermodynamic potential that takes into account both the internal energy of a system and the work that can be done by the system.

The relationship between the Gibbs free energy and the equation of state is given by the equation you mentioned: V=(\frac{∂G}{∂P})_T. This equation is known as the Maxwell relation and it relates the change in volume of a system (V) to the change in Gibbs free energy (G) with respect to pressure (P) at constant temperature (T).

This relationship is derived from the fundamental thermodynamic equation: dG = -SdT + VdP. By taking the partial derivative of this equation with respect to pressure at constant temperature, we get the Maxwell relation.

In simpler terms, this equation means that at constant temperature, the change in volume of a system is related to the change in Gibbs free energy with respect to pressure. This relationship is useful in understanding the behavior of a system under different conditions, such as changes in pressure or temperature.

I hope this explanation helps in understanding the relationship between the Gibbs free energy and the equation of state. It is a fundamental concept in thermodynamics and is essential in understanding the behavior of physical systems.