# Van der Waals gas - free energy

• Quelsita
In summary, the conversation discusses the use of the free energy equation in the context of the van der Waals gas. The equation is introduced as F = U – TS, where U is the internal energy, T is the absolute temperature, and S is the entropy. The conversation then delves into the first law of thermodynamics and how it relates to the free energy equation. The question of where the inequality in the first law comes from is raised, and the use of the inequality is discussed in the context of the process not being reversible. The conversation also mentions the use of L=pdV and the relationship between deltaF and dV.
Quelsita
For this problem, I'm really jst trying to figure out everything that is going on and then I can simply follow through with the derivatives once I know what I'm working with.

Q: For the van der Waals gas, introduce the free energy as F = U – TS and verify that its derivatives over V and T give the correct expressions for p and S.

I know the eqaution for free energy is the Helmholz free energy where
F=( Hem.) free energy
U= Internal energy of the system
T= absolute temperature (K)
S= Entropy

From the first law of thermo. :
L=-$$\Delta$$U +Q (L being the external work)

since the system is in thermal contact w/ an environment at constant temperature and transforming from a state A to B:

$$\int$$(dQ/T) $$\leq$$S(B)-S(A)

and since T is constant throughout I can say that:
Q= $$\int$$(dQ) $$\leq$$ T{S(B)-S(A)}

my question is, where did the inequality come from??

Then, I can plug in Q to the First law?:
L$$\leq$$ -$$\Delta$$U+ T{S(B)-S(A)}
L$$\leq$$ U(A)-U(B) + T{S(B)-S(A)}
again, why is an inequality used?
where does the volume and pressure come into play?

ok, is the inequality used because the process is not reversible?
can L=pdV be applied somewhere?

also, I found in our text that deltaF=(dF/dV) dV and it states that this was found using:
L< F(A)-F(B)= -deltaF...how is this so?

## 1. What is Van der Waals gas?

Van der Waals gas is a type of gas that takes into account the intermolecular forces between particles, which leads to deviations from the ideal gas law. It was introduced by Dutch scientist Johannes Diderik van der Waals in the late 19th century.

## 2. How is Van der Waals gas different from an ideal gas?

Unlike an ideal gas, Van der Waals gas takes into account the volume occupied by the particles and the attractive forces between them. This results in a higher pressure and lower volume than predicted by the ideal gas law.

## 3. What is free energy in the context of Van der Waals gas?

Free energy is a measure of the energy that is available to do work. In the context of Van der Waals gas, it is the difference between the total energy of the gas and the energy that is lost due to intermolecular forces.

## 4. Why is it important to study Van der Waals gas and its free energy?

Understanding Van der Waals gas and its free energy is important in many practical applications, such as predicting the behavior of real gases and designing efficient processes for gas separation and purification. It also provides insights into the behavior of other non-ideal systems.

## 5. How is free energy related to the stability of Van der Waals gas?

The free energy of a system is a measure of its stability. In the case of Van der Waals gas, a lower free energy indicates a more stable state, as the system has less energy available to do work and is closer to its equilibrium state.

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