# Phase Transitions in the Van Der Waals Gas

• GravityX
In summary, the conversation discussed the concept of isothermal compressibility and its relationship to the free energy becoming minimal. It was mentioned that the equation ##\kappa_T=\frac{1}{V}\Bigl( \frac{\partial^2 F}{\partial V^2} \Bigr)^{-1}_T## can be used to solve the second task. It was also noted that if the free energy becomes minimal, the isothermal compressibility would be negative and left-curved. It was then clarified that for the system to be stable, ##\kappa_T## must always be positive. Lastly, it was mentioned that for the sketched isotherms, only E would not pose a problem as it is continuously
GravityX
Homework Statement
What does a minimum free energy mean for isothermal compressibility?
Relevant Equations
##\kappa_T=-\frac{1}{V}\Bigl( \frac{\partial V}{\partial P} \Bigr)_T=\frac{1}{V}\Bigl( \frac{\partial^2 F}{\partial V^2} \Bigr)^{-1}_T##
Hi,

I am not quite sure if I have understood the second task correctly, but I proceeded as follows.

It's about what happens to the isothermal compressibility when the free energy becomes minimal. In the first task there was already the equation ##\kappa_T=\frac{1}{V}\Bigl( \frac{\partial^2 F}{\partial V^2} \Bigr)^{-1}_T## and I assume that this was not given without reason, but that one can use it to solve the second task.

If the free energy becomes minimal, then surely it means that the difference with ##F_2## < ##F_1## is therefore ##F_2-F_1## negative and thus also its derivative. Then the isothermal compressibility would be negative and thus left-curved.

If the free energy is minimal for an isotherm, it means that the 2nd derivative of F with respect to V is positive.

GravityX

That would then mean that ##\kappa_T## would be curved to the left. If I have understood correctly, then ##\kappa_T## must always be positive for the system to be stable.

The problem says "For which of
the sketched isotherms does this pose a problem?" I would say for all but E, as this is continuously left curved.

## 1. What is a phase transition in the Van Der Waals gas?

A phase transition in the Van Der Waals gas is a change in the physical state of the gas, such as from liquid to gas or from gas to solid, due to changes in temperature and pressure.

## 2. How does the Van Der Waals equation describe phase transitions in the gas?

The Van Der Waals equation takes into account the intermolecular forces between gas particles, which play a crucial role in phase transitions. It predicts a critical point at which the gas can no longer be distinguished as a liquid or gas, and undergoes a phase transition.

## 3. What is the critical point in the Van Der Waals gas?

The critical point is the temperature and pressure at which the gas can no longer be distinguished as a liquid or gas. At this point, the gas undergoes a phase transition and exhibits unique properties such as critical opalescence and a divergent heat capacity.

## 4. How does the Van Der Waals gas differ from an ideal gas in terms of phase transitions?

An ideal gas follows the ideal gas law, which assumes no intermolecular forces between particles. Therefore, it does not exhibit phase transitions. The Van Der Waals gas, on the other hand, takes into account intermolecular forces and can undergo phase transitions.

## 5. What are some real-world applications of understanding phase transitions in the Van Der Waals gas?

Understanding phase transitions in the Van Der Waals gas is crucial in many industries, such as in the production of liquefied natural gas and refrigeration systems. It also plays a role in understanding the behavior of fluids in geological processes, such as the formation of oil and gas reservoirs.

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