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It seems you've uploaded a new image with a problem labeled "Problem 3." From what I can interpret, it looks like a set of physics-related questions involving topics like free energy, phase transitions, thermodynamic potentials, and possibly quantum mechanics. Here's a breakdown of what I can extract and analyze from the image:

### Problem 3 Overview:

1. **Part (a):** It asks for the free energy \( F \) of a system given as a function of \( T \), \( V \), and \( N \) (temperature, volume, and particle number respectively).
   - Free energy \( F \) could be referring to Helmholtz free energy \( F = U - TS \), where \( U \) is the internal energy and \( S \) is the entropy.

2. **Part (b):** It involves calculating an entropy expression \( S \), likely by using thermodynamic relations such as:
   \[
   S = - \left( \frac{\partial F}{\partial T} \right)_V
   \]

3. **Part (c):** Refers to finding specific heat \( C_V \), which might involve:
   \[
   C_V = T \left( \frac{\partial S}{\partial T} \right)_V
   \]
   or using a relation involving internal energy \( U \).

4. **Part (d):** Involves calculating something related to thermal expansion or some thermodynamic potential involving pressure \( P \).

5. **Part (e):** Likely the equation of state or another thermodynamic relation.

Would you like me to work through a specific part of this problem or all of them? I can derive expressions or solve any part you're interested in.

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Here are 5 questions you can consider to build your understanding:

1. What are the differences between Helmholtz and Gibbs free energies?
2. How do you derive entropy from free energy in thermodynamics?
3. What is the physical significance of specific heat at constant volume?
4. How does free energy change during a phase transition?
5. What is the relation between thermal expansion and entropy?

**Tip:** When working with free energy, always identify whether you're dealing with Helmholtz or Gibbs free energy, as they have different forms and applications based on the system's constraints.

What is the free energy of a system, entropy, specific heat, and related thermodynamic potentials?

Explore how to calculate the free energy, entropy, and specific heat of a system using thermodynamic relations. This problem involves core concepts such as Helmholtz free energy, entropy, and thermodynamic potentials, suitable for university-level students.

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